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Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science pp 533–562Cite as

Birkhäuser

Convergence and Regularization of Sampling Series

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Abstract

This chapter consists of an overview of recent results concerning the convergence and regularization of the quintessential sampling series, the cardinal sine series. Conditions, that go beyond those associated with the standard theory, are formulated that ensure reconstruction by this series. The conditions are one of two types: (i) on the coefficients or samples or (ii) on the functions or signals being reconstructed. A class of regularization methods, that in effect consist of mollifications of cardinal sine series, is discussed. One of the highlights is a result that shows how piecewise polynomial splines and certain variants can be regarded as such mollifications.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    W. R. Madych

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  1. W. R. Madych
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Correspondence to W. R. Madych .

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Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Madych, W.R. (2017). Convergence and Regularization of Sampling Series. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55556-0_5

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