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Multidimensional chromatic derivatives and series expansions

Authors: Aleksandar Ignjatovic and Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 139 (2011), 3513-3525
MSC (2010): Primary 41A58, 42C15; Secondary 94A12, 94A20
Published electronically: February 17, 2011
MathSciNet review: 2813383
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Abstract: Chromatic derivatives and series expansions of bandlimited functions have recently been introduced as an alternative representation to the Taylor series, and they have been shown to be more useful in practical signal processing applications than in the Taylor series. Although chromatic series were originally introduced for bandlimited functions, they have now been extended to a larger class of functions. The $ n$-th chromatic derivative of an analytic function is a linear combination of the $ k$-th ordinary derivatives with $ 0\leq k\leq n,$ where the coefficients of the linear combination are based on a suitable system of orthogonal polynomials. The goal of this article is to extend chromatic derivatives and series to higher dimensions. This is of interest not only because the associated multivariate orthogonal polynomials have much reacher structure than in the univariate case, but also because we believe that the multidimensional case will find natural applications to fields such as image processing and analysis.

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Additional Information

Aleksandar Ignjatovic
Affiliation: School of Computer Science and Engineering, University of New South Wales, Sydney, Australia

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614

Keywords: Chromatic derivatives, chromatic expansions, signal representation
Received by editor(s): January 24, 2010
Received by editor(s) in revised form: August 24, 2010
Published electronically: February 17, 2011
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.