G. Mastroianni

Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey

In this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes, i.e., with Lebesgue constant having logarithmic behaviour. A new result on Lagrange interpolation based on the zeros of the associated Jacobi polynomials and on suitable additional nodes is given.

Lagrange Interpolation at Laguerre Zeros in Some Weighted Uniform Spaces

We introduce an interpolatory process essentially based on the Laguerre zeros and we prove that it is an optimal process in some weighted uniform spaces.

Truncated Quadrature Rules Over

We propose replacing the classical Gauss--Laguerre quadrature formula by a truncated version of it, obtained by ignoring the last part of its nodes. This has the effect of obtaining optimal orders of convergence. Corresponding quadrature rules with kernels are then considered and optimal error estimates are derived also for them. These rules are finally used to define stable Nystrom-type interpolants for a second kind of integral equation on the real semiaxis whose solutions decay exponentially at

(0,\infty)

\infty

and Nyström-Type Methods

.

Best Approximation and Moduli of Smoothness for Doubling Weights

In this paper we relate the rate of weighted polynomial approximation to some weighted moduli of smoothness for so-called doubling weights. We shall also consider the problem in a more restrictive sense for generalized Jacobi weights with zeros in the interval of approximation. These zeros constitute a special problem that has not been resolved so far in the literature.

Pointwise simultaneous approximation by rational operators

Abstract We obtain pointwise simultaneous approximation estimates for rational operators which are not possible by algebraic polynomials.

Convergence properties of a class of product formulas for weakly singular integral equations

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type f K(x,y)f(x)dx, -l? -1 , and f (x) has algebraic singularities of the form (1 ? x)a, a > -1 , we prove that the uniform rate of convergence of the rules is O(m log2 m) in the case of the first kernel, and O(m 2-a2v logim) if v 0, for the second, where m is the number of points in the quadrature rule.

Saturation of the Shepard operators

We discuss the Shepard operators Sn (f; x) in this paper and establish the saturation of the sequence {Snf}n-1∞, as well as investigate some related questions.

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

SummaryIn a previous paper the authors proposed a modified Gaussian ruleФ*m(wf;t)to compute the integral Π(wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence {Ф*m(wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rulesФm(wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.