absolute convergence of the Fourier coeff for Hölder continuous function - MathOverflow most recent 30 from http://mathoverflow.net2013-05-21T14:00:11Zhttp://mathoverflow.net/feeds/question/48499http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/48499/absolute-convergence-of-the-fourier-coeff-for-holder-continuous-functionabsolute convergence of the Fourier coeff for Hölder continuous functionlapordge2010-12-06T22:10:19Z2012-05-29T14:59:20Z
<p>I saw the following theorem in the wiki page:
<a href="http://en.wikipedia.org/wiki/H%C3%B6lder_condition" rel="nofollow">http://en.wikipedia.org/wiki/H%C3%B6lder_condition</a></p>
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<p>if $f$ satisfies the $\alpha$-Hölder condition
$| f(x) - f(y) | \leq C \, |x - y|^{\alpha}$ for some $\alpha>1/2$,
then </p>
<p>$||f||_{A} = \sum_i |c_{i}|\leq C c_{\alpha}$</p>
<p>where $c_{\alpha}$ only depends on $\alpha$</p>
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<p>But I could not find a reference or a proof for this theorem.
Can anybody provide me a ref for this? Thanks a lot!</p>
http://mathoverflow.net/questions/48499/absolute-convergence-of-the-fourier-coeff-for-holder-continuous-function/48507#48507Answer by TCL for absolute convergence of the Fourier coeff for Hölder continuous functionTCL2010-12-06T22:37:33Z2010-12-06T22:37:33Z<p>The proof is outlined in Stein-Shakarchi's book Fourier Analysis, Chapter 3, Exercise 16. </p>