Christoffel-type functions for m-orthogonal polynomials for Freud weights

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Abstract

This paper gives upper and lower bounds of the Christoffel-type functions λjn(Wm,m;x),j=m-2,m-4,,m-2[m/2], for the m-orthogonal polynomials for a Freud weight W=e-Q, which are given as follows. Let an=an(Q) be the nth Mhaskar–Rahmanov–Saff number, φn(x)=max{n-2/3,1-|x|/an}, and d>0. Assume that QC(R) is even, QC[0,),Q(x)>0,x(0,),Q(0)=0, and for some A,B>1A(xQ(x))Q(x)B,x(0,).Then for xRλjn(Wm,m;x)cannj+1W(x)mφn(x)-1/2,miseven,j=0,cannj+1W(x)motherwise,and for |x|an(1+dn-2/3)λjn(Wm,m;x)cannj+1W(x)mφn(x)(1-m)/2.

MSC

41A55

Keywords

m-orthogonal polynomials
Christoffel-type functions
Freud weights

Cited by (0)

The research is supported in part by the National Natural Science Foundation of China (No. 10571049) and by Hunan Provincial Natural Science Foundation of China (No. 05JJ30011).