Christoffel Functions And Mean Convergence For Lagrange Interpolation For Exponential Weights

The study of the orthogonal polynomials on infinite interval and mean con-vergnece Lagrange interpolation has been a major and hot point in approximationtheory at present. There are three interesting results in this paper.The first result provides estimations for lower bound for the quadrature sumsof general weights on infinite interval. It plays a fundamental role in orthogonalpolynomials on infinite interval and mean convergence for Lagrange interpolation.Theorem A. Let dμand dν, be measures onâ… and 0＜p＜∞. Letâ–³(?)â… and letΩ=[c, d] (?)â–³, -∞＜c＜d＜+∞, satisfy that if integral from n=c′to d′dμ(x)=integral from n=c to d dμ(x)with c≤c′≤d′≤d then c′=c and d′=d. Suppose that the zero of the largestmodulus of P_n(dμ) is o(n). Then forδ＞0 small enough and n large enough, {(γn-1)/γn sum from x_kn∈△=1λ_kn|P_n-1(x_kn|}^p integral from n=â–³|P_n(x)|^pdν(x)≥δ^p/(2(12)^p)σ(dν,Ω;δ)integral from n=Ωdν(x).Moreover, if integral from n=Ωdν(x)＞0, then (?){(γn-1)/γn sum from n=x_kn∈△λ_kn|P_n-1(x_kn|}^p integral from n=â–³|P_n(x)|^pdν(x)＞0.The second result provides precise estimations for quadrature sum of exponen-tial weights, which are the basis of the research of orthogonal polynomials associatedwith exponential weights and the convergence for interpolation on its zeros. It is oftheoretical significance. For simplicity for 0＜p≤2, case A means that p=2 and c=a or d=b, andcase B otherwise.Theorem B. Let W∈F(liP1／2+) with -a=b and let Q be even. Assumethatâ–³(?)â… is an interval and 0＜p≤2. Thensum from n=x_kn∈△λ_knW(x_kn)^-P～(?)Theorem C. Let W∈F(liP1／2+) with -a=b and let Q be even. Assumethatâ–³(?)â… is an interval and 0＜p≤2. Thensum from n=x_kn∈△λ_kn|P_n-1(x_kn|^p～(?).Theorem D. Let W∈F(liP1／2+) with -a=b and let Q be even. Assumethatâ–³(?)â… is an interval. Thensum from n=x_kn∈△1／|P′_n(x_kn)|～a_n^1／2.The third result gives a new necessary condition of mean convergence of La-grange interpolation on zeros of the orthogonal polynomial associated with expo-nential weights.Theorem E. Let dμand dνbe measures onâ… . Letâ–³=(c, d), -∞＜c＜d＜∞, and 0＜p＜∞. Then‖L_n(X)‖C₀(â–³)→L_dν^P≥c(â–³, p)‖1／1+|x| sum from n=x_kn∈△|(x-x_kn)e_kn(x)|‖_{dν, p}and‖L_n(X)‖C₀(â–³)→L_dν^P≥c(â–³, p) sum from n=x_kn∈△1／|w′_n(x_kn)|‖w_n(x)／1+|x|‖_{dν, p} Moreover, under the assumptions of Theorem A we have‖L_n(dμ)‖C_o(â–³)→L_dν^p≥c(dν,â–³, p)[integral from n=â–³|P_n(x)|dν(x)]^-1‖P_n(x)／1+|x|‖_{dν, p}Theorem F. Let W∈F(lip1／2+) with -a=b and let Q be even. Let0＜p＜∞. Suppose that u≥0 and u∈C(â… ). ThenTheorem G. Let W∈F(liP1／2+) with -a=b and let Q be even. Letâ–³(?)â… be a finite interval and let 0＜p＜∞. Assume that u≥0 and u∈C(â… ). Supposethat (?)integral from n=â… |L_n(W², f; x)-f(x)|^pu(x)dx=0holds for every f∈C₀(â–³). Then integral from n=â… [(1+|x|W(x)]^-pu(x)dx＜∞.