absolute convergence of the Fourier coeff for Hölder continuous function

Question

I saw the following theorem in the wiki page: http://en.wikipedia.org/wiki/H%C3%B6lder_condition

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if $f$ satisfies the $\alpha$-Hölder condition $| f(x) - f(y) | \leq C \, |x - y|^{\alpha}$ for some $\alpha>1/2$, then

$||f||_{A} = \sum_i |c_{i}|\leq C c_{\alpha}$

where $c_{\alpha}$ only depends on $\alpha$

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But I could not find a reference or a proof for this theorem. Can anybody provide me a ref for this? Thanks a lot!

Answer 1

The proof is outlined in Stein-Shakarchi's book Fourier Analysis, Chapter 3, Exercise 16.