Extended lagrange interpolation in L1 spaces

Let w(x) = e-xβ xα, w (x) = xw(x) and denote by fpm(w)gm; fpn(w )gn the corresponding sequences of orthonormal polynomials. The zeros of the polynomial Q2m+1 = pm+1(w)pm(w ) are simple and are sufficiently far among them. Therefore it is possible to construct an interpolation process essentially based on the zeros of Q2m+1, which is called "Extended Lagrange Interpolation". Here we study the convergence of this interpolation process in suitable weighted L1 spaces. This study completes the results given by the authors in previous papers in weighted Lpu ((0; +∞)), for 1 ≤ p ≤ ∞ Moreover an application of the proposed interpolation process in order to construct an efficient product quadrature scheme for weakly singular integrals is given.Grant: The Authors are grateful to the referees for the accurate reading and for the interesting comments which let them improve the quality of the paper.