absolute convergence of the Fourier coeff for Hölder continuous function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:00:11Z http://mathoverflow.net/feeds/question/48499 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48499/absolute-convergence-of-the-fourier-coeff-for-holder-continuous-function absolute convergence of the Fourier coeff for Hölder continuous function lapordge 2010-12-06T22:10:19Z 2012-05-29T14:59:20Z <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> <hr> <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> <hr> <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#48507 Answer by TCL for absolute convergence of the Fourier coeff for Hölder continuous function TCL 2010-12-06T22:37:33Z 2010-12-06T22:37:33Z <p>The proof is outlined in Stein-Shakarchi's book Fourier Analysis, Chapter 3, Exercise 16. </p>