ME 234, Lp spaces, Convolution, and Fourier transforms

1. Assume f ∈ L1. Let f denote its Fourier transform. Show that

∥

∥

∥f∥

∥

∥

∞

≤ ‖f‖1

Remark: Written out, this simply means

supω∈R

∣

∣

∣f(ω)∣

∣

∣ ≤∫

∞

−∞

|f(t)| dt

2. Show that for f, g ∈ L2, the function

〈f, g〉 :=∫

∞

−∞

f(t)g(t)dt

is an inner product. You should assume that the functions f and g are real valued.

How would you change the definition if f and g map R to C?

3. Suppose that f and g are in L2. Show that f ⋆ g ∈ L∞,

‖f ⋆ g‖∞

≤ ‖f‖2‖g‖

2

and

limt→∞

(f ⋆ g)(t) = 0

4. Suppose h and u are in L1. In class we showed that h ⋆ u ∈ L1. The Fourier

transform (using just the integral defintion) is well-defined for all L1 functions,

yielding a nice bounded, continuous function. Show that for all ω ∈ R,

(h ⋆ u)ˆ(ω) = h(ω)u(ω)

Hint: You should first justify that the order of integration can be exchanged in

the same manner that we did in class.

5. Suppose h ∈ L1 and u ∈ L2. In class we showed that h ⋆ u ∈ L2 and ‖h ⋆ u‖2≤

‖h‖1‖u‖

2.

Show that

(h ⋆ u)ˆ = hu

Hint: Recall how Fourier transform on L2 (the Plancheral transform) is defined.

6. Let our domain be I := [0 ∞). Define V := L2 × L∞, so an element v ∈ V is a

pair of functions, v = (v1, v2) with v1 ∈ L2, v2 ∈ L∞.

1

Addition in V is defined obviously: if v, w ∈ V , then v = (v1, v2) and w = (w1, w2)

for some v1, w1 ∈ L2, v2, w2 ∈ L∞. Define v + w ∈ V as

v + w := (v1 + w1, v2 + w2)

Scalar multiplication in V is similarly defined: for v ∈ V and α ∈ F (ie. R or C),

define αv ∈ V as

αv := (αv1, αv2)

where v1 ∈ L2 and v2 ∈ L∞ are the components of v. Finally, for v ∈ V , define

‖v‖ := max {‖v1‖2, ‖v2‖∞}

(a) Show that ‖·‖ is a norm.

(b) Suppose g ∈ L1

⋂

L2 and h ∈ L1. Define a linear operator A : V → L∞ by

A(v) := g ⋆ v1 + h ⋆ v2

Show that

‖A‖L∞←V = ‖g‖

2+ ‖h‖

1

2