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THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS

  • JOSEF DICK (a1)
Abstract
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 nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] N. Aronszajn , ‘Theory of reproducing kernels’, Trans. Amer. Math. Soc. 68 (1950), 337404.

[2] H. E. Chrestenson , ‘A class of generalised Walsh functions’, Pacific J. Math. 5 (1955), 1731.

[3] J. Dick , ‘Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions’, SIAM J. Numer. Anal. 45 (2007), 21412176.

[4] J. Dick , ‘Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order’, SIAM J. Numer. Anal. 46 (2008), 15191553.

[5] J. Dick and F. Pillichshammer , ‘Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces’, J. Complexity 21 (2005), 149195.

[6] N. J. Fine , ‘On the Walsh functions’, Trans. Amer. Math. Soc. 65 (1949), 372414.

[7] K. Niederdrenk , Die endliche Fourier- und Walshtransformation mit einer Einführung in die Bildverarbeitung (Vieweg, Braunschweig, 1982).

[9] W. R. Wade , ‘Recent developments in the theory of Walsh series’, Internat. J. Math. Math. Sci. 5 (1982), 625673.

[10] G. Wahba , Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 59 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990).

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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