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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Chromatic expansions in function spaces
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by Ahmed I. Zayed PDF
Trans. Amer. Math. Soc. 366 (2014), 4097-4125 Request permission

Abstract:

Chromatic series expansions of bandlimited functions have recently been introduced in signal processing with promising results. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function, but unlike Taylor series, chromatic series have a better rate of convergence and more practical applications.

The $n$-th chromatic derivative $K^n(f)$ of an analytic function $f(t)$ is a linear combination of the ordinary derivatives $f^{(k)}(t), 0\leq k\leq n,$ where the coefficients of the combination are based on systems of orthogonal polynomials. In addition to their practical applications, chromatic series expansions have useful theoretical and mathematical applications. For example, functions in the Paley-Wiener space can be completely characterized by their chromatic series expansions associated with the Legendre polynomials.

The purpose of this paper is to show that chromatic series expansions can be used to characterize other important function spaces. We show that functions in weighted Bergman spaces $\mathfrak {B}_\gamma$ can be characterized by their chromatic series expansions that use chromatic derivatives associated with the Laguerre polynomials, while functions in the Bargmann-Segal-Foch space $\mathfrak {F}$ can be characterized by their chromatic series expansions that use chromatic derivatives associated with the Hermite polynomials. Another goal of this article is to show that each one of these spaces has an orthonormal basis that is generated from one single function $\psi$ by applying successive chromatic derivatives to it, that is, both $\mathfrak {B}_\gamma$ and $\mathfrak {F}$ have an orthonormal basis of the form $\left \{K^n\psi \right \}_{n=0}^\infty .$

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Additional Information
  • Ahmed I. Zayed
  • Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
  • Email: azayed@condor.depaul.edu
  • Received by editor(s): August 17, 2011
  • Received by editor(s) in revised form: July 20, 2012
  • Published electronically: March 24, 2014
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 4097-4125
  • MSC (2010): Primary 41A58, 42C15; Secondary 44A15, 42B35
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05991-6
  • MathSciNet review: 3206453