Extension of the Dirichlet-Jordan Convergence Criterion for Exponential Weights

Abstract

The well-known Dirichlet-Jordan Criterion for Fourier series states that the trigonometric Fourier series of a 2π periodic function f having bounded variation converges to ½[f(x + 0) + f(x – 0)] for every x and this convergence is uniform on every closed interval on which f is continuous (see Theorem 2.8.1 in [3]). We extend this criterion to orthonormal polynomial expansions, and treat the even case of a more general class of exponential weights, on the real line.