Local Approximations Based on Orthogonal Differential Operators (original) (raw)
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Chromatic derivatives and series expansions of bandlimited functions have recently been introduced as an alternative representation to Taylor series and they have been shown to be more useful in practical signal processing applications than 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 kth ordinary derivatives with 0 ≤ k ≤ 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 multidimensional case will find natural applications to fields such as image processing and analy...
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Abstract We present a detailed motivation for the notions of chromatic derivatives and chromatic expansions. Chromatic derivatives are special, numerically robust linear differential operators; chromatic expansions are the associated local expansions, which possess the best features of both the Taylor and the Nyquist expansions. We give a simplified treatment of some of the basic properties of chromatic derivatives and chromatic expansions which are relevant for applications.
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Chromatic expansions in function spaces
Transactions of the American Mathematical Society, 2014
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 ≤ k ≤ 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 B γ 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 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 ψ by applying successive chromatic derivatives to it, that is, both B γ and F have an orthonormal basis of the form {K n ψ} ∞ n=0 .
Chromatic Derivatives, chromatic expansions and associated function spaces
East Journal on Approximations, 2009
The Nyquist(WhittakerKotelnikovShannon) expan-sion f(t) = ∑∞ n=−∞ f(n) sin π(t − n)/π(t − n) of a π-band limited signal of finite energy f(t) ∈ BL(π) is of global nature, because it requires samples of the sig-nal at integers of arbitrarily large absolute value. On the ...
CHROMATIC SERIES EXPANSIONS IN SEVERAL VARIABLES
ABSTRACT Chromatic derivatives and series expansions of bandlimited functions have recently been introduced as an alternative representation to Taylor series and they have been shown to be more useful in practical signal processing applications than Taylor series. The n-th chromatic derivative of an analytic function is a linear combination of its kth ordinary derivatives with 0≤ k≤ n, where the coefficients of the linear combination are based on a suitable system of orthogonal polynomials.
Chromatic Derivatives and Approximations in Practice—Part I: A General Framework
IEEE Transactions on Signal Processing, 2018
Chromatic derivatives are special, numerically robust differential operators that preserve spectral features of a signal; the associated chromatic approximations accurately capture local features of a signal. For this reason they allow digital processing of continuous time signals often superior to processing of discrete samples of such signals. We introduce a new concept of "matched filter" chromatic approximations, where the underlying basis functions are chosen to match the spectral profile of the signals being approximated. We then derive a collection of formulas and theorems that form a general framework for practical applications of chromatic derivatives and approximations. In the second part of this paper, we use such a general framework in several case studies of such applications that aim to illustrate how chromatic derivatives and approximations can be used in signal processing with an intention of motivating DSP engineers to find applications of these novel concepts in their own practice. Index Terms-Chromatic derivatives, chromatic expansions, digital representation and processing of continuous time signals.
The aim of this paper is to investigate the quality of approximation of almost time and almost band-limited functions by its expansion in three classical orthogonal polynomials bases: the Hermite, Legendre and Chebyshev bases. As a corollary, this allows us to obtain the quality of approximation in the L 2 −Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions. Some numerical examples are given to illustrate the different results of this work.