Measure, Integration & Real Analysis, 2021a

342 Chapter 11 Fourier Analysis
Hilbert space theory tells us that if f is in the closure in L 2 (∂D) of span{z n } n∈Z ,
then
f = ∑ 〈 f , z n 〉z n ,
n∈Z
where the infinite sum above converges as an unordered sum in the norm of L 2 (∂D)
(see 8.58). The inner product 〈 f , z n 〉 above equals
∫
∂D
f (z)z n dσ(z).
Because |z n | = 1 for every z ∈ ∂D, the integral above makes sense not only for
f ∈ L 2 (∂D) but also for f in the larger space L 1 (∂D). Thus we make the following
definition.
11.7 Definition Fourier coefficient; ̂f (n); Fourier series
Suppose f ∈ L 1 (∂D).
• For n ∈ Z, the n th Fourier coefficient of f is denoted ̂f (n) and is defined by
∫
̂f (n) =
∂D
f (z)z n dσ(z) =
• The Fourier series of f is the formal sum
∫ π
−π
∞
∑
̂f (n)z n .
n=−∞
f (e it )e −int dt
2π .
As we will see, Fourier analysis helps describe the sense in which the Fourier
series of f represents f .
11.8 Example Fourier coefficients
• Suppose h is an analytic function on an open set that contains the closed unit
disk D. Then h has a power series representation
h(z) =
∞
∑ a n z n ,
n=0
where the sum on the right converges uniformly on D to h. Because uniform
convergence on ∂D implies convergence in L 2 (∂D), 8.58(b) and 11.6 now imply
that
{
a n if n ≥ 0,
(h| ∂D )̂(n) =
0 if n < 0
for all n ∈ Z. In other words, for functions analytic on an open set containing
D, the Fourier series is the same as the Taylor series.
Measure, Integration & Real Analysis, by Sheldon Axler