Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

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Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L 2 . Sam Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Shlomo Sternberg September 18, 2014 Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calc

Transcript of Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Page 1: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Math212a1406The Fourier TransformThe Laplace transform

The spectral theorem for bounded self-adjointoperators, functional calculus form

The Mellin Transform

Shlomo Sternberg

September 18, 2014

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 2: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

1 Conventions, especially about 2π.

2 Basic facts about the Fourier transform acting on S.

3 The Fourier transform on L2.

4 Sampling.

5 The Heisenberg Uncertainty Principle.

6 Tempered distributions.Examples of Fourier transforms of elements of S ′.

7 The Laplace transform.

8 The spectral theorem for bounded self-adjoint operators,functional calculus form.

9 The Mellin trransformDirichlet series and their special values

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 3: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The space S.

The space S consists of all functions on R which are infinitelydifferentiable and vanish at infinity rapidly with all their derivativesin the sense that

‖f ‖m,n := supx∈R{|xmf (n)(x)|} <∞.

The ‖ · ‖m,n give a family of semi-norms on S making S into aFrechet space - that is, a vector space space whose topology isdetermined by a countable family of semi-norms.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 4: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The measure on R.

We use the measure1√2π

dx

on R and so define the Fourier transform of an element of S by

f (ξ) :=1√2π

Rf (x)e−ixξdx

and the convolution of two elements of S by

(f ? g)(x) :=1√2π

Rf (x − t)g(t)dt.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 5: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

We are allowed to differentiate 1√2π

∫R f (x)e−ixξdx with respect to

ξ under the integral sign since f (x) vanishes so rapidly at ∞. Weget

d

(1√2π

Rf (x)e−ixξdx

)=

1√2π

R(−ix)f (x)e−ixξdx .

So the Fourier transform of (−ix)f (x) is ddξ f (ξ).

Integration by parts (with vanishing values at the end points) gives

1√2π

Rf ′(x)e−ixξdx = (iξ)

1√2π

Rf (x)e−ixξdx .

So the Fourier transform of f ′ is (iξ)f (ξ).

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 6: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

We are allowed to differentiate 1√2π

∫R f (x)e−ixξdx with respect to

ξ under the integral sign since f (x) vanishes so rapidly at ∞. Weget

d

(1√2π

Rf (x)e−ixξdx

)=

1√2π

R(−ix)f (x)e−ixξdx .

So the Fourier transform of (−ix)f (x) is ddξ f (ξ).

Integration by parts (with vanishing values at the end points) gives

1√2π

Rf ′(x)e−ixξdx = (iξ)

1√2π

Rf (x)e−ixξdx .

So the Fourier transform of f ′ is (iξ)f (ξ).

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 7: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

We are allowed to differentiate 1√2π

∫R f (x)e−ixξdx with respect to

ξ under the integral sign since f (x) vanishes so rapidly at ∞. Weget

d

(1√2π

Rf (x)e−ixξdx

)=

1√2π

R(−ix)f (x)e−ixξdx .

So the Fourier transform of (−ix)f (x) is ddξ f (ξ).

Integration by parts (with vanishing values at the end points) gives

1√2π

Rf ′(x)e−ixξdx = (iξ)

1√2π

Rf (x)e−ixξdx .

So the Fourier transform of f ′ is (iξ)f (ξ).

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 8: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The Fourier transform maps S to S.

Putting these two facts together gives

The Fourier transform is well defined on S and[(

d

dx

)m

((−ix)nf )

]ˆ= (iξ)m

(d

)n

f ,

as follows by differentiation under the integral sign and byintegration by parts. This shows that the Fourier transform mapsS to S.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 9: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Convolution goes to multiplication.

(f ? g )(ξ) =1

∫ ∫f (x − t)g(t)dte−ixξdx

=1

∫ ∫f (u)g(t)e−i(u+t)ξdudt

=1√2π

Rf (u)e−iuξdu

1√2π

Rg(t)e−itξdt

so(f ? g ) = f g .

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 10: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Scaling.

For any f ∈ S and a > 0 define Saf by (Sa)f (x) := f (ax). Thensetting u = ax so dx = (1/a)du we have

(Saf )(ξ) =1√2π

Rf (ax)e−ixξdx

=1√2π

R(1/a)f (u)e−iu(ξ/a)du

so(Saf ) = (1/a)S1/a f .

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 11: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Fourier transform of a Gaussian is a Gaussian.

The polar coordinate trick evaluates

1√2π

Re−x

2/2dx = 1.

The integral1√2π

Re−x

2/2−xηdx

converges for all complex values of η, uniformly in any compactregion. Hence it defines an analytic function of η that can beevaluated by taking η to be real and then using analyticcontinuation.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 12: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The Fourier transform of the unit Gaussian is the unitGaussian.

For real η we complete the square and make a change of variables:

1√2π

Re−x

2/2−xηdx =1√2π

Re−(x+η)2/2+η2/2dx

= eη2/2 1√

Re−(x+η)2/2dx

= eη2/2.

Setting η = iξ gives

n = n if n(x) := e−x2/2.

We will make much use of this equation over the next few slides.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 13: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Scaling the unit Gaussian.

n = n if n(x) := e−x2/2.

If we set a = ε in our scaling equation and define ρε := Sεn so

ρε(x) = e−ε2x2/2,

then

(ρε)(x) =1

εe−x

2/2ε2.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 14: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

(ρε)(x) =1

εe−x

2/2ε2.

Notice that for any g ∈ S we have (by a change of varialbes)

R(1/a)(S1/ag)(ξ)dξ =

Rg(ξ)dξ

so setting a = ε we conclude that

1√2π

R(ρε)(ξ)dξ = 1

for all ε.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 15: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

1√2π

R(ρε)(ξ)dξ = 1

Letψ := ψ1 := (ρ1)

andψε := (ρε).

Then

ψε(η) =1

εψ(ηε

)

so

(ψε ? g)(ξ)− g(ξ) =1√2π

R[g(ξ − η)− g(ξ)]

1

εψ(ηε

)dη =

=1√2π

R[g(ξ − εζ)− g(ξ)]ψ(ζ)dζ.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 16: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

(ψε ? g)(ξ)− g(ξ) =1√2π

R[g(ξ − η)− g(ξ)]

1

εψ(ηε

)dη =

=1√2π

R[g(ξ − εζ)− g(ξ)]ψ(ζ)dζ.

Since g ∈ S it is uniformly continuous on R, so that for any δ > 0we can find ε0 so that the above integral is less than δ in absolutevalue for all 0 < ε < ε0. In short,

‖ψε ? g − g‖∞ → 0, as ε→ 0.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 17: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The multiplication formula.

This says that

Rf (x)g(x)dx =

Rf (x)g(x)dx

for any f , g ∈ S. Indeed the left hand side equals

1√2π

R

Rf (y)e−ixydyg(x)dx .

We can write this integral as a double integral and theninterchange the order of integration which gives the right handside.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 18: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The inversion formula.

This says that for any f ∈ S

f (x) =1√2π

Rf (ξ)e ixξdξ.

To prove this, we first observe that for any h ∈ S the Fouriertransform of x 7→ e iηxh(x) is just ξ 7→ h(ξ − η) as follows directlyfrom the definition. Taking g(x) = e itxe−ε

2x2/2 in themultiplication formula gives

1√2π

Rf (t)e itxe−ε

2t2/2dt =1√2π

Rf (t)ψε(t−x)dt = (f ?ψε)(x).

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 19: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

1√2π

Rf (t)e itxe−ε

2t2/2dt =1√2π

Rf (t)ψε(t−x)dt = (f ?ψε)(x).

We know that the right hand side approaches f (x) as ε→ 0. Also,e−ε

2t2/2 → 1 for each fixed t, and in fact uniformly on anybounded t interval. Furthermore, 0 < e−ε

2t2/2 ≤ 1 for all t. Sochoosing the interval of integration large enough, we can take theleft hand side as close as we like to 1√

∫R f (x)e ixtdt by then

choosing ε sufficiently small. �

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 20: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Plancherel’s theorem.

Letf (x) := f (−x).

Then the Fourier transform of f is given by

1√2π

Rf (−x)e−ixξdx =

1√2π

Rf (u)e iuξdu = f (ξ)

so(f ) = f .

Thus(f ? f ) = |f |2.

The inversion formula applied to f ? f and evaluated at 0 gives

(f ? f )(0) =1√2π

R|f |2dx .

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 21: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

(f ? f )(0) =1√2π

R|f |2dx .

The left hand side of this equation is

1√2π

Rf (x)f (0− x)dx =

1√2π

R|f (x)|2dx .

Thus we have proved Plancherel’s formula

1√2π

R|f (x)|2dx =

1√2π

R|f (x)|2dx .

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 22: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Extending the Fourier transform to L2.

1√2π

R|f (x)|2dx =

1√2π

R|f (x)|2dx .

Define L2(R) to be the completion of S with respect to the L2

norm given by the left hand side of the above equation. Since S isdense in L2(R) we conclude that the Fourier transform extends tounitary isomorphism of L2(R) onto itself.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 23: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The Poisson summation formula.

This says that for any g ∈ S we have

k∈Zg(2πk) =

1√2π

m∈Zg(m).

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 24: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Proof. Let h(x) :=∑

k g(x + 2πk) so h is a smooth function,periodic of period 2π and

h(0) =∑

k

g(2πk).

Expand h into a Fourier series h(x) =∑

m ame imx where

am =1

∫ 2π

0h(x)e−imxdx =

1

Rg(x)e−imxdx =

1√2π

g(m).

Setting x = 0 in the Fourier expansion

h(x) =1√2π

∑g(m)e imx

gives

h(0) =1√2π

m

g(m). �

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 25: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The Shannon sampling theorem.

Let f ∈ S be such that its Fourier transform is supported in theinterval [−π, π]. Then a knowledge of f (n) for all n ∈ Zdetermines f . This theorem is the basis for all digital samplingused in information technology. More explicitly,

f (t) =1

π

∞∑

n=−∞f (n)

sinπ(n − t)

n − t. (1)

Proof. Let g be the periodic function (of period 2π) whichextends f , the Fourier transform of f . So

g(τ) = f (τ), τ ∈ [−π, π]

and is periodic.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 26: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Expand g into a Fourier series:

g =∑

n∈Zcne inτ ,

where

cn =1

∫ π

−πg(τ)e−inτdτ =

1

∫ ∞

−∞f (τ)e−inτdτ,

or, by the Fourier inversion formula,

cn =1

(2π)12

f (−n).

But

f (t) =1

(2π)12

∫ ∞

−∞f (τ)e itτdτ =

1

(2π)12

∫ π

−πg(τ)e itτdτ =

1

(2π)12

∫ π

−π

∑ 1

(2π)12

f (−n)e i(n+t)τdτ.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 27: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

f (t) =1

(2π)12

∫ ∞

−∞f (τ)e itτdτ =

1

(2π)12

∫ π

−πg(τ)e itτdτ =

1

(2π)12

∫ π

−π

∑ 1

(2π)12

f (−n)e i(n+t)τdτ.

Replacing n by −n in the sum, and interchanging summation andintegration, which is legitimate since the f (n) decrease very fast,this becomes

f (t) =1

n

f (n)

∫ π

−πe i(t−n)τdτ.

But∫ π

−πe i(t−n)τdτ =

e i(t−n)τ

i(t − n)

∣∣∣∣∣

π

−π

=e i(t−n)π − e i(t−n)π

i(t − n)= 2

sinπ(n − t)

n − t.�

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 28: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Rescaling the Shannon sampling theorem.

It is useful to reformulate this via rescaling so that the interval[−π, π] is replaced by an arbitrary interval symmetric about theorigin: In the engineering literature the frequency λ is defined by

ξ = 2πλ.

Suppose we want to apply (1) to g = Saf . We know that theFourier transform of g is (1/a)S1/a f and

supp S1/a f = a · supp f .

So ifsupp f ⊂ [−2πλc , 2πλc ]

we want to choose a so that a2πλc ≤ π or

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 29: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

a ≤ 1

2λc. (2)

For a in this range (1) says that

f (ax) =1

π

∑f (na)

sinπ(x − n)

x − n,

or setting t = ax ,

f (t) =∞∑

n=−∞f (na)

sin(πa (t − na)πa (t − na)

. (3)

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 30: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The Nyquist rate.

This holds in L2 under the assumption that f satisfiessupp f ⊂ [−2πλc , 2πλc ]. We say that f has finite bandwidth oris bandlimited with bandlimit λc . The critical value ac = 1/2λc isknown as the Nyquist sampling interval and (1/a) = 2λc isknown as the Nyquist sampling rate. Thus the Shannonsampling theorem says that a band-limited signal can be recoveredcompletely from a set of samples taken at a rate ≥ the Nyquistsampling rate.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

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Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

The Heisenberg Uncertainty Principle.

Let f ∈ S(R) with∫|f (x)|2dx = 1. We can think of x 7→ |f (x)|2

as a probability density on the line. The mean of this probabilitydensity is

xm :=

∫x |f (x)|2dx .

If we take the Fourier transform, then Plancherel says that

∫|f (ξ)|2dξ = 1

as well, so it defines a probability density with mean

ξm :=

∫ξ|f (ξ)|2dξ.

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The Heisenberg Uncertainty Principle.

Suppose for the moment that these means both vanish. TheHeisenberg Uncertainty Principle says that

(∫|xf (x)|2dx

)(∫|ξ f (ξ)|2dξ

)≥ 1

4.

In other words, if Var(f ) denotes the variance of the probabilitydensity |f |2 with similar notation for f then

Var(f ) · Var(f ) ≥ 1

4.

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Proof.

Write −iξf (ξ) as the Fourier transform of f ′ and use Plancherel towrite the second integral as

∫|f ′(x)|2dx . Then the Cauchy -

Schwarz inequality says that the left hand side is ≥ the square of

∫|xf (x)f ′(x)|dx ≥

∣∣∣∣∫

Re(xf (x)f ′(x))dx

∣∣∣∣ =

1

2

∣∣∣∣∫

x(f (x)f ′(x) + f (x)f ′(x))dx

∣∣∣∣

=1

2

∣∣∣∣∫

xd

dx|f |2dx

∣∣∣∣ =1

2

∣∣∣∣∫−|f |2dx

∣∣∣∣ =1

2.

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The general case.

If f has norm one but the mean of the probability density |f |2 isnot necessarily zero (and similarly for for its Fourier transform) theHeisenberg uncertainty principle says that

(∫|(x − xm)f (x)|2dx

)(∫|(ξ − ξm)f (ξ)|2dξ

)≥ 1

4.

The general case is reduced to the special case by replacing f (x) by

f (x + xm)e iξmx .

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The topology on S.

The space S was defined to be the collection of all smoothfunctions on R such that

‖f ‖m,n := supx{|xmf (n)(x)|} <∞.

The collection of these norms define a topology on S which ismuch finer that the L2 topology: We declare that a sequence offunctions {fk} approaches g ∈ S if and only if

‖fk − g‖m,n → 0

for every m and n.A linear function on S which is continuous with respect to thistopology is called a tempered distribution.

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The space of tempered distributions is denoted by S ′. Forexample, every element f ∈ S defines a linear function on S by

φ 7→ 〈φ, f 〉 =1√2π

Rφ(x)f (x)dx .

But this last expression makes sense for any element f ∈ L2(R), orfor any piecewise continuous function f which grows at infinity nofaster than any polynomial. For example, if f ≡ 1, the linearfunction associated to f assigns to φ the value

1√2π

Rφ(x)dx .

This is clearly continuous with respect to the topology of S butthis function of φ does not make sense for a general element φ ofL2(R).

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The Dirac delta function.

Another example of an element of S ′ is the Dirac δ-function whichassigns to φ ∈ S its value at 0. This is an element of S ′ but makesno sense when evaluated on a general element of L2(R).

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Defining the Fourier transform of a tempered distribution.

If f ∈ S, then the Plancherel formula formula implies that itsFourier transform F(f ) = f satisfies

(φ, f ) = (F(φ),F(f )).

But we can now use this equation to define the Fourier transformof an arbitrary element of S ′: If ` ∈ S ′ we define F(`) to be thelinear function

F(`)(ψ) := `(F−1(ψ)).

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Examples of Fourier transforms of elements of S′.

The Fourier transform of the constant 1.

If ` corresponds to the function f ≡ 1, then

F(`)(ψ) =1√2π

R(F−1ψ)(ξ)dξ = F

(F−1ψ

)(0) = ψ(0).

So the Fourier transform of the function which is identically one isthe Dirac δ-function.

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Examples of Fourier transforms of elements of S′.

The Fourier transform of the δ function.

If δ denotes the Dirac δ-function, then

(F(δ)(ψ) = δ(F−1(ψ)) =((F−1(ψ)

)(0) =

1√2π

Rψ(x)dx .

So the Fourier transform of the Dirac δ function is the functionwhich is identically one.

In fact, this last example follows from thepreceding one: If m = F(`) then

(F(m)(φ) = m(F−1(φ)) = `(F−1(F−1(φ)).

ButF−2(φ)(x) = φ(−x).

So if m = F(`) then F(m) = ˘ where

˘(φ) := `(φ(−•)).

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Examples of Fourier transforms of elements of S′.

The Fourier transform of the δ function.

If δ denotes the Dirac δ-function, then

(F(δ)(ψ) = δ(F−1(ψ)) =((F−1(ψ)

)(0) =

1√2π

Rψ(x)dx .

So the Fourier transform of the Dirac δ function is the functionwhich is identically one.In fact, this last example follows from thepreceding one: If m = F(`) then

(F(m)(φ) = m(F−1(φ)) = `(F−1(F−1(φ)).

ButF−2(φ)(x) = φ(−x).

So if m = F(`) then F(m) = ˘ where

˘(φ) := `(φ(−•)).

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Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

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Examples of Fourier transforms of elements of S′.

The Fourier transform of the function x

This assigns to every ψ ∈ S the value

1√2π

∫ψ(ξ)e ixξxdξdx =

1√2π

∫ψ(ξ)

1

i

d

(e ixξ)

dξdx =

i1√2π

∫dψ(ξ)

dxe ixξdξdx = i

(F(

(F−1

(dψ(ξ)

dx

)))(0)

= iδ

(dψ(ξ)

dx

).

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Examples of Fourier transforms of elements of S′.

For an element of S we have

∫dφ

dx· f dx = − 1√

∫φ

df

dxdx .

So we define the derivative of an ` ∈ S ′ by

d`

dx(φ) = `

(−dφ

dx

).

Thus the Fourier transform of x is −i dδdx .

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Definition of the (one sided) Laplace transformThe inversion problem.

Let f be a (possibly vector valued) bounded piecewisedifferentiable function, so that the integral

F (z) =

∫ ∞

0e−zt f (t)dt

converges for z with <z > 0. F is called the Laplace transformof f . The inversion problem is to reconstruct f from F .

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The Laplace transform as a Fourier transform.

Let f be a bounded piecewise differentiable function defined on[0,∞). Let c > 0, z = c + iξ and h the function given by

h(t) =

{(2π)

12 e−ct f (t) t ≥ 0

0 t < 0

Then h is integrable and

h(ξ) =

∫ ∞

0e−zt f (t)dt = F (z), z = c + iξ.

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h(ξ) =

∫ ∞

0e−zt f (t)dt = F (z), z = c + iξ.

If the function F were integrable over the line Γ given by <z = c,then the Fourier inversion formula would say that

1

2πi

ΓezxF (z)dz =

1

2πecx∫ ∞

−∞e ixξh(ξ)dξ = f (x) for x ≥ 0.

The condition that the function F be integrable over Γ, which isthe same as the condition that h be integrable over R, would implythat h is continuous. But h will have a jump at 0 (if f (0) 6= 0). Sowe need to be careful about the above formula expressing f interms of F .

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The philosophy we have been pushing up until now in today’slecture has been to pass to tempered distributions. But forapplications that I have in mind (to the theory of semi-groups ofoperators) later on in this course, I need to go back to 19thcentury mathematics - more precisely to the analogue for theFourier transform of Dirichlet’s theorem about Fourier series thatwe proved in Lecture 2.

Recall that Dirichlet proved the convergence of the symmetricFourier sum

∑n−n ake ikx to 1

2 (f (x+) + f (x−)) under theassumptions that the periodic function f is piecewise differentiable.The analogue of the limit of the symmetric sum for the case of anintegral is the “Cauchy principal value”:

limR→∞

∫ R

−R.

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Fourier inversion a la Dirichlet.

Theorem

Let h ∈ L(R) be bounded and such that there is a finite number ofreal numbers a1, . . . ar such that h is differentiable on(−∞, a1), (a1, a2), . . . (ar ,∞) with bounded derivative (and rightand left handed derivatives at the endpoints). Then for any x ∈ Rwe have

1

2[h(x+) + h(x−)] = lim

R→∞

1

(2π)12

∫ R

−Re ixξh(ξ)dξ.

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In the proof of this theorem we may take x = 0 by a shift ofvariables. So we want to evaluate the limit of

1√2π

∫ R

−Rh(ξ)dξ =

1

∫ ∞

−∞h(t)

∫ R

−Re−iξtdξdt

=1

∫ ∞

−∞h(t)

e iRt − e−iRt

itdt =

1

π

∫ ∞

−∞h(t)

sin Rt

tdt

=1

π

∫ ∞

0[h(t) + h(−t)]

sin Rt

tdt

where the interchange of the order of integration in the firstequation is justified by the assumption that h is absolutelyintegrable.

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The next fact that we will use is the evaluation of the “Dirichletintegral” ∫ ∞

0

sin t

tdt =

π

2.

There are many ways of establishing this classical result. For aproof using integration by parts see Wikipedia under “Dirichletintegral”. An alternative proof can be given via a contour integral.In fact, this evaluation will also be a consequence of what follows:it is clear that the integral converges, so if we let k denote thevalue and carry the k throughout the proof, we will find thatk = π

2 since the above formula is a special case of our Laplaceinversion formula for the Heaviside function.

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The proof of the theorem will proceed by integration by parts: Let

s(y) :=

∫ ∞

y

sin t

tdt

so

s ′(y) = −sin y

y.

Let H(x) := h(x) + h(−x) so we are interested in evaluating thelimit of

v(R) := −R

∫ ∞

0H(u)s ′(Ru)du

as R →∞. The function H is piecewise differentiable with a finitenumber of points of non-differentiablity where the right and lefthanded derivatives exist as in the theorem. We break the integralon the right up into the sum of the integrals over intervals ofdifferentiability.

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For example, the last integral will contribute

−R

∫ ∞

pH(u)s ′(Ru)du := −R lim

q→∞

∫ q

pH(u)s ′(Ru)du.

Integration by parts gives

−R

∫ q

pH(u)s ′(Ru)du = −[H(q)s(Rq)−H(p)s(Rp)]+

∫ q

pH ′(u)s(Ru)du.

We are assuming that H and H ′ are bounded. Since s(Ru)→ 0 asR →∞ and s(Rq)→ 0 as q →∞, the contribution of these termstend to zero.

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The same integration by parts argument applies to any interval ofthe form (a, b) where a 6= 0. For the interval (0, c) we have

−R

∫ c

0H(u)s ′(Ru)du = −H(c)s(Rc)+H(0)s(0)+

∫ c

0H ′(u)s(Ru)du.

The first and third terms tend to zero as before, and sinces(0) = π

2 we are left with π2 H(0) proving the theorem. �

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Comment.

The hypotheses on h in the theorem are far too strong. In fact, forfor a scalar valued function h, all that need be assumed is that h isin L and is locally of bounded variation. Even weaker hypotheseswork. See for example Widder, The Laplace Transform.

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The Mellin inversion formula.

In any event, we have proved the Mellin inversion formula

f (x) =1

2πi

ΓezxF (z)dz

where the left hand side is interpreted to mean 12 (f (x+) + f (x−))

and the “contour integral” on the right is interpreted as a Cauchyprincipal value.

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The exponential series for a bounded operator

Suppose that B is a bounded operator on a Banach space B. Forexample, any linear operator on a finite dimensional space. Thenthe series

etB =∞∑

0

tk

k!Bk

converges for any t. (We will concentrate on t real, and eventuallyon t ≥ 0 when we get to more general cases.) Convergence isguaranteed as a result of the convergence of the usual exponentialseries in one variable. (There are serious problems with thisdefinition from the point of view of numerical implementationwhich we will not discuss here.)

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The standard proof using the binomial formula shows that

e(s+t)B = esB · etB .

Also, the standard proof for the usual exponential series shows thatthe operator valued function t 7→ etB is differentiable (in theuniform topology) and that

d

dt

(etB)

= B · etB = etB · B.

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The unitary group associated with a bounded self-adjointoperator

In particular, let A be a bounded self-adjoint operator on a Hilbertspace H and take B = iA. So let

U(t) := e itA.

Then t 7→ U(t) defines a one parameter group of boundedtransformations on H. Furthermore,

d

dt(U(t)U(t)∗) = U(t)(iA + (iA)∗)U(t)∗ = 0,

and since U(0)U(0)∗ = I we conclude that

U(t)U(t)∗ ≡ I .

In words: the operators U(t) are unitary.Shlomo Sternberg

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The functional calculus for functions in S.

Recall the Fourier inversion formula for functions f ∈ S which saysthat

f (x) =1√2π

Rf (t)e itxdt.

If we replace x by A and write U(t) instead of e itA this suggeststhat we define

f (A) :=1√2π

Rf (t)U(t)dt. (4)

We want to check that this assignment f 7→ f (A) has theproperties that we would expect from a “functional calculus”. Thismap is clearly linear in f since the Fourier transform is. We nowcheck that it is multiplicative:

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Checking that (fg)(A) = f (A)g(A).

To check this we use fact that the Fourier transform takesmultiplication into convolution, i.e. that (fg ) = f ? g so

(fg)(A) =1

R

Rf (t − s)g(s)U(t)dsdt

=1

R

Rf (r)g(s)U(r + s)drds

=1

R

Rf (r)g(s)U(r)U(s)drds

= f (A)g(A).

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Checking that the map f 7→ f (A) sends f 7→ (f (A))∗.

For the standard Fourier transform we know that the Fouriertransform of f is given by

f (ξ) = f (−ξ).

Substituting this into the right hand side of (4) gives

1√2π

Rf (−t)U(t)dt =

1√2π

Rf (−t)U∗(−t)dt

=

(1√2π

Rf (−t)U(−t)dt

)∗

= (f (A))∗

by making the change of variables s = −t.Shlomo Sternberg

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Checking that ‖f (A)‖ ≤ ‖f ‖∞.

Let ‖f ‖∞ denote the sup norm of f , and let c > ‖f ‖∞. Define gby

g(s) := c −√

c2 − |f (s)|2.So g is a real element of S and

g 2 = c2 − 2c√

c2 − |f |2 + c2 − |f |2

= 2cg − f f

so

f f − 2cg + g 2 = 0.

So by our previous results,

f (A)∗f (A)− cg(A)− cg(A)∗ + g(A)∗g(A) = 0.

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f (A)∗f (A)− cg(A)− cg(A)∗ + g(A)∗g(A) = 0

i.e.f (A)∗f (A) + (c − g(A))∗(c − g(A) = c2.

So for any v ∈ H we have

‖f (A)v‖2 ≤ ‖f (A)v‖2 + ‖(c − g(A))v‖2 = c2‖v‖2

proving that‖f (A)‖ ≤ ‖f ‖∞. (5)

Shlomo Sternberg

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Enlarging the functional calculus to continuous functionsvanishing at infinity.

The inequality

‖f (A)‖ ≤ ‖f ‖∞ (5)

allows us to extend the functional calculus to all continuousfunctions vanishing at infinity. Indeed if f is an element of L1 sothat its inverse Fourier transform f is continuous and vanishes atinfinity (by Riemann-Lebesgue) we can approximate f in the ‖ · ‖∞norm by elements of S and so the formula (4) applies to f .

We will denote the space of continuous functions vanishing atinfinity by C0(R).

Shlomo Sternberg

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Preview of coming attractions.

We will be devoting eight or nine lectures to generalizing this resultin two different ways: We will extend the result from bounded tounbounded self-adjoint operators. Of course this will require us todefine “unbounded self-adjoint operators”.

We will also greatly extend the class of functions to which thefunctional calculus is defined. For example, suppose that we haveextended the calculus so as to include functions of the form 1I

where I is an interval on the real line. Since 1I is real valued, weconclude that 1I (A) = 1I (A)∗, i.e. it is self-adjoint. Since 12

I = 1I ,we conclude that 1I (A)2 = 1I (A). In other words, 1I (A) is aprojection. We will examine the meaning of the image of thisprojection.

Shlomo Sternberg

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But our first item of business will be to try to understand moredeeply the meaning of

etB

and its relation to the resolvent. This will take at least threelectures.

Shlomo Sternberg

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Temporary change in notation

The rest of this lecture is a bonus: A look at some applications ofFourier analysis to analytic number theory, specifically to theRiemann zeta function. I will follow verbatim an article by Zagierwhich appeared in the book Quantum field theory by Zeidler.Many of the slides are photographic copies from the book.The number theorists use a different convention for the Fouriertransform than the one that we have been using and will continueto use after this interruption. They define the Fourier transform as

f (y) = F(f )(y) =

Rf (x)e−2πixydx

so that the Poisson summation formula has the more symmetricallooking form ∑

n∈Zf (n) =

n∈Zf (n).

As we are following Zagier, we will use this convention for the restof this lecture.

Consider the exponential map sending R→ R+ taking sum intoproduct and make the change of variable t = ex , s = −2πiy andφ(t) = f (et) in the definition of the Fourier transform to get

Shlomo Sternberg

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The Mellin transform

φ(s) =

∫ ∞

0φ(t)ts−1dt. (1)

This is the Mellin transform. If φ is a function on the positive realaxis which is piecewise continuous and decays rapidly at both 0and ∞, this integral converges for any complex value of s anddefines a holomorphic function of s. The following small table, inwhich α denotes a complex number and λ a positive real number,shows how φ changes when φ is modified in various simple ways:

6.7 The Mellin Transformation and Other Useful Analytic Techniques 305

and to the two survey articles on the Casimir effect written by B. Duplantier andR. Balian. These articles are contained in the following collection:

B. Duplantier and V. Rivasseau (Eds.), Vacuum Energy – Renormaliza-tion. Poincare Seminar 2002, Birkhauser, Basel, 2003.

Experimental results about the Casimir effect can be found in

T. Ederth, Template-stripped gold surfaces with 0.4-nm rms roughnesssuitable for force measurements: Applications to the Casimir force in the20-100-nm range, Physical Reviews A62 (6) (2000), 062104.

6.7 Appendix: The Mellin Transformation and OtherUseful Analytic Techniques by Don Zagier

The Mellin transformation is a magic wand.Folklore

The following material is not sufficiently well known to a broad audience. The readershould note that the tools to be described are extremely useful. These tools enlargethe arsenal of weapons used in mathematical physics. They allow interesting appli-cations concerning the asymptotic behavior of functions occurring in mathematicsand physics.

6.7.1 The Generalized Mellin Transformation

The Mellin transformation is a basic tool for analyzing the behavior of many impor-tant functions in mathematics and mathematical physics, such as the zeta functionsoccurring in number theory and in connection with various spectral problems in-cluding the Casimir effect. We describe it first in its simplest form and then explainhow this basic definition can be extended to a much wider class of functions, im-portant for many applications.

Let ϕ : ]0, ∞[→ C be a function on the positive real axis which is reasonablysmooth (actually, continuous or even piecewise continuous would be enough) anddecays rapidly at both 0 and ∞, i.e., the function tAϕ(t) is bounded on ]0, ∞[ forany A ∈ R. Then the integral

ϕ(s) =

Z ∞

0

ϕ(t)ts−1dt (6.31)

converges for any complex value s and defines a holomorphic function of s called theMellin transform of ϕ(t). The following small table, in which α denotes a complexnumber and λ a positive real number shows how ϕ(s) changes when ϕ(t) is modifiedin various simple ways:

ϕ(λt) tαϕ(t) ϕ(tλ) ϕ(t−1) ϕ′(t)

λ−sϕ(s) ϕ(s + α) λ−sϕ(λ−1s) ϕ(−s) (1 − s)ϕ(s − 1)(6.32)

(2)We extend the definition of the Mellin transform:Shlomo Sternberg

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Extending the definition of the Mellin transform

Let us start with the frequently occurring case where ϕ(t) is of rapid decay at infinity and isC∞ at zero, i.e., it has an asymptotic expansion ϕ(t) ∼ ∑∞

n=0 antn as t → 0. (Recall that this

means that the difference ϕ(t) − ∑N−1n=0 antn is O(tN ) as t → 0 for any integer N ≥ 0; it is not

required that the series∑

antn be convergent for any positive t.) Then for s with %(s) > 0 andany positive integer N the integral (1) converges and can be decomposed as follows:

ϕ(s) =

∫ 1

0

ϕ(t) ts−1 dt +

∫ ∞

1

ϕ(t) ts−1 dt

=

∫ 1

0

(ϕ(t) −

N−1∑

n=0

antn)

ts−1 dt +N−1∑

n=0

an

n + s+

∫ ∞

1

ϕ(t) ts−1 dt .

The first integral on the right converges in the larger half-plane %(s) > −N and the second for alls ∈ C, so we deduce that ϕ(s) has a meromorphic continuation to %(s) > −N with simple polesof residue an at s = −n (n = 0, . . . , N − 1) and no other singularities. Since this holds for every n,it follows that the Mellin transform ϕ(s) in fact has a meromorphic continuation to all of C withsimple poles of residue an at s = −n (n = 0, 1, 2, . . . ) and no other poles. The same argumentshows that, more generally, if ϕ(t) is of rapid decay at infinity and has an asymptotic expansion

ϕ(t) ∼∞∑

j=1

aj tαj (t → 0) (3)

as t tends to zero, where the αj are real numbers tending to +∞ as j → ∞ or complex num-bers with real parts tending to infinity, then the function ϕ(s) defined by the integral (1) for%(s) > − minj %(αj) has a meromorphic extension to all of C with simple poles of residue aj

at s = −αj (j = 1, 2, . . . ) and no other poles. Yet more generally, we can allow terms of theform tα(log t)m with λ ∈ C and m ∈ Z≥0 in the asymptotic expansion of ϕ(t) at t = 0 andeach such term contributes a pole with principal part (−1)mm!/(s + α)m+1 at s = −α, because∫ 1

0tα+s−1(log t)m dt = (∂/∂α)m

∫ 1

0tα+s−1 dt = (−1)mm!/(α + s)m+1 for %(s + α) > 0.

By exactly the same considerations, or by replacing ϕ(t) by ϕ(t−1), we find that if ϕ(t) is ofrapid decay (faster than any power of t) as t → 0 but has an asymptotic expansion of the form

ϕ(t) ∼∞∑

k=1

bk tβk (t → ∞) (4)

at infinity, where now the exponents βk are complex numbers whose real parts tend to −∞, thenthe function ϕ(s), originally defined by (1) in a left half-plane %(s) < − maxk %(βk), extendsmeromorphically to the whole complex s-plane with simple poles of residue −bk at s = −βk andno other poles. (More generally, again as before, we can allow terms bktβk(log t)nk in (3) whichthen produce poles with principal parts (−1)nk+1nk! bk/(s + βk)nk+1 at s = −βk.)

Now we can use these ideas to define ϕ(s) for functions which are not small either at 0 or at ∞,even when the integral (1) does not converge for any value of s. We simply assume that ϕ(t) isa smooth (or continuous) function on (0, ∞) which has asymptotic expansions of the forms (3)and (4) at zero and infinity, respectively. (Again, we could allow more general terms with powersof log t in the expansions, as already explained, but the corresponding modifications are easy andfor simplicity of expression we will assume expansions purely in powers of t.) For convenience weassume that the numbering s such that %(α1) ≤ %(α2) ≤ · · · and %(β1) ≥ %(β2) ≥ · · · . Then, for

2

simple poles of residue an at s = −n, n = 0, 1, 2, . . . and no other poles.Shlomo Sternberg

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Extending the definition of the Mellin transform, 2

The same argument shows that, more generally, if φ is of rapid decay atinfinity and has an asymptotic expansion

Let us start with the frequently occurring case where ϕ(t) is of rapid decay at infinity and isC∞ at zero, i.e., it has an asymptotic expansion ϕ(t) ∼ ∑∞

n=0 antn as t → 0. (Recall that this

means that the difference ϕ(t) − ∑N−1n=0 antn is O(tN ) as t → 0 for any integer N ≥ 0; it is not

required that the series∑

antn be convergent for any positive t.) Then for s with %(s) > 0 andany positive integer N the integral (1) converges and can be decomposed as follows:

ϕ(s) =

∫ 1

0

ϕ(t) ts−1 dt +

∫ ∞

1

ϕ(t) ts−1 dt

=

∫ 1

0

(ϕ(t) −

N−1∑

n=0

antn)

ts−1 dt +N−1∑

n=0

an

n + s+

∫ ∞

1

ϕ(t) ts−1 dt .

The first integral on the right converges in the larger half-plane %(s) > −N and the second for alls ∈ C, so we deduce that ϕ(s) has a meromorphic continuation to %(s) > −N with simple polesof residue an at s = −n (n = 0, . . . , N − 1) and no other singularities. Since this holds for every n,it follows that the Mellin transform ϕ(s) in fact has a meromorphic continuation to all of C withsimple poles of residue an at s = −n (n = 0, 1, 2, . . . ) and no other poles. The same argumentshows that, more generally, if ϕ(t) is of rapid decay at infinity and has an asymptotic expansion

ϕ(t) ∼∞∑

j=1

aj tαj (t → 0) (3)

as t tends to zero, where the αj are real numbers tending to +∞ as j → ∞ or complex num-bers with real parts tending to infinity, then the function ϕ(s) defined by the integral (1) for%(s) > − minj %(αj) has a meromorphic extension to all of C with simple poles of residue aj

at s = −αj (j = 1, 2, . . . ) and no other poles. Yet more generally, we can allow terms of theform tα(log t)m with λ ∈ C and m ∈ Z≥0 in the asymptotic expansion of ϕ(t) at t = 0 andeach such term contributes a pole with principal part (−1)mm!/(s + α)m+1 at s = −α, because∫ 1

0tα+s−1(log t)m dt = (∂/∂α)m

∫ 1

0tα+s−1 dt = (−1)mm!/(α + s)m+1 for %(s + α) > 0.

By exactly the same considerations, or by replacing ϕ(t) by ϕ(t−1), we find that if ϕ(t) is ofrapid decay (faster than any power of t) as t → 0 but has an asymptotic expansion of the form

ϕ(t) ∼∞∑

k=1

bk tβk (t → ∞) (4)

at infinity, where now the exponents βk are complex numbers whose real parts tend to −∞, thenthe function ϕ(s), originally defined by (1) in a left half-plane %(s) < − maxk %(βk), extendsmeromorphically to the whole complex s-plane with simple poles of residue −bk at s = −βk andno other poles. (More generally, again as before, we can allow terms bktβk(log t)nk in (3) whichthen produce poles with principal parts (−1)nk+1nk! bk/(s + βk)nk+1 at s = −βk.)

Now we can use these ideas to define ϕ(s) for functions which are not small either at 0 or at ∞,even when the integral (1) does not converge for any value of s. We simply assume that ϕ(t) isa smooth (or continuous) function on (0, ∞) which has asymptotic expansions of the forms (3)and (4) at zero and infinity, respectively. (Again, we could allow more general terms with powersof log t in the expansions, as already explained, but the corresponding modifications are easy andfor simplicity of expression we will assume expansions purely in powers of t.) For convenience weassume that the numbering s such that %(α1) ≤ %(α2) ≤ · · · and %(β1) ≥ %(β2) ≥ · · · . Then, for

2

Shlomo Sternberg

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Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Extending the definition of the Mellin transform, 3

Let us start with the frequently occurring case where ϕ(t) is of rapid decay at infinity and isC∞ at zero, i.e., it has an asymptotic expansion ϕ(t) ∼ ∑∞

n=0 antn as t → 0. (Recall that this

means that the difference ϕ(t) − ∑N−1n=0 antn is O(tN ) as t → 0 for any integer N ≥ 0; it is not

required that the series∑

antn be convergent for any positive t.) Then for s with %(s) > 0 andany positive integer N the integral (1) converges and can be decomposed as follows:

ϕ(s) =

∫ 1

0

ϕ(t) ts−1 dt +

∫ ∞

1

ϕ(t) ts−1 dt

=

∫ 1

0

(ϕ(t) −

N−1∑

n=0

antn)

ts−1 dt +N−1∑

n=0

an

n + s+

∫ ∞

1

ϕ(t) ts−1 dt .

The first integral on the right converges in the larger half-plane %(s) > −N and the second for alls ∈ C, so we deduce that ϕ(s) has a meromorphic continuation to %(s) > −N with simple polesof residue an at s = −n (n = 0, . . . , N − 1) and no other singularities. Since this holds for every n,it follows that the Mellin transform ϕ(s) in fact has a meromorphic continuation to all of C withsimple poles of residue an at s = −n (n = 0, 1, 2, . . . ) and no other poles. The same argumentshows that, more generally, if ϕ(t) is of rapid decay at infinity and has an asymptotic expansion

ϕ(t) ∼∞∑

j=1

aj tαj (t → 0) (3)

as t tends to zero, where the αj are real numbers tending to +∞ as j → ∞ or complex num-bers with real parts tending to infinity, then the function ϕ(s) defined by the integral (1) for%(s) > − minj %(αj) has a meromorphic extension to all of C with simple poles of residue aj

at s = −αj (j = 1, 2, . . . ) and no other poles. Yet more generally, we can allow terms of theform tα(log t)m with λ ∈ C and m ∈ Z≥0 in the asymptotic expansion of ϕ(t) at t = 0 andeach such term contributes a pole with principal part (−1)mm!/(s + α)m+1 at s = −α, because∫ 1

0tα+s−1(log t)m dt = (∂/∂α)m

∫ 1

0tα+s−1 dt = (−1)mm!/(α + s)m+1 for %(s + α) > 0.

By exactly the same considerations, or by replacing ϕ(t) by ϕ(t−1), we find that if ϕ(t) is ofrapid decay (faster than any power of t) as t → 0 but has an asymptotic expansion of the form

ϕ(t) ∼∞∑

k=1

bk tβk (t → ∞) (4)

at infinity, where now the exponents βk are complex numbers whose real parts tend to −∞, thenthe function ϕ(s), originally defined by (1) in a left half-plane %(s) < − maxk %(βk), extendsmeromorphically to the whole complex s-plane with simple poles of residue −bk at s = −βk andno other poles. (More generally, again as before, we can allow terms bktβk(log t)nk in (3) whichthen produce poles with principal parts (−1)nk+1nk! bk/(s + βk)nk+1 at s = −βk.)

Now we can use these ideas to define ϕ(s) for functions which are not small either at 0 or at ∞,even when the integral (1) does not converge for any value of s. We simply assume that ϕ(t) isa smooth (or continuous) function on (0, ∞) which has asymptotic expansions of the forms (3)and (4) at zero and infinity, respectively. (Again, we could allow more general terms with powersof log t in the expansions, as already explained, but the corresponding modifications are easy andfor simplicity of expression we will assume expansions purely in powers of t.) For convenience weassume that the numbering s such that %(α1) ≤ %(α2) ≤ · · · and %(β1) ≥ %(β2) ≥ · · · . Then, for

2

Shlomo Sternberg

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Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Extending the definition of the Mellin transform, 4

Let us start with the frequently occurring case where ϕ(t) is of rapid decay at infinity and isC∞ at zero, i.e., it has an asymptotic expansion ϕ(t) ∼ ∑∞

n=0 antn as t → 0. (Recall that this

means that the difference ϕ(t) − ∑N−1n=0 antn is O(tN ) as t → 0 for any integer N ≥ 0; it is not

required that the series∑

antn be convergent for any positive t.) Then for s with %(s) > 0 andany positive integer N the integral (1) converges and can be decomposed as follows:

ϕ(s) =

∫ 1

0

ϕ(t) ts−1 dt +

∫ ∞

1

ϕ(t) ts−1 dt

=

∫ 1

0

(ϕ(t) −

N−1∑

n=0

antn)

ts−1 dt +N−1∑

n=0

an

n + s+

∫ ∞

1

ϕ(t) ts−1 dt .

The first integral on the right converges in the larger half-plane %(s) > −N and the second for alls ∈ C, so we deduce that ϕ(s) has a meromorphic continuation to %(s) > −N with simple polesof residue an at s = −n (n = 0, . . . , N − 1) and no other singularities. Since this holds for every n,it follows that the Mellin transform ϕ(s) in fact has a meromorphic continuation to all of C withsimple poles of residue an at s = −n (n = 0, 1, 2, . . . ) and no other poles. The same argumentshows that, more generally, if ϕ(t) is of rapid decay at infinity and has an asymptotic expansion

ϕ(t) ∼∞∑

j=1

aj tαj (t → 0) (3)

as t tends to zero, where the αj are real numbers tending to +∞ as j → ∞ or complex num-bers with real parts tending to infinity, then the function ϕ(s) defined by the integral (1) for%(s) > − minj %(αj) has a meromorphic extension to all of C with simple poles of residue aj

at s = −αj (j = 1, 2, . . . ) and no other poles. Yet more generally, we can allow terms of theform tα(log t)m with λ ∈ C and m ∈ Z≥0 in the asymptotic expansion of ϕ(t) at t = 0 andeach such term contributes a pole with principal part (−1)mm!/(s + α)m+1 at s = −α, because∫ 1

0tα+s−1(log t)m dt = (∂/∂α)m

∫ 1

0tα+s−1 dt = (−1)mm!/(α + s)m+1 for %(s + α) > 0.

By exactly the same considerations, or by replacing ϕ(t) by ϕ(t−1), we find that if ϕ(t) is ofrapid decay (faster than any power of t) as t → 0 but has an asymptotic expansion of the form

ϕ(t) ∼∞∑

k=1

bk tβk (t → ∞) (4)

at infinity, where now the exponents βk are complex numbers whose real parts tend to −∞, thenthe function ϕ(s), originally defined by (1) in a left half-plane %(s) < − maxk %(βk), extendsmeromorphically to the whole complex s-plane with simple poles of residue −bk at s = −βk andno other poles. (More generally, again as before, we can allow terms bktβk(log t)nk in (3) whichthen produce poles with principal parts (−1)nk+1nk! bk/(s + βk)nk+1 at s = −βk.)

Now we can use these ideas to define ϕ(s) for functions which are not small either at 0 or at ∞,even when the integral (1) does not converge for any value of s. We simply assume that ϕ(t) isa smooth (or continuous) function on (0, ∞) which has asymptotic expansions of the forms (3)and (4) at zero and infinity, respectively. (Again, we could allow more general terms with powersof log t in the expansions, as already explained, but the corresponding modifications are easy andfor simplicity of expression we will assume expansions purely in powers of t.) For convenience weassume that the numbering s such that %(α1) ≤ %(α2) ≤ · · · and %(β1) ≥ %(β2) ≥ · · · . Then, for

2any T > 0 (formerly we took T = 1, but the extra freedom of being able to choose any value of Twill be very useful later) we define two “half-Mellin transforms” ϕ≤T (s) and ϕ≥T (s) by

ϕ≤T (s) =

∫ T

0

ϕ(t) ts−1 dt(!(s) > −!(α1)

),

ϕ≥T (s) =

∫ ∞

T

ϕ(t) ts−1 dt(!(s) < −!(β1)

).

Just as before, we see that for each integer J ≥ 1 the function ϕ≤T (s) extends by the formula

ϕ≤T (s) =

∫ T

0

(ϕ(t) −

J∑

j=1

ajtαj

)ts−1 dt +

J∑

j=1

aj

s + αjT s+αj

to the half-plane !(s) > −!(αJ+1) and hence, letting J → ∞, that ϕ≤T (s) is a meromorphicfunction of s with simple poles of residue aj at s = −αj (j = 1, 2, . . . ) and no other poles.Similarly, ϕ≥T (s) extends to a meromorphic function whose only poles are simple ones of residue−bk at s = −βk. We now define

ϕ(s) = ϕ≤T (s) + ϕ≥T (s) . (4 12 )

This is a meromorphic function of s and is independent of the choice of T , since the effect of

changing T to T ′ is simply to add the everywhere holomorphic function∫ T ′

Tϕ(t) ts−1 dt to ϕ≤T (s)

and subtract the same function from ϕ≥T (s), not affecting the sum of their analytic continuations.

In summary, if ϕ(t) is a function of t with asymptotic expansions as a sum of powers of t (or ofpowers of t multiplied by integral powers of log t) at both zero and infinity, then we can define in acanonical way a Mellin transform ϕ(s) which is meromorphic in the entire s-plane and whose polesreflect directly the coefficients in the asymptotic expansions of ϕ(t). This definition is consistentwith and has the same properties (2) as the original definition (1). We end this section by givingtwo simple examples, while Sections 2 and 3 will give further applications of the method.

Example 1. Let ϕ(t) = tα, where α is a complex number. Then ϕ has an asymptotic expansion (3)at 0 with a single term α1 = α, a1 = 1, and an asymptotic expansion (4) at ∞ with a single termβ1 = α, b1 = 1. We immediately find that ϕ≤T (s) = T s+α/(s + α) for !(s + α) > 0 andϕ≥T (s) = −T s+α/(s + α) for !(s + α) < 0, so that, although the original Mellin transformintegral (1) does not converge for any value of s, the function ϕ(s) defined as the sum of themeromorphic continuations of ϕ≤T (s) and ϕ≥T (s) makes sense, is independent of T , and in fact isidentically zero. More generally, we find that ϕ(s) ≡ 0 whenever ϕ(t) is a finite linear combinationof functions of the form tα logm t with α ∈ C, m ∈ Z≥0. (These are exactly the functions whoseimages ϕλ(t) = ϕ(λt) under the action of the multiplicative group R+ span a finite-dimensionalspace.) In particular, we see that the generalized Mellin transformation is no longer injective.

Example 2. Let ϕ(t) = e−t. Here the integral (1) converges for !(s) > 0 and defines Euler’sgamma-function Γ(s). From the fact that ϕ(t) is of rapid decay at infinity and has the asymptotic(here even convergent) expansion

∑∞n=0(−t)n/n! at zero, we deduce that Γ(s) = ϕ(s) has a mero-

morphic continuation to all s with a simple pole of residue (−1)n/n! at s = −n (n = 0, 1, . . . )and no other poles. Of course, in this special case these well-known properties can also be de-duced from the functional equation Γ(s + 1) = sΓ(s) (proved for !(s) > 0 by integration byparts in the integral defining Γ(s)), N applications of which gives the meromorphic extensionΓ(s) = s−1(s + 1)−1 · · · (s + N − 1)−1Γ(s + N) of Γ(s) to the half-plane !(s) > −N .

From the first of the properties listed in (2), we find the following formula, which we will usemany times:

ϕ(t) = e−λt ⇒ ϕ(s) = Γ(s) λ−s (λ > 0) . (5)

3

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

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Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Extending the definition of the Mellin transform, 5

any T > 0 (formerly we took T = 1, but the extra freedom of being able to choose any value of Twill be very useful later) we define two “half-Mellin transforms” ϕ≤T (s) and ϕ≥T (s) by

ϕ≤T (s) =

∫ T

0

ϕ(t) ts−1 dt(!(s) > −!(α1)

),

ϕ≥T (s) =

∫ ∞

T

ϕ(t) ts−1 dt(!(s) < −!(β1)

).

Just as before, we see that for each integer J ≥ 1 the function ϕ≤T (s) extends by the formula

ϕ≤T (s) =

∫ T

0

(ϕ(t) −

J∑

j=1

ajtαj

)ts−1 dt +

J∑

j=1

aj

s + αjT s+αj

to the half-plane !(s) > −!(αJ+1) and hence, letting J → ∞, that ϕ≤T (s) is a meromorphicfunction of s with simple poles of residue aj at s = −αj (j = 1, 2, . . . ) and no other poles.Similarly, ϕ≥T (s) extends to a meromorphic function whose only poles are simple ones of residue−bk at s = −βk. We now define

ϕ(s) = ϕ≤T (s) + ϕ≥T (s) . (4 12 )

This is a meromorphic function of s and is independent of the choice of T , since the effect of

changing T to T ′ is simply to add the everywhere holomorphic function∫ T ′

Tϕ(t) ts−1 dt to ϕ≤T (s)

and subtract the same function from ϕ≥T (s), not affecting the sum of their analytic continuations.

In summary, if ϕ(t) is a function of t with asymptotic expansions as a sum of powers of t (or ofpowers of t multiplied by integral powers of log t) at both zero and infinity, then we can define in acanonical way a Mellin transform ϕ(s) which is meromorphic in the entire s-plane and whose polesreflect directly the coefficients in the asymptotic expansions of ϕ(t). This definition is consistentwith and has the same properties (2) as the original definition (1). We end this section by givingtwo simple examples, while Sections 2 and 3 will give further applications of the method.

Example 1. Let ϕ(t) = tα, where α is a complex number. Then ϕ has an asymptotic expansion (3)at 0 with a single term α1 = α, a1 = 1, and an asymptotic expansion (4) at ∞ with a single termβ1 = α, b1 = 1. We immediately find that ϕ≤T (s) = T s+α/(s + α) for !(s + α) > 0 andϕ≥T (s) = −T s+α/(s + α) for !(s + α) < 0, so that, although the original Mellin transformintegral (1) does not converge for any value of s, the function ϕ(s) defined as the sum of themeromorphic continuations of ϕ≤T (s) and ϕ≥T (s) makes sense, is independent of T , and in fact isidentically zero. More generally, we find that ϕ(s) ≡ 0 whenever ϕ(t) is a finite linear combinationof functions of the form tα logm t with α ∈ C, m ∈ Z≥0. (These are exactly the functions whoseimages ϕλ(t) = ϕ(λt) under the action of the multiplicative group R+ span a finite-dimensionalspace.) In particular, we see that the generalized Mellin transformation is no longer injective.

Example 2. Let ϕ(t) = e−t. Here the integral (1) converges for !(s) > 0 and defines Euler’sgamma-function Γ(s). From the fact that ϕ(t) is of rapid decay at infinity and has the asymptotic(here even convergent) expansion

∑∞n=0(−t)n/n! at zero, we deduce that Γ(s) = ϕ(s) has a mero-

morphic continuation to all s with a simple pole of residue (−1)n/n! at s = −n (n = 0, 1, . . . )and no other poles. Of course, in this special case these well-known properties can also be de-duced from the functional equation Γ(s + 1) = sΓ(s) (proved for !(s) > 0 by integration byparts in the integral defining Γ(s)), N applications of which gives the meromorphic extensionΓ(s) = s−1(s + 1)−1 · · · (s + N − 1)−1Γ(s + N) of Γ(s) to the half-plane !(s) > −N .

From the first of the properties listed in (2), we find the following formula, which we will usemany times:

ϕ(t) = e−λt ⇒ ϕ(s) = Γ(s) λ−s (λ > 0) . (5)

3

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 74: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Extending the definition of the Mellin transform, 6

any T > 0 (formerly we took T = 1, but the extra freedom of being able to choose any value of Twill be very useful later) we define two “half-Mellin transforms” ϕ≤T (s) and ϕ≥T (s) by

ϕ≤T (s) =

∫ T

0

ϕ(t) ts−1 dt(!(s) > −!(α1)

),

ϕ≥T (s) =

∫ ∞

T

ϕ(t) ts−1 dt(!(s) < −!(β1)

).

Just as before, we see that for each integer J ≥ 1 the function ϕ≤T (s) extends by the formula

ϕ≤T (s) =

∫ T

0

(ϕ(t) −

J∑

j=1

ajtαj

)ts−1 dt +

J∑

j=1

aj

s + αjT s+αj

to the half-plane !(s) > −!(αJ+1) and hence, letting J → ∞, that ϕ≤T (s) is a meromorphicfunction of s with simple poles of residue aj at s = −αj (j = 1, 2, . . . ) and no other poles.Similarly, ϕ≥T (s) extends to a meromorphic function whose only poles are simple ones of residue−bk at s = −βk. We now define

ϕ(s) = ϕ≤T (s) + ϕ≥T (s) . (4 12 )

This is a meromorphic function of s and is independent of the choice of T , since the effect of

changing T to T ′ is simply to add the everywhere holomorphic function∫ T ′

Tϕ(t) ts−1 dt to ϕ≤T (s)

and subtract the same function from ϕ≥T (s), not affecting the sum of their analytic continuations.

In summary, if ϕ(t) is a function of t with asymptotic expansions as a sum of powers of t (or ofpowers of t multiplied by integral powers of log t) at both zero and infinity, then we can define in acanonical way a Mellin transform ϕ(s) which is meromorphic in the entire s-plane and whose polesreflect directly the coefficients in the asymptotic expansions of ϕ(t). This definition is consistentwith and has the same properties (2) as the original definition (1). We end this section by givingtwo simple examples, while Sections 2 and 3 will give further applications of the method.

Example 1. Let ϕ(t) = tα, where α is a complex number. Then ϕ has an asymptotic expansion (3)at 0 with a single term α1 = α, a1 = 1, and an asymptotic expansion (4) at ∞ with a single termβ1 = α, b1 = 1. We immediately find that ϕ≤T (s) = T s+α/(s + α) for !(s + α) > 0 andϕ≥T (s) = −T s+α/(s + α) for !(s + α) < 0, so that, although the original Mellin transformintegral (1) does not converge for any value of s, the function ϕ(s) defined as the sum of themeromorphic continuations of ϕ≤T (s) and ϕ≥T (s) makes sense, is independent of T , and in fact isidentically zero. More generally, we find that ϕ(s) ≡ 0 whenever ϕ(t) is a finite linear combinationof functions of the form tα logm t with α ∈ C, m ∈ Z≥0. (These are exactly the functions whoseimages ϕλ(t) = ϕ(λt) under the action of the multiplicative group R+ span a finite-dimensionalspace.) In particular, we see that the generalized Mellin transformation is no longer injective.

Example 2. Let ϕ(t) = e−t. Here the integral (1) converges for !(s) > 0 and defines Euler’sgamma-function Γ(s). From the fact that ϕ(t) is of rapid decay at infinity and has the asymptotic(here even convergent) expansion

∑∞n=0(−t)n/n! at zero, we deduce that Γ(s) = ϕ(s) has a mero-

morphic continuation to all s with a simple pole of residue (−1)n/n! at s = −n (n = 0, 1, . . . )and no other poles. Of course, in this special case these well-known properties can also be de-duced from the functional equation Γ(s + 1) = sΓ(s) (proved for !(s) > 0 by integration byparts in the integral defining Γ(s)), N applications of which gives the meromorphic extensionΓ(s) = s−1(s + 1)−1 · · · (s + N − 1)−1Γ(s + N) of Γ(s) to the half-plane !(s) > −N .

From the first of the properties listed in (2), we find the following formula, which we will usemany times:

ϕ(t) = e−λt ⇒ ϕ(s) = Γ(s) λ−s (λ > 0) . (5)

3

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 75: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Extending the definition of the Mellin transform, 7 - theGamma function

any T > 0 (formerly we took T = 1, but the extra freedom of being able to choose any value of Twill be very useful later) we define two “half-Mellin transforms” ϕ≤T (s) and ϕ≥T (s) by

ϕ≤T (s) =

∫ T

0

ϕ(t) ts−1 dt(!(s) > −!(α1)

),

ϕ≥T (s) =

∫ ∞

T

ϕ(t) ts−1 dt(!(s) < −!(β1)

).

Just as before, we see that for each integer J ≥ 1 the function ϕ≤T (s) extends by the formula

ϕ≤T (s) =

∫ T

0

(ϕ(t) −

J∑

j=1

ajtαj

)ts−1 dt +

J∑

j=1

aj

s + αjT s+αj

to the half-plane !(s) > −!(αJ+1) and hence, letting J → ∞, that ϕ≤T (s) is a meromorphicfunction of s with simple poles of residue aj at s = −αj (j = 1, 2, . . . ) and no other poles.Similarly, ϕ≥T (s) extends to a meromorphic function whose only poles are simple ones of residue−bk at s = −βk. We now define

ϕ(s) = ϕ≤T (s) + ϕ≥T (s) . (4 12 )

This is a meromorphic function of s and is independent of the choice of T , since the effect of

changing T to T ′ is simply to add the everywhere holomorphic function∫ T ′

Tϕ(t) ts−1 dt to ϕ≤T (s)

and subtract the same function from ϕ≥T (s), not affecting the sum of their analytic continuations.

In summary, if ϕ(t) is a function of t with asymptotic expansions as a sum of powers of t (or ofpowers of t multiplied by integral powers of log t) at both zero and infinity, then we can define in acanonical way a Mellin transform ϕ(s) which is meromorphic in the entire s-plane and whose polesreflect directly the coefficients in the asymptotic expansions of ϕ(t). This definition is consistentwith and has the same properties (2) as the original definition (1). We end this section by givingtwo simple examples, while Sections 2 and 3 will give further applications of the method.

Example 1. Let ϕ(t) = tα, where α is a complex number. Then ϕ has an asymptotic expansion (3)at 0 with a single term α1 = α, a1 = 1, and an asymptotic expansion (4) at ∞ with a single termβ1 = α, b1 = 1. We immediately find that ϕ≤T (s) = T s+α/(s + α) for !(s + α) > 0 andϕ≥T (s) = −T s+α/(s + α) for !(s + α) < 0, so that, although the original Mellin transformintegral (1) does not converge for any value of s, the function ϕ(s) defined as the sum of themeromorphic continuations of ϕ≤T (s) and ϕ≥T (s) makes sense, is independent of T , and in fact isidentically zero. More generally, we find that ϕ(s) ≡ 0 whenever ϕ(t) is a finite linear combinationof functions of the form tα logm t with α ∈ C, m ∈ Z≥0. (These are exactly the functions whoseimages ϕλ(t) = ϕ(λt) under the action of the multiplicative group R+ span a finite-dimensionalspace.) In particular, we see that the generalized Mellin transformation is no longer injective.

Example 2. Let ϕ(t) = e−t. Here the integral (1) converges for !(s) > 0 and defines Euler’sgamma-function Γ(s). From the fact that ϕ(t) is of rapid decay at infinity and has the asymptotic(here even convergent) expansion

∑∞n=0(−t)n/n! at zero, we deduce that Γ(s) = ϕ(s) has a mero-

morphic continuation to all s with a simple pole of residue (−1)n/n! at s = −n (n = 0, 1, . . . )and no other poles. Of course, in this special case these well-known properties can also be de-duced from the functional equation Γ(s + 1) = sΓ(s) (proved for !(s) > 0 by integration byparts in the integral defining Γ(s)), N applications of which gives the meromorphic extensionΓ(s) = s−1(s + 1)−1 · · · (s + N − 1)−1Γ(s + N) of Γ(s) to the half-plane !(s) > −N .

From the first of the properties listed in (2), we find the following formula, which we will usemany times:

ϕ(t) = e−λt ⇒ ϕ(s) = Γ(s) λ−s (λ > 0) . (5)

3

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 76: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Dirichlet series and their special values

We now look at functions φ for which the Mellin transform definedin Section 1 is related to a Dirichlet series. The key formula is (5),because it allows us to convert Dirichlet series into exponentialseries, which are much simpler.

2. Dirichlet series and their special values

In this section we look at functions ϕ(t) for which the Mellin transform defined in Section 1 isrelated to a Dirichlet series. The key formula is (5), because it allows us to convert Dirichlet seriesinto exponential series, which are much simpler.

Example 3. Define ϕ(t) for t > 0 by ϕ(t) = 1/(et − 1). This function is of rapid decay at infinityand has an asymptotic expansion (actually convergent for t < 2π)

1

et − 1=

1

t +t2

2+

t3

6+ · · ·

=∞∑

r=0

Br

r!tr−1 (6)

with certain rational coefficients B0 = 1, B1 = − 12 , B2 = 1

6 , . . . called Bernoulli numbers. Fromthe results of Section 1 we know that the Mellin transform ϕ(s), originally defined for "(s) > 1by the integral (1), has a meromorphic continuation to all s with simple poles of residue Br/r!at s = 1 − r (r = 0, 1, 2, . . . ). On the other hand, since et > 1 for t > 0, we can expand ϕ(t)as a geometric series e−t + e−2t + e−3t + · · · , so (5) gives (first in the region of convergence)ϕ(s) = Γ(s)ζ(s), where

ζ(s) =∞∑

m=1

1

ms("(s) > 1) (7)

is the Riemann zeta function. Since Γ(s), as we have seen is also meromorphic, with simple polesof residue (−1)n/n! at non-positive integral arguments s = −n and no other poles, and sinceΓ(s) (as is well-known and easily proved) never vanishes, we deduce that ζ(s) has a meromorphiccontinuation to all s with a unique simple pole of residue 1/Γ(1) = 1 at s = 1 and that its values atnon-positive integral arguments are rational numbers expressible in terms of the Bernoulli numbers:

ζ(−n) = (−1)n Bn+1

n + 1(n = 0, 1, 2, . . . ) . (8)

Example 4. To approach ζ(s) in another way, we choose for ϕ(t) the theta function

ϑ(t) =

∞∑

n=−∞e−πn2t (t > 0) . (9)

(The factor π in the exponent has been included for later convenience.) We can write this out as

ϑ(t) = 1 + 2 e−πt + 2 e−4πt + · · · , (10)

and since the generalized Mellin transform of the function 1 is identically 0 by Example 1, wededuce from (5) that ϕ(s) = 2 ζ∗(2s), where

ζ∗(s) = π−s/2 Γ(s/2) ζ(s) . (11)

To obtain the analytic properties of ζ(s) from the results of Section 1, we need the asymptotics ofϑ(t) at zero and infinity. They follow immediately from the following famous result, due to Jacobi:

4

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 77: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Dirichlet series and their special values

Euler’s calculation of the zeta function at negative integers

2. Dirichlet series and their special values

In this section we look at functions ϕ(t) for which the Mellin transform defined in Section 1 isrelated to a Dirichlet series. The key formula is (5), because it allows us to convert Dirichlet seriesinto exponential series, which are much simpler.

Example 3. Define ϕ(t) for t > 0 by ϕ(t) = 1/(et − 1). This function is of rapid decay at infinityand has an asymptotic expansion (actually convergent for t < 2π)

1

et − 1=

1

t +t2

2+

t3

6+ · · ·

=∞∑

r=0

Br

r!tr−1 (6)

with certain rational coefficients B0 = 1, B1 = − 12 , B2 = 1

6 , . . . called Bernoulli numbers. Fromthe results of Section 1 we know that the Mellin transform ϕ(s), originally defined for "(s) > 1by the integral (1), has a meromorphic continuation to all s with simple poles of residue Br/r!at s = 1 − r (r = 0, 1, 2, . . . ). On the other hand, since et > 1 for t > 0, we can expand ϕ(t)as a geometric series e−t + e−2t + e−3t + · · · , so (5) gives (first in the region of convergence)ϕ(s) = Γ(s)ζ(s), where

ζ(s) =∞∑

m=1

1

ms("(s) > 1) (7)

is the Riemann zeta function. Since Γ(s), as we have seen is also meromorphic, with simple polesof residue (−1)n/n! at non-positive integral arguments s = −n and no other poles, and sinceΓ(s) (as is well-known and easily proved) never vanishes, we deduce that ζ(s) has a meromorphiccontinuation to all s with a unique simple pole of residue 1/Γ(1) = 1 at s = 1 and that its values atnon-positive integral arguments are rational numbers expressible in terms of the Bernoulli numbers:

ζ(−n) = (−1)n Bn+1

n + 1(n = 0, 1, 2, . . . ) . (8)

Example 4. To approach ζ(s) in another way, we choose for ϕ(t) the theta function

ϑ(t) =

∞∑

n=−∞e−πn2t (t > 0) . (9)

(The factor π in the exponent has been included for later convenience.) We can write this out as

ϑ(t) = 1 + 2 e−πt + 2 e−4πt + · · · , (10)

and since the generalized Mellin transform of the function 1 is identically 0 by Example 1, wededuce from (5) that ϕ(s) = 2 ζ∗(2s), where

ζ∗(s) = π−s/2 Γ(s/2) ζ(s) . (11)

To obtain the analytic properties of ζ(s) from the results of Section 1, we need the asymptotics ofϑ(t) at zero and infinity. They follow immediately from the following famous result, due to Jacobi:

4

For n = 1 we find ζ(−1) = − 112

which was our “mysterious” formula

1 + 2 + 3 + · · · = −1

12

in Lecture 1.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 78: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Dirichlet series and their special values

Enter the theta function

2. Dirichlet series and their special values

In this section we look at functions ϕ(t) for which the Mellin transform defined in Section 1 isrelated to a Dirichlet series. The key formula is (5), because it allows us to convert Dirichlet seriesinto exponential series, which are much simpler.

Example 3. Define ϕ(t) for t > 0 by ϕ(t) = 1/(et − 1). This function is of rapid decay at infinityand has an asymptotic expansion (actually convergent for t < 2π)

1

et − 1=

1

t +t2

2+

t3

6+ · · ·

=∞∑

r=0

Br

r!tr−1 (6)

with certain rational coefficients B0 = 1, B1 = − 12 , B2 = 1

6 , . . . called Bernoulli numbers. Fromthe results of Section 1 we know that the Mellin transform ϕ(s), originally defined for "(s) > 1by the integral (1), has a meromorphic continuation to all s with simple poles of residue Br/r!at s = 1 − r (r = 0, 1, 2, . . . ). On the other hand, since et > 1 for t > 0, we can expand ϕ(t)as a geometric series e−t + e−2t + e−3t + · · · , so (5) gives (first in the region of convergence)ϕ(s) = Γ(s)ζ(s), where

ζ(s) =∞∑

m=1

1

ms("(s) > 1) (7)

is the Riemann zeta function. Since Γ(s), as we have seen is also meromorphic, with simple polesof residue (−1)n/n! at non-positive integral arguments s = −n and no other poles, and sinceΓ(s) (as is well-known and easily proved) never vanishes, we deduce that ζ(s) has a meromorphiccontinuation to all s with a unique simple pole of residue 1/Γ(1) = 1 at s = 1 and that its values atnon-positive integral arguments are rational numbers expressible in terms of the Bernoulli numbers:

ζ(−n) = (−1)n Bn+1

n + 1(n = 0, 1, 2, . . . ) . (8)

Example 4. To approach ζ(s) in another way, we choose for ϕ(t) the theta function

ϑ(t) =

∞∑

n=−∞e−πn2t (t > 0) . (9)

(The factor π in the exponent has been included for later convenience.) We can write this out as

ϑ(t) = 1 + 2 e−πt + 2 e−4πt + · · · , (10)

and since the generalized Mellin transform of the function 1 is identically 0 by Example 1, wededuce from (5) that ϕ(s) = 2 ζ∗(2s), where

ζ∗(s) = π−s/2 Γ(s/2) ζ(s) . (11)

To obtain the analytic properties of ζ(s) from the results of Section 1, we need the asymptotics ofϑ(t) at zero and infinity. They follow immediately from the following famous result, due to Jacobi:

4

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 79: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Dirichlet series and their special values

Jacobi’s functional equation of the theta function

Proposition 1. The function ϑ(t) satisfies the functional equation

ϑ(t) =1√tϑ(1

t

)(t > 0) . (12)

Proof. Formula (12) is a special case of the Poisson summation formula, which says that∑

n∈Zf(n) =

n∈Zf(n) (13)

for any sufficiently well-behaved (i.e., smooth and small at infinity) function f : R → C, where

f(y) =∫ ∞

−∞ f(x) e2πixy dx is the Fourier transform of f . (To prove this, note that the function

F (x) =∑

n∈Z f(n+x) is periodic with period 1, so has a Fourier expansion F (x) =∑

m∈Z cme2πimx

with cm =∫ 1

0F (x)e−2πimxdx = f(−m). Now set x = 0.) Now consider the function ft(x) =

e−πtx2

. Its Fourier transform is given by

ft(y) =

∫ ∞

−∞e−πtx2+2πixy dx = e−πy2/t

∫ ∞

−∞e−πt(x+iy/t)2 dx =

c√tf1/t(y) ,

where c > 0 is the constant c =∫ ∞

−∞ e−πx2

dx. Applying (13) with f = ft therefore gives ϑ(t) =

ct−1/2ϑ(1/t), and taking t = 1 in this formula gives c = 1 and proves equation (12). !

Now we find from (10) that ϑ(t) has the asymptotic expansions ϑ(t) = 1+O(t−N ) as t → ∞ andϑ(t) = t−1/2 + O(tN ) as t → 0, where N > 0 is arbitrary. It follows from the results of Section 1

that its Mellin transform ϑ(s) has a meromorphic extension to all s with simple poles of residue 1

and −1 at s = 1/2 and s = 0, respectively, and no other poles. From the formula ζ∗(s) = 12 ϑ(s/2)

we deduce that the function ζ∗(s) defined in (11) is meromorphic having simple poles of residue 1and 0 at s = 1 and s = 0 and no other poles and hence (using once again that Γ(s) has simplepoles at non-positive integers and never vanishes) that ζ(s) itself is holomorphic except for a singlepole of residue 1 at s = 1 and vanishes at negative even arguments s = −2, −4, . . . . This is weakerthan (8), which gives a formula for ζ(s) at all non-positive arguments (and also shows the vanishingat negative even integers because it is an exercise to deduce from the definition (6) that Br vanishesfor odd r > 1). The advantage of the second approach to ζ(s) is that from equation (12) and theproperties of Mellin transforms listed in (2) we immediately deduce the famous functional equation

ζ∗(1 − s) = ζ∗(1 − s) (14)

of the Riemann zeta-function which was discovered (for integer values %= 0, 1 of s) by Euler in 1749and proved (for all complex values %= 0, 1 of s) by Riemann in 1859 by just this argument.

We next generalize the method of Example 3. Consider a generalized Dirichlet series

L(s) =

∞∑

m=1

cm λ−sm (15)

where the λm are real numbers satisfying 0 < λ1 < λ2 < · · · and growing at least as fast as somepositive power of m. (This is an ordinary Dirichlet series if λm = m for all m.) Assume thatthe series converges for at least one value s0 of s. Then it automatically converges in a half-plane(for instance, if λm = m then the fact that cm = O(ms0) implies convergence in the half-plane&(s) > &(s0) + 1) and the associated exponential series

ϕ(t) =

∞∑

m=1

cm e−λmt (t > 0), (16)

converges for all positive values of t. We then have:5

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

Page 80: Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform

Dirichlet series and their special values

The functional equation of the zeta function

Proposition 1. The function ϑ(t) satisfies the functional equation

ϑ(t) =1√tϑ(1

t

)(t > 0) . (12)

Proof. Formula (12) is a special case of the Poisson summation formula, which says that∑

n∈Zf(n) =

n∈Zf(n) (13)

for any sufficiently well-behaved (i.e., smooth and small at infinity) function f : R → C, where

f(y) =∫ ∞

−∞ f(x) e2πixy dx is the Fourier transform of f . (To prove this, note that the function

F (x) =∑

n∈Z f(n+x) is periodic with period 1, so has a Fourier expansion F (x) =∑

m∈Z cme2πimx

with cm =∫ 1

0F (x)e−2πimxdx = f(−m). Now set x = 0.) Now consider the function ft(x) =

e−πtx2

. Its Fourier transform is given by

ft(y) =

∫ ∞

−∞e−πtx2+2πixy dx = e−πy2/t

∫ ∞

−∞e−πt(x+iy/t)2 dx =

c√tf1/t(y) ,

where c > 0 is the constant c =∫ ∞

−∞ e−πx2

dx. Applying (13) with f = ft therefore gives ϑ(t) =

ct−1/2ϑ(1/t), and taking t = 1 in this formula gives c = 1 and proves equation (12). !

Now we find from (10) that ϑ(t) has the asymptotic expansions ϑ(t) = 1+O(t−N ) as t → ∞ andϑ(t) = t−1/2 + O(tN ) as t → 0, where N > 0 is arbitrary. It follows from the results of Section 1

that its Mellin transform ϑ(s) has a meromorphic extension to all s with simple poles of residue 1

and −1 at s = 1/2 and s = 0, respectively, and no other poles. From the formula ζ∗(s) = 12 ϑ(s/2)

we deduce that the function ζ∗(s) defined in (11) is meromorphic having simple poles of residue 1and 0 at s = 1 and s = 0 and no other poles and hence (using once again that Γ(s) has simplepoles at non-positive integers and never vanishes) that ζ(s) itself is holomorphic except for a singlepole of residue 1 at s = 1 and vanishes at negative even arguments s = −2, −4, . . . . This is weakerthan (8), which gives a formula for ζ(s) at all non-positive arguments (and also shows the vanishingat negative even integers because it is an exercise to deduce from the definition (6) that Br vanishesfor odd r > 1). The advantage of the second approach to ζ(s) is that from equation (12) and theproperties of Mellin transforms listed in (2) we immediately deduce the famous functional equation

ζ∗(1 − s) = ζ∗(1 − s) (14)

of the Riemann zeta-function which was discovered (for integer values %= 0, 1 of s) by Euler in 1749and proved (for all complex values %= 0, 1 of s) by Riemann in 1859 by just this argument.

We next generalize the method of Example 3. Consider a generalized Dirichlet series

L(s) =

∞∑

m=1

cm λ−sm (15)

where the λm are real numbers satisfying 0 < λ1 < λ2 < · · · and growing at least as fast as somepositive power of m. (This is an ordinary Dirichlet series if λm = m for all m.) Assume thatthe series converges for at least one value s0 of s. Then it automatically converges in a half-plane(for instance, if λm = m then the fact that cm = O(ms0) implies convergence in the half-plane&(s) > &(s0) + 1) and the associated exponential series

ϕ(t) =

∞∑

m=1

cm e−λmt (t > 0), (16)

converges for all positive values of t. We then have:5

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform