Determination of a jump by Fourier and Fourier-Chebyshev series

TL;DR

This paper generalizes methods for detecting jump discontinuities in functions using Fourier and Fourier-Chebyshev series, extending previous results to broader classes of functions with harmonic bounded variation.

Contribution

It extends existing techniques for jump detection to functions of harmonic bounded variation without restrictions on discontinuity count, and introduces new results using Fourier-Chebyshev series.

Findings

Generalized jump detection to harmonic bounded variation functions.

Derived new results using Fourier-Chebyshev series tails.

Extended previous work without finiteness assumptions.

Abstract

By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. Two results on determination of jump discontinuities by means of the tails of integrated Fourier-Chebyshev series are derived.