Interlacing properties of Laguerre zeros and some applications. A survey

Let w(x) be a Generalized Laguerre weight and denote by fpm(w)gm the corresponding sequence of orthonormal polynomials. The starting point is that the polynomial Q2m+1 = pm+1(w)pm(w+1) has simple zeros and also well distributed in some sense. In view of this property two dierent applications are described: the extended interpolation polynomial L2m+2(w;w+1; f), dened as the Lagrange polynomial interpolating a given function f at the zeros of Q2m+1 and on the extra points am, being am the Maskar-Rackmano-Sa number w.r.t. w. For this process will be estimati the Lebesgue constants in some weighted uniform spaces [31]. The second application deals with the approximation of the Hilbert transform by a suitable Lagrange interpolating polynomial [32].