Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function - a reference request

Question

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to $\frac{1}{2}(f(x^+)+f(x^-))$. I have seen a proof in Zygmund's book but I don't understand it as it apparently requires some number theory and also not self contained. Please suggest me a reference (book or even a web location) which gives a self contained proof without any reference to number theory. I guess the original proof by Dirichlet uses number theory.Thank you. Kindly note that I expect more than "any introductory book on fourier analysis" type of an answer.

Answer 1

Try Chapter 11 of Apostol's Mathematical Analysis, 2nd Ed.

Theorem 11.12 is the particular result you are after.

Answer 2

Let me suggest the following (among many others):

Theorem 1 in chapter 2, section 4 of

Harmonic Analysis: A Gentle Introduction Carl L. DeVito, JONES AND BARTLETT PUBLISHERS, 2007

Theorem 2.1 in

Princeton Lectures in Analysis

I Fourier Analysys An Introduction

E. Stein, R. Shakarchi,

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