# Fourier series convergence - holder continuity and differentiability

1. Oct 21, 2011

### hlin818

1. The problem statement, all variables and given/known data

Given each of the functions f below, describe the set of points at which the Fourier
series converges to f.

b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x)

2. Relevant equations

Theorem: If f(x) is absolutely integrable, then its fourier series converges to f at the points where f is also holder continuous or differentiable.

3. The attempt at a solution

I managed to prove that f(x) is holder continuous on [-pi,pi] so by the theorem above since f(x) is absolutely integrable its fourier series converges to f(x) at every point on this interval. But f(x) is not differentiable at x=0. What did I analyze incorrectly? Or in this context are holder continuity and differentiability not equivalent conditions (i.e. one can fail and the other can hold)?

Last edited: Oct 21, 2011
2. Oct 21, 2011

### hlin818

I may be wrong about the holder condition, but it looks to me like f(x) is holder continuous as long as the exponent in the condition is equal to or less than 1/2.

Last edited: Oct 21, 2011
3. Oct 22, 2011

### hlin818

Sorry about the bump, but this question is killing me. I don't feel like the book explained this convergence criteria well at all.

Last edited: Oct 22, 2011
4. Oct 23, 2011

final bump