The decay of the Walsh coefficients of smooth functions

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.

References in corpus (2)

• Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order
• Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions

Cited by in corpus (13)

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• Optimal order quadrature error bounds for infinite-dimensional higher order digital sequences
• An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands
• Richardson extrapolation of polynomial lattice rules
• Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients
• Quasi-Monte Carlo integration using digital nets with antithetics
• The Mean Square Quasi-Monte Carlo Error for Digitally Shifted Digital Nets
• Construction of interlaced polynomial lattice rules for infinitely differentiable functions
• Richardson extrapolation allows truncation of higher order digital nets and sequences