Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients

Abstract

We establish formulas for the $b$-adic Walsh coefficients of functions in $C^{\alpha }[0,1]$ for an integer $\alpha \ge 1$ and give upper bounds on the Walsh coefficients of these functions. We also study the Walsh coefficients of periodic and non-periodic functions in reproducing kernel Hilbert spaces.

Walsh series, Walsh coefficient, Sobolev space, smooth function

References in corpus (2)

• The decay of the Walsh coefficients of smooth functions
• Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration

Cited by in corpus (6)

• Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions
• A universal median quasi-Monte Carlo integration
• An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands
• Quasi-Monte Carlo integration using digital nets with antithetics
• Recent advances in higher order quasi-Monte Carlo methods
• Construction of interlaced polynomial lattice rules for infinitely differentiable functions