The decay of the Walsh coefficients of smooth functions

TL;DR

This paper establishes upper bounds on Walsh coefficients for functions with fractional bounded variation derivatives, explores their behavior in various Hilbert spaces, and confirms the bounds are optimal through lower bound analysis.

Contribution

It provides the first comprehensive bounds on Walsh coefficients for functions with fractional variation derivatives and analyzes their optimality.

Findings

Upper bounds on Walsh coefficients for fractional variation functions

Analysis of Walsh coefficients in different Hilbert spaces

Lower bounds demonstrating the bounds are tight

Abstract

We give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and non-periodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.