在Freud正交多项式零点处的Grünwald插值算子的收敛性

Let Wβ(x)=exp(-1/2|x|β)be the Freud weight and pn(x) ∈пn be the sequence of orthogonal polynomials with respect to W2β(x),that is,∫∞-∞pn(x)pm(x)W2β(x)dx={0,1, n≠m, n=m.It is known that all the zeros of pn(x)are distributed on the whole real line.The present paper investigates the convergence of Gr(u)nwald interpolatory operators based on the zeros of orthogonal polynomials for the Freud weights.We prove that,if we take the zeros of Freud polynomials as the interpolation nodes,then Gn(f,x)→,f(x),n→∞ holds for every x ∈(-∞,∞),where f(x) is any continous function on the real line satisfying |f(x)|=O(exp(1/2|x|β)).

The Convergence of Grünwald Interpolation Operator on the Zeros of Freud Orthogonal Polynomials

Let Wβ(x)=exp(-1/2|x|β)be the Freud weight and pn(x) ∈пn be the sequence of orthogonal polynomials with respect to W2β(x),that is,∫∞-∞pn(x)pm(x)W2β(x)dx={0,1, n≠m, n=m.It is known that all the zeros of pn(x)are distributed on the whole real line.The present paper investigates the convergence of Gr(u)nwald interpolatory operators based on the zeros of orthogonal polynomials for the Freud weights.We prove that,if we take the zeros of Freud polynomials as the interpolation nodes,then Gn(f,x)→,f(x),n→∞ holds for every x ∈(-∞,∞),where f(x) is any continous function on the real line satisfying |f(x)|=O(exp(1/2|x|β)).