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THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  02 July 2009

JOSEF DICK
JOSEF DICK*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia (email: josef.dick@unsw.edu.au)
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Abstract

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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 nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

Keywords

waveletWalsh seriesWalsh coefficientsobolev spacesmooth function

MSC classification

Secondary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 430 - 453
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Aronszajn, N., ‘Theory of reproducing kernels’, Trans. Amer. Math. Soc. 68 (1950), 337–404.CrossRefGoogle Scholar
[2]Chrestenson, H. E., ‘A class of generalised Walsh functions’, Pacific J. Math. 5 (1955), 17–31.CrossRefGoogle Scholar
[3]Dick, J., ‘Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions’, SIAM J. Numer. Anal. 45 (2007), 2141–2176.CrossRefGoogle Scholar
[4]Dick, J., ‘Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order’, SIAM J. Numer. Anal. 46 (2008), 1519–1553.CrossRefGoogle Scholar
[5]Dick, J. and Pillichshammer, F., ‘Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces’, J. Complexity 21 (2005), 149–195.CrossRefGoogle Scholar
[6]Fine, N. J., ‘On the Walsh functions’, Trans. Amer. Math. Soc. 65 (1949), 372–414.CrossRefGoogle Scholar
[7]Niederdrenk, K., Die endliche Fourier- und Walshtransformation mit einer Einführung in die Bildverarbeitung (Vieweg, Braunschweig, 1982).CrossRefGoogle Scholar
[8]Pirsic, G., ‘Embedding theorems and numerical integration of Walsh series over groups’, PhD Thesis, University of Salzburg, 1997.Google Scholar
[9]Wade, W. R., ‘Recent developments in the theory of Walsh series’, Internat. J. Math. Math. Sci. 5 (1982), 625–673.CrossRefGoogle Scholar
[10]Wahba, G., Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 59 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990).CrossRefGoogle Scholar
[11]Walsh, J. L., ‘A closed set of normal orthogonal functions’, Amer. J. Math. 55 (1923), 5–24.CrossRefGoogle Scholar