Measure, Integration & Real Analysis, 2021a

356 Chapter 11 Fourier Analysis
Now the convergence of the Fourier series of f ∈ L 2 (∂D) to f follows immediately
from standard Hilbert space theory [see 8.63(a)] and the previous result. Thus
with no further proof needed, we have the following important result.
11.31 convergence of Fourier series in the norm of L 2 (∂D)
Suppose f ∈ L 2 (∂D). Then
f =
∞
∑
̂f (n)z n ,
n=−∞
where the infinite sum converges to f in the norm of L 2 (∂D).
The next example is a spectacular application
of Hilbert space theory and the
orthonormal basis {z n } n∈Z of L 2 (∂D).
The evaluation of ∑ ∞ n=1 1 had been an
n
open question until Euler 2
discovered in
1734 that this infinite sum equals π2
6 .
Euler’s proof, which would not be
considered sufficiently rigorous by
today’s standards, was quite
different from the technique used in
the example below.
1
11.32 Example
1 2 + 1 2 2 + 1 π2
+ ···=
32 6
Define f ∈ L 2 (∂D) by f (e it )=t for t ∈ (−π, π]. Then ̂f (0) = ∫ π
For n ∈ Z \{0},wehave
̂f (n) =
∫ π
−π
te −int dt
2π
= te−int ] t=π
−2πin
+ 1 ∫ π
e −int dt
t=−π in −π 2π
−π t dt
2π = 0.
= (−1)n i
,
n
where the second line above follows from integration by parts. The equation above
implies that
11.33
Also,
∞
∑
n=−∞
11.34 ‖ f ‖ 2 2 = ∫ π
| ̂f (n)| 2 = 2
−π
∞
∑
n=1
1
n 2 .
t 2 dt
2π = π2
3 .
Parseval’s identity [8.63(c)] implies that the left side of 11.33 equals the left side of
11.34. Setting the right side of 11.33 equal to the right side of 11.34 shows that
∞
∑
n=1
1
n 2 = π2
6 .
Measure, Integration & Real Analysis, by Sheldon Axler