Complex Analysis of Real Functions VI: On the Convergence of Fourier Series

TL;DR

This paper introduces a compact Hilbert transform variant to derive explicit formulas for Fourier series partial sums and remainders, establishing a new integral-based convergence criterion.

Contribution

It provides a novel integral expression for Fourier series remainders using a compact Hilbert transform, offering a new convergence condition.

Findings

Explicit formulas for Fourier partial sums and remainders

Integral-based necessary and sufficient convergence condition

Extension of classical Fourier analysis results

Abstract

We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms of Dirichlet integrals. The expression for the remainder is written in terms of a similar type of integral. Since the asymptotic limit of the remainder being zero is a necessary and sufficient condition for the convergence of the series, this same condition on the asymptotic behavior of the corresponding integrals constitutes such a necessary and sufficient condition.