Local Approximations Based on Orthogonal Differential Operators

Abstract

Let M be a symmetric positive definite moment functional and let \(\{P_n^{\cal M}(\omega)\}_{n\in {\Bbb N}}\) be the family of orthonormal polynomials that corresponds to M. We introduce a family of linear differential operators \({\cal K}^n =(-i)^nP_n^{\cal M}(i\frac{d}{dt})\), called the chromatic derivatives associated with M, which are orthonormal with respect to a suitably defined scalar product. We consider a Taylor type expansion of an analytic function f(t), with the values f(n) (t0) of the derivatives replaced by the values \({\cal K}^n[f](t_0)\) of these orthonormal operators, and with monomials (t − t0)n/n! replaced by an orthonormal family of "special functions" of the form \((−1)^n{\cal K}^n[m](t-t_0)\), where \(m(t) = \sum_{n=0}^{\infty} (−1)^n{\cal M}(\omega^{2n}) t^{2n}/(2n)!\). Such expansions are called the chromatic expansions. Our main results relate the convergence of the chromatic expansions to the asymptotic behavior of the coefficients appearing in the three term recurrence satisfied by the corresponding family of orthogonal polynomials PMn(ω). Like the truncations of the Taylor expansion, the truncations of a chromatic expansion at t = t0 of an analytic function f(t) approximate f(t) locally, in a neighborhood of t0. However, unlike the values of f(n)(t0), the values of the chromatic derivatives Kn[f](t0) can be obtained in a noise robust way from sufficiently dense samples of f(t). The chromatic expansions have properties which make them useful in fields involving empirically sampled data, such as signal processing.