Orthogonal polynomials for refinable linear functionals

Abstract

Abstract. 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(n 2 ) rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.

An efficient and reliable quadrature algorithm for integration with respect to binomial measures

“…The construction of these recurrence coefficients for the µ α -orthogonal polynomials can be done applying the analysis in [21,19]. In Tables 3.2 and 3.3 we report the first of them as function of α.…”

“…The relation (1.2) gives also that the functional L[f ] ≡ fdµ α is a refinable linear functional, as defined in [19], with the Bernoulli shifts as stretch-shift operators and mask [2(1 − α), 2α].…”

“…In particular, in [21] a stable algorithm is described for the recurrence coefficients of orthogonal polynomials with respect to a general balanced measure, while in [19] is proposed the same procedure for orthogonal polynomials with respect to a refinable linear functional. The quadrature with respect to refinable linear functions has been considered because of its connection with the wavelet transformation, and also the composite case is discussed in some papers, see [3,16,18] and references therein.…”

Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

“…A different technique was needed and this was found [5,12] see also [11] avoiding the perilous passage through the moments' mountain range. To further exploit the analogy with the magnificent ambience of this meeting, one could say that the circumventing path was traced in a succession of demanding, yet well conditioned mountain passes, like those encircling the Sella Group in the Dolomites.…”

“…The algorithm that we describe now is a recursive evaluation of the triangular matrices n and of the Jacobi matrix J σ . For simplicity of notation, in the following we shall let the indices k and r run freely, while assuming that n k,r = 0 unless 0 ≤ k+ r ≤ n. Of fundamental importance are the following formulae: φ β (s) p n (μ; φ β (s)) = k,r n k,r δ P k (μ; s) p r (σ ; β) +δ P r (σ ; β) p k (μ; s) , (11) where we have put…”