Laura Gori

On weight functions which admit explicit Gauss-Tura´n quadrature formulas

The main purpose of this paper is the construction of explicit Gauss-Turan 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.

Quadrature rules for rational functions

Summary. 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.

Positive Refinable Operators

Recently, linear positive operators of Bernstein–Schoenberg type, relative to B-splines bases, have been considered. The properties of these operators are derived mainly from the total positivity of normalized B-spline bases.In this paper we shall construct a generalization of the operator considered in [15] by means of normalized totally positive bases generated by a particular class of totally positive scaling functions. Next, we shall study its approximation properties. Our results can be established also for more general sequences of normalized totally positive bases.

On the Galerkin Method Based on a Particular Class of Scaling Functions

The aim of this paper is to provide a large class of scaling functions for which the convergence analysis for the Galerkin method developed in [9] is applicable, whereas in that paper the only scaling functions considered for practical applications are B-splines and a few of the orthonormal Daubechies scaling functions. The functions considered here, were recently introduced in [12] where it was proved that they satisfy many properties making them interesting for the applications. In particular, here we show that the use of these functions has some advantages with respect to other basis functions.

Original article: Numerical evaluation of certain hypersingular integrals using refinable operators

This paper concerns the construction of quadrature rules based on the use of suitable refinable quasi-interpolatory operators, for the numerical evaluation of Hadamard finite-part integrals. Convergence analysis of the obtained rules is developed and numerical examples are included.

On the evaluation of Hilbert transforms by means of a particular class of Turán quadrature rules

A method for evaluating Hilbert transforms, by means of Turán quadrature rules with generalized Gegenbauer weights, is presented. The main feature of these integration formulas is the independence of the nodes of their multiplicity and thus of the precision degree. The error is analyzed both from a real and a complex perspective; in this context a new representation of the remainder term of the quadrature rules with multiple nodes for the evaluation of Hilbert transforms, valid not only for the particular class of weight functions here considered, is presented. A few numerical examples are provided.

On a class of shape-preserving refinable functions with dilation 3

We analyze the properties of a class of shape-preserving refinable functions with dilation M=3. We give an algorithm to construct totally positive bases with optimal shape-preserving properties on a finite interval. Bernstein-type bases on [0,1] are also treated. Moreover, semiorthogonal wavelets associated with these refinable functions are constructed. Finally, a detailed example is described.

Quadrature Rules Based on s-Orthogonal Polynomials for Evaluating Integrals with Strong Singularities

Integrals defined by Cauchy principal values or by Hadamard finite parts are involved in many boundary integral equations; in this paper we present a method based on the use of Gauss-Turan quadrature formulas, that is, formulas with multiple nodes, for approximating these kinds of singular or hypersingular integrals. We show that the rules here proposed are particularly effective when the s-orthogonal polynomials involved in the construction of the method are invariant with respect to s. Finally, we introduce another type of integration rules, also with multiple nodes, for numerically evaluating hypersingular integrals.

Refinable interpolatory and quasi-interpolatory operators

In this paper, we present a new class of operators, which are refinable, quasi-interpolatory and satisfy some interpolation conditions. The refinability is achieved by using as functional bases the B-bases corresponding to totally positive refinable functions. We analyze the main properties of the constructed refinable operators and give some convergence results. Some examples are also displayed.

A review of some numerical methods for the solution of Cauchy singular integral equations

SuntoViene presentata una breve rassegna di alcuni metodi per la soluzione numerica di equazioni integrali singolari di tipo Cauchy; alcuni di essi sono su basi polinomiali, altri su basi di splines. In particolare, è esposto, in grandi linee, un recente metodo che fa uso di splines di tipo proiettore. Alcuni esempi numerici, tratti dalla letteratura, completano la panoramica.