Orthogonal polynomials for refinable linear functionals

A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires O(n2) rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic. 1. Refinable linear functionals Let N be a positive integer and P be the linear space of all polynomials with real coeffients. Denote by Ej : P → P the stretch-shift operator defined by (1.1) Ejf(x) = f ( x+ j 2 ) , j = 0, 1, . . . , N. We say that the linear functional L : P → R is refinable if there exists an N -tuple γ = (γ0, γ1, . . . , γN ) of real numbers, called a mask, such that (1.2) L[f ] = 1 2 N ∑ j=0 γjL[Ejf ]. We consider only the case where L[e0] is nonzero, where e0 denotes the constant function e0(x) = 1, in which case one may assume without loss of generality that L[e0] = 1. Clearly (put f = e0) the mask coefficients satisfy