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

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 mm-orthogonal polynomials for a Freud weight W=e-QW=e-Q, which are given as follows. Let an=an(Q)an=an(Q) be the nnth Mhaskar–Rahmanov–Saff number, φn(x)=max{n-2/3,1-|x|/an}φn(x)=max{n-2/3,1-|x|/an}, and d>0d>0. Assume that Q∈C(R)Q∈C(R) is even, Q″∈C[0,∞),Q′(x)>0,x∈(0,∞),Q(0)=0, and for some A,B>1A,B>1A⩽(xQ′(x))′Q′(x)⩽B,x∈(0,∞).Then for x∈Rx∈Rλjn(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)|x|⩽an(1+dn-2/3)λjn(Wm,m;x)⩽cannj+1W(x)mφn(x)(1-m)/2.