On the convergence of regular families of cardinal interpolators
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On the convergence of regular families of cardinal interpolators Jeff Ledford
Received: 20 September 2013 / Accepted: 8 May 2014 © Springer Science+Business Media New York 2014
Abstract In this article, we develop a one parameter family of cardinal interpolants associated to a function φ. The main theorem gives conditions on φ and the parameter that allow Paley-Wiener functions to be recovered from their samples on the integer lattice Zn . Our methods unify previous work done on splines and the Gaussian and provide new examples of this phenomenon, including two families of multiquadrics. Keywords Cardinal interpolation · Multiquadric interpolation · Multivariate interpolation · Paley-Wiener functions Mathematics Subject Classification (2010) 41A05 · 41A30 · 41A63
1 Introduction In this article we prove a result similar to the ones found in [2, 4–7]. These results may be thought of as outgrowths of Schoenberg’s work on splines (see for instance [8]). The general set up is as follows. Suppose that we have data {f (j ) : j ∈ Zn }, which we think of as samples of a particular function; we wish to interpolate this data with a parametrized interpolant that depends upon a single function φ. In each case, what is investigated is what happens as the parameter approaches a limit. For instance in [4], polyharmonic cardinal splines are studied as the degree tends to infinity. In [6], parametrized Gaussian interpolants are used in place of the spline interpolants and Communicated by: T. Lyche J. Ledford () University of Connecticut, Storrs, CT 06269, USA e-mail: [email protected] J. Ledford Virginia Commonwealth University, Richmond, VA 23284 USA
J. Ledford
similar convergence results hold when the parameter tends to 0. The goal of this paper is to unify the procedures in those and similar papers in order to provide conditions on an interpolator which allow one to recover Paley-Wiener functions from their samples on the integer lattice by allowing the parameter to approach a limiting value. This article is organized as follows. In the next section, several preliminary definitions are laid out. We define what is meant by cardinal interpolator in the third section and introduce the corresponding fundamental function. The fourth section deals with properties of the fundamental function, while in the fifth section regular families are introduced and the main theorem is proved. The final section consists of three examples of regular families of cardinal interpolators. Of particular interest is the third example, that of the general multiquadric φ(x) = (x2 + c2 )α .
2 Definitions and basic facts In this section, we collect several definitions and general facts that will be used in what follows. Our methods require the use of the Fourier transform, for which we adopt the following convention. Definition 1 If f ∈ L1 (Rn ), we define its Fourier transform, denoted fˆ(ξ ), to be: fˆ(ξ ) = (2π)−n/2 f (x)e−ix·ξ dx, (1) where x · ξ =
Rn
n
j =1 xj ξj .
Definition 2 For a function f ∈ C ∞ (Rn ), and multi-indices α, β, let f
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