It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turán, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. Mathematics Subject Classification (1991): 65D32 IntroductionThe idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received some attention in recent years; see, e.g., [9], [10], [11], [2], [4]. These "rational" quadrature rules have proven to be quite effective if the poles are chosen so as to simulate the poles present in the integrand; see [3] for an application to integrals occurring in solid state physics. The work so far has been exclusively centered around quadrature rules of Gaussian or Newton-Cotes type. Here we construct rational versions of other important quadrature rules, specifically the Gauss-Kronrod and the Gauss-Turán rule, and Cauchy principal value quadrature rules. It is found that the accuracy is enhanced similarly as has been observed for Gauss-type quadrature rules.

A two-parameter class of refinable functions is considered and Gaussian quadrature rules having these functions as weight functions. A discretization method is described for generating the recursion coefficients of the required orthogonal polynomials. Numerical results are also presented. © 2000 Academic Press A CLASS OF REFINABLE FUNCTIONSA refinable function φ is a solution of a two-scale difference equationwhere a j are real numbers satisfyingis called a refinement equation, and a = {a j } j ∈Z the mask. Refinable functions occur in many different branches of analysis, notably in multiresolution approximation and wavelet analysis. The existence of a solution of (1.1) is ensured under suitable assumptions on the mask a (see, for example, [2,8] and references therein). In particular, the existence of a unique refinable function φ associated with a given mask a was proved in [4] in the case when a satisfies, in addition to (1.2), the following conditions, depending on an integer parameter m ≥ 2:(i) a is compactly supported on j = 0, 1, . . ., m + 1 (i.e., a j = 0 for j / ∈ [0, m + 1], with a 0 a m+1 = 0);

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Abstract. The main purpose of this paper is the construction of explicit Gauss-Turán quadrature formulas: they are relative to some classes of weight functions, which have the peculiarity that the corresponding s-orthogonal polynomials, of the same degree, are independent of s. These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev coefficients are given.