A General framework for local interpolation

Abstract

Summary. We present a general framework for the construction of local interpolation methods with a given approximation order. Some applications to multivariate spline spaces are presented.

Applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets

“…In this section, we demonstrate how the blending formula introduced in [6] can be used to construct a local interpolatory spline approximant, with arbitrary order (≤ n) of exactness on polynomial reproduction, by matching an explicitly constructed optimally local spline approximant with the optimally local spline interpolant discussed in Section 1. In addition, it is shown that the resulting approximation operator is, for 3 ≤ n ≤ 5, the optimally local operator for which existence and uniqueness was established by Dahmen, Goodman, and Micchelli [11].…”

“…For n ≥ 4, however, the interpolation property (2.25) does not hold; but for this general case, we can employ the Chui-Diamond blending formula [6] as in the bottom line of the construction (2.26) below. The main result of this section is as follows.…”

“…It does not, however, give immediately the spline wavelets on bounded intervals introduced in [7], and a somewhat more general consideration by Lyche and Mørken [21]. The idea behind our constructive scheme for compactly supported interpolants with any desirable order of approximation is originated from [6]. However, since we are interested in smallest possible and specific supports as designated by the index ν, great care has to be taken in the construction of the quasi-interpolants that give a tight match with the optimally local interpolation operator L n,ν .…”

Fast integral wavelet transform on a dense set of the time-scale domain

“…1, we only need the computation of c M (or f M ∈ V M ) using the digital samples f (k/2 M − ), k ∈ Z Z, of f for some appropriate nonnegative integer which depends upon the scaling function φ. For this purpose, we follow the local interpolation scheme for optimal-order approximation introduced in [6], where optimality is achieved by reproducing all polynomials of the highest degree relative to the MRA. Suppose that the compactly supported scaling function φ has m th order of approximation in the sense that the Fourier transform of φ satisfies the Strang-Fix condition…”

“…For this purpose, we also propose a somewhat different scheme which can be considered as an extension of the Mallat algorithm. To compute the IWT with the inter-octave scales, we again rely on the FIR optimal-order spline interpolation algorithm introduced in [6]. The two-scale relations of the transformed B-splines in the inter-octave scales, as a result of this real-time operation, remain unchanged, so that the computational complexity of this scheme does not increase with the increasing number of desirable scale parameters.…”