(Solution) - Let f R R be periodic and a -(2025 Original AI-Free Solution)

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Let f : R ? R be periodic and a > 0. Suppose that f is Lipschitz of order a; that is, there is a constant M > 0 such that

for all x, h ? R.
a) Prove that
holds for each h ? R.
b) If h = ?/2n+l, prove that sin2 kh > 1/2 for all k ? [2n-1, 2n].
c) Combine parts a) and b) to prove that
for» = 1, 2, 3,....
d) Assuming
(see Exercise 11.7.9), prove that if f is Lipschitz of order a for some a > 1/2, then Sf converges absolutely and uniformly on R.
e) Prove that if f: R ? R is periodic and continuously differentiable, then Sf converges absolutely and uniformly on R.