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Transcript of Math212a1406 The Fourier Transform The Laplace transform ... shlomo/212a/06.pdf OutlineConventions,

  • 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

    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 calculus form The Mellin Transform

  • 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 trransform Dirichlet 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

  • 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 infinitely differentiable and vanish at infinity rapidly with all their derivatives in the sense that

    ‖f ‖m,n := sup x∈R {|xmf (n)(x)|}

  • 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 measure 1√ 2π

    dx

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

    f̂ (ξ) := 1√ 2π

    R f (x)e−ixξdx

    and the convolution of two elements of S by

    (f ? g)(x) := 1√ 2π

    R f (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

  • 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 ∞. We get

    d

    ( 1√ 2π

    R f (x)e−ixξdx

    ) =

    1√ 2π

    R (−ix)f (x)e−ixξdx .

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

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

    1√ 2π

    R f ′(x)e−ixξdx = (iξ)

    1√ 2π

    R f (x)e−ixξdx .

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

    Shlomo Sternberg

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

  • 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 ∞. We get

    d

    ( 1√ 2π

    R f (x)e−ixξdx

    ) =

    1√ 2π

    R (−ix)f (x)e−ixξdx .

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

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

    1√ 2π

    R f ′(x)e−ixξdx = (iξ)

    1√ 2π

    R f (x)e−ixξdx .

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

    Shlomo Sternberg

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

  • 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 ∞. We get

    d

    ( 1√ 2π

    R f (x)e−ixξdx

    ) =

    1√ 2π

    R (−ix)f (x)e−ixξdx .

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

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

    1√ 2π

    R f ′(x)e−ixξdx = (iξ)

    1√ 2π

    R f (x)e−ixξdx .

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

    Shlomo Sternberg

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

  • 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 f̂ ,

    as follows by differentiation under the integral sign and by integration by parts. This shows that the Fourier transform maps S 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

  • 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

    R f (u)e−iuξdu

    1√ 2π

    R g(t)e−itξdt

    so (f ? g )̂ = f̂ ĝ .

    Shlomo Sternberg

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

  • 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). Then setting u = ax so dx = (1/a)du we have

    (Saf )̂(ξ) = 1√ 2π

    R f (ax)e−ixξdx

    = 1√ 2π

    R (1/a)f (u)e−iu(ξ/a)du

    so (Saf )̂ = (1/a)S1/a f̂ .

    Shlomo Sternberg

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

  • 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 coordina