Error estimates of Lagrange interpolation and orthonormal expansions for Freud weights

Abstract

Let S n [ f] be the nth partial sum of the orthonormal polynomials expansion with respect to a Freud weight. Then we obtain sufficient conditions for the boundedness of S n [ f] and discuss the speed of the convergence of S n [ f] in weighted L p space. We also find sufficient conditions for the boundedness of the Lagrange interpolation polynomial L n [ f], whose nodal points are the zeros of orthonormal polynomials with respect to a Freud weight. In particular, if W( x)=e −(1/2) x 2 is the Hermite weight function, then we obtain sufficient conditions for the inequalities to hold: ∥(S n[f]−f) (k)Wu b∥ L p( R) ⩽C 1 n r−k∥f (r)Wu B∥ L p( R) and ∥(L n[f]−f) (k)Wu b∥ L p( R) ⩽C 1 n r−k∥f (r)W(1+x 2) r/3u B∥ L p( R) , where u γ(x)=(1+|x|) γ, γ∈ R and k=0,1,2…, r.