Chromatic expansions in function spaces

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 $\left \{K^n\psi \right \}_{n=0}^\infty .$