ON THE COMPUTATION OF GRAMIAN MATRICES FOR REFINABLE BASES ON THE INTERVAL

In this paper, we discuss a simple method for the numerical computation of Gramian matrices, that appears within the construction of multiresolution analysis on the interval. The presented approach covers all, the orthogonal, the semiorthogonal and the biorthogonal cases. We consider constructions, which are based on translates of a known pair of compactly supported functions ϕ,[Formula: see text] inside the interval, and supplemented by boundary functions. Using the given two-scale-coefficients of these boundary functions, we can reduce the problem to a linear system of equations. Furthermore, we discuss conditions providing that these systems are uniquely solvable. In particular, no integrals have to be computed numerically. Finally, using the Schoenberg spline basis on the interval as an example, we show how to apply the method to a well-known problem.

Convolution operators with trigonometric spline kernels

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].

A property of Bernstein-Schoenberg spline operators

Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].