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

TL;DR

The paper develops two complementary representations for the b-adic Walsh coefficients of smooth functions, introducing the auxiliary function W(k) and its relatives to express coefficients in terms of derivatives. It derives explicit univariate and multivariate bounds for |f̂(k)| in terms of derivative norms, base b, and frequency weights μ, refining prior results and providing sharper constants, including in dyadic and Sobolev settings. The work further analyzes the Walsh coefficients of Bernoulli polynomials and extends the bounds to Sobolev spaces and to periodic/nonperiodic reproducing-kernel Hilbert spaces, yielding both exact formulas and practical decay estimates. These results have direct implications for quasi-Monte Carlo methods and the analysis of smooth function spaces using Walsh expansions.

Abstract

We establish formulas for the $b$-adic Walsh coefficients of functions in $C^α[0,1]$ for an integer $α\geq 1$ 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.

Theorems & Definitions (48)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Lemma 3.2