Incoronata Notarangelo

A Lagrange-type projector on the real line

We introduce an interpolation process based on some of the zeros of the mth generalized Freud polynomial. Convergence results and error estimates are given. In particular we show that, in some important function spaces, the interpolating polynomial behaves like the best approximation. Moreover the stability and the convergence of some quadrature rules are proved.

Gaussian quadrature rules with exponential weights on (−1, 1)

We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergence of a Nyström method for Fredholm integral equations of the second kind. Finally, some numerical examples are shown.

Embedding theorems with an exponential weight on the real semiaxis

Abstract We state embedding theorems between spaces of functions defined on the real semi-axis, which can grow exponentially both at 0 and at +∞.

On an interpolation process of Lagrange-Hermite type

We consider a Lagrange-Hermite polynomial, interpolating a func- tion at the Jacobi zeros and, with its first (r 1) derivatives, at the points ±1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted L p -spaces, 1 < p < 1, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ±1. Moreover, we give optimal estimates for the error of this process also in the weighted uniform metric.

Approximation of the Hilbert Transform on the Real Line Using Freud Weights

Some quadrature rules based on the zeros of Freud polynomials are suggested to compute Cauchy principal value integrals on the real line. In this paper, the main effort is to prove the stability and the convergence of the proposed rules. Error estimates in a weighted uniform norm are stated and some numerical tests are described.

On Nyström and Product Integration Methods for Fredholm Integral Equations

The aim of this paper is to combine classical ideas for the theoretical investigation of the Nystrom method for second kind Fredholm integral equations with recent results on polynomial approximation in weighted spaces of continuous functions on bounded and unbounded intervals, where also zeros of polynomials w.r.t. exponential weights are used.