Orthogonal polynomials and Gaussian quadrature for refinable weight functions (Q704214)

A function \(\phi:\mathbb{R}\rightarrow \mathbb{R}\) is a weight function if the moments \[ \mu_{k}=\int_{-\infty}^{\infty} x^{k} \phi(x) dx\tag{1} \] exist for \(k=0,1,2, \ldots\) and \(\mu_{0}\neq 0\). It is a refinable function if it satisfies a refinement equation, which in the simplest case (where typically \(\phi\) has support \([0,n]\)) takes the form \[ \phi(x)=\sum_{j=0}^{n}\gamma_{j}\phi(2x-j),\;\;x\in \mathbb{R}, \] where the \((n+1)\)-tuple of real coefficients \(\gamma=(\gamma_{0},\gamma_{1},\ldots,\gamma_{n})\) is known as the mask of the function. In this paper the authors show how to find the moments (1) of \(\phi\) in \(O(N^2n)\) rational operations without making any approximations whatsoever -- no numerical integration, no evaluations of \(\phi\). The recursion coefficients can therefore in principle be evaluated exactly in a symbolic language, and when the mask coefficients are rational, the recursion coefficients are also rational. The authors also show how to find the modified moments of \(\phi\) with respect to a system of monic orthogonal polynomials. In particular, when the system of polynomials is the system of Legendre polynomials shifted to \([0,n]\), these moments are related in a very simple way to the coefficients in the expansion of \(\phi\) as a Legendre series. Several applications and numerical examples are given.