Suppose that f R R is periodic on R, integrable on , , and that ak(f) 0 for k 0, 1, a) Prove that (Skf)(0) (Sjf)(0) for all k j 0 b) Prove that SNf(0) 22Nf(0) for N N c) Prove that k 1 ak(f) d) Suppose that f is also even Prove that f must be continuous and Sf converges uniformly and absolutely on ...

a Let k j 0 Since the coefficients are nonnegative b By part a c By Exer ...

Suppose that f: R R is periodic on R, integrable on [-, ], and that ak(f)

Question:

Suppose that f: R → R is periodic on R, integrable on [-π, π], and that ak(f) > 0 for k = 0, 1,....
a) Prove that (Skf)(0) > (Sjf)(0) for all k > j > 0.
b) Prove that SNf(0) < 2σ2Nf(0) for N ∈ N.
c) Prove that ∑∞k=1 |ak(f)| < ∞.
d) Suppose that f is also even. Prove that f must be continuous and Sf converges uniformly and absolutely on R.