Measure, Integration & Real Analysis, 2021a

354 Chapter 11 Fourier Analysis
12 Define f : ∂D → R by
⎧
⎪⎨ 1 if Im z > 0,
f (z) = −1 if Im z < 0,
⎪⎩
0 if Im z = 0.
(a) Show that if n ∈ Z, then
(b) Show that
̂f (n) =
(P r f )(z) = 2 π
{
−
2i
nπ
if n is odd,
0 if n is even.
arctan
2r Im z
1 − r 2
for every r ∈ [0, 1) and every z ∈ ∂D.
(c) Verify that lim r↑1 (P r f )(z) = f (z) for every z ∈ ∂D.
(d) Prove that P r f does not converge uniformly to f on ∂D.
Measure, Integration & Real Analysis, by Sheldon Axler