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

Abstract

Abstract. The objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary (nonuniform) knots. Both construction procedures are based on certain "optimally local" interpolatory fundamental spline functions which are not required to possess any approximation property.

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

“…In this section we briefly introduce the family of order-n non-uniform local interpolatory fundamental splines proposed by Chui and De Villiers [9]. This family has the capability of efficiently generating optimal quality interpolants, due to its features of arbitrary knot-spacing, compact support and polynomial precision.…”

“…However, this proposal was limited to the case of uniform knots. Six years later the explicit formulation of the coefficients of locally-supported interpolatory fundamental splines with arbitrary knots was provided [9]. We will now start by briefly summarizing these results, as they will be taken as a starting point for the theory developed in the remainder of the paper.…”

“…Given an ordered sequence of real values X := {x i } i∈Z with ... < x i < x i+1 < ... , define the knot-sequence T := {t i } i∈Z with ... < t i < t i+1 < ... such that t 2i := x i . In [9] the problem of interpolation from a space of spline functions with sample points at X is addressed by the definition of a sequence of order n ≥ 3 functions v ρ,µ,i , i ∈ Z, which, for positive integers ρ and µ satisfying 1…”

“…Since in [9] the authors considered only the case of simple knots, the functions in S n,T are in C n−2 (R). The problem of suitably defining the knotpartition as well as the influence of the free parameters µ and ρ in determining the shape of the basis will be addressed in detail in Section 2.2.…”

“…In Section 2 we introduce the family of order-n nonuniform local interpolatory fundamental splines proposed in [9] by Chui and De Villiers (Section 2.1) and we recall its distinctive features (Section 2.2); successively we discuss the issue of designing refinement equations upon the selected family members (Section 2.3). In Section 3 we present in detail the NULI 4-point scheme originated from the order-3 NULIFS (Sections 3.1 and 3.2) and analyze its main properties concerning polynomial precision, smoothness order and support width (Section 3.3).…”