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 ˆˆ R.
a) Prove that

holds for each h ˆˆ R.
b) If h = Ï€/2n+l, prove that sin2 kh > 1/2 for all k ˆˆ [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.

ST 1/2 2"-1 1/2 2"-1