Error estimates of Lagrange interpolation and orthonormal expansions for Freud weights

摘要

Let Sn[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 Sn[f] and discuss the speed of the convergence of Sn[f] in weighted Lp space. We also find sufficient conditions for the boundedness of the Lagrange interpolation polynomial Ln[f], whose nodal points are the zeros of orthonormal polynomials with respect to a Freud weight. In particular, if W(x)=e−(1/2)x2 is the Hermite weight function, then we obtain sufficient conditions for the inequalities to hold:∥(Sn[f]−f)(k)Wub∥Lp(R)⩽C1nr−k∥f(r)WuB∥Lp(R)and∥(Ln[f]−f)(k)Wub∥Lp(R)⩽C1nr−k∥f(r)W(1+x2)r/3uB∥Lp(R),where uγ(x)=(1+|x|)γ,γ∈R and k=0,1,2…,r.