5. [BERNSTEIN] Let f : R -> R be periodic and ex > O. Suppose that f is Lipschitz of order ex; i.e., there is a constant M > 0 such that for all x, hER.

(a) Prove that If(x + h) - f(x)1 :::; MlhlQ 1j7r 00 :;;: -7r If(x + h) - f(x - h)12 dx = 4 t;(a~

(f) + b~(f)) sin2 kh

(b) If h = 7f/2n +1, prove that sin2 kh ~ 1/2 for all k E [2n - 1,2n J.

(c) Combine parts

(a) and

(b) to prove that for n = 1,2,3, ....

(d) Assuming that (see Exercise 9, p. 380), prove that if I is Lipschitz of order 0: for some 0: > 1/2, then SI converges absolutely and uniformly on R.

(e) Prove that if I : R -+ R is periodic and continuously differentiable, then S I converges absolutely and uniformly on R.