Measure, Integration & Real Analysis, 2021a

360 Chapter 11 Fourier Analysis
11.44 Fourier coefficients of a convolution
Suppose f , g ∈ L 1 (∂D). Then
for every n ∈ Z.
Proof
11.45
( f ∗ g)̂(n) = ̂f (n) ĝ(n)
First note that if w ∈ ∂D and n ∈ Z, then
∫
∫
g(zw)z n dσ(z) = g(ζ)ζ n w n dσ(ζ) =w n ĝ(n),
∂D
∂D
where the first equality comes from the substitution ζ = zw (equivalent to z = ζw),
which is justified by the rotation invariance of σ.
Now
∫
( f ∗ g)̂(n) = ( f ∗ g)(z)z n dσ(z)
∂D
∫ ∫
= z n f (w)g(zw) dσ(w) dσ(z)
∂D ∂D
∫ ∫
= f (w) g(zw)z n dσ(z) dσ(w)
∂D ∂D
∫
= f (w)w n ĝ(n) dσ(w)
∂D
= ̂f (n) ĝ(n),
where the interchange of integration order in the third equality is justified by the same
steps used in the proof of 11.37 and the fourth equality above is justified by 11.45.
The next result could be proved by appropriate uses of Tonelli’s Theorem and
Fubini’s Theorem. However, the slick proof technique used in the proof below should
be useful in dealing with some of the exercises.
11.46 convolution is associative
Suppose f , g, h ∈ L 1 (∂D). Then ( f ∗ g) ∗ h = f ∗ (g ∗ h).
Proof
Similarly,
Suppose n ∈ Z. Using 11.44 twice, we have
( ( f ∗ g) ∗ h
)̂(n) =(f ∗ g)̂(n)ĥ(n) = ̂f (n)ĝ(n)ĥ(n).
(
f ∗ (g ∗ h)
)̂(n) = ̂f (n)(g ∗ h)̂(n) =
̂f (n)ĝ(n)ĥ(n).
Hence ( f ∗ g) ∗ h and f ∗ (g ∗ h) have the same Fourier coefficients. Because
functions in L 1 (∂D) are determined by their Fourier coefficients (see 11.43), this
implies that ( f ∗ g) ∗ h = f ∗ (g ∗ h).
Measure, Integration & Real Analysis, by Sheldon Axler