From: Ronald Bruck Subject: Re: elementary proof of convergence od fourier series of x^2n ??? Date: Thu, 11 Jan 2001 22:08:42 -0800 Newsgroups: sci.math Summary: Proving pointwise convergence of Fourier series In article , "stalking butler" wrote: :hi there sci.math. : :i wondered if any of you were aware of a simple proof of the convergence of :the fourier series of x^2n to f(x)=x^2n. : :i suppose proving that the series converge is rather trivial, but proving :that they converge to f(x)=x^2n is not... : :by the way, my math level is calculus, for now... : :any help greatly aprreciated, thanks... : :this proof would be the last step in my proof of sum_1_infinity_(1/m^2) = :pi^2/6 Good for you!--the research habit is learned early. Rudin's "Principles of Mathematical Analysis" contains a proof of the fact that if a function f is Lipschitz, then the Fourier series of f converges to f pointwise. It's more precise than this; you only need Lipschitz at a point x, i.e. there exists M such that |f(y)-f(x)| <= M*|x-y| for all y near x. It's Theorem 8.14 in the Third Edition. It's pretty direct; all the cleverness is in the manipulation of the Dirichlet kernel, and he lays it all out. Exercises 12 and 13 of that chapter are what you're trying to do (but using the Parseval identity, which is a little easier than trying to do a pointwise convergence of a Fourier series). Alternatively, you can go to the master, Zygmund. I don't have his two-volume set with me, so I can't say EXACTLY where the result is, but this is probably a little deep for you with only calculus. For example, start by looking at the Fourier coefficients of f(x) = x (and then modify the example, as will be obvious to you). The Parseval identity is \sum_n |c_n|^2 = 1/(2\pi) \int_{-\pi}^\pi |f(t)|^2 dt. --Ron Bruck -- Due to University fiscal constraints, .sigs may not be exceed one line.