Principles of Fourier Analysis, 2nd Edition

14

Convergence and Fourier’s Conjecture: The Proofs

As promised, here we will go into the details of verifying the basic theorem on pointwise convergence. In addition, as also promised, we will carefully examine the behavior of the partial sums of the Fourier series of certain saw functions both on intervals away from the discontinuities (to verify “nearly uniform convergence”) and on intervals containing points of discontinuity (to study Gibbs phenomenon). And, finally, we will construct a periodic function that, though continuous, has a divergent Fourier series at one point.

14.1  Basic Theorem on Pointwise Convergence

Our first big goal is to prove the following theorem (which is the same as theorem 13.1 on page 156).

Theorem 14.1 (basic ...