Chromatic Derivatives, Chromatic Expansions and Associated Spaces

This paper presents the basic properties of chromatic derivatives and chromatic expansions and provides an appropriate motivation for introducing these notions. Chromatic derivatives are special, numerically robust linear differential operators which correspond to certain families of orthogonal polynomials. Chromatic expansions are series of the corresponding special functions, which possess the best features of both the Taylor and the Shannon expansions. This makes chromatic derivatives and chromatic expansions applicable in fields involving empirically sampled data, such as digital signal and image processing. 1. Extended Abstract Let BL(π) be the space of continuous L functions with the Fourier transform supported within [−π, π] (i.e., the space of π band limited signals of finite energy), and let P n (ω) be obtained by normalizing and scaling the Legendre polynomials, so that 1 2π ∫ π −π P n (ω) P L m(ω)dω = δ(m− n). We consider linear differential operators Kn = (−i)P n ( i d dt ) ; for such operators and every f ∈ BL(π), K[f ](t) = 1 2π ∫ π −π i P n (ω)f̂(ω)e idω. We show that for f ∈ BL(π) the values of Kn[f ](t) can be obtained in a numerically accurate and noise robust way from samples of f(t), even for differential operators Kn of high order. Operators Kn have the following remarkable properties, relevant for applications in digital signal processing. Proposition 1.1. Let f : R → R be a restriction of any entire function; then the following are equivalent: (a) ∑∞ n=0 Kn[f ](0) < ∞; (b) for all t ∈ R the sum ∑∞ n=0 Kn[f ](t) converges, and its values are independent of t ∈ R; (c) f ∈ BL(π). 2000 Mathematics Subject Classification. 41A58, 42C15, 94A12, 94A20.