Higher-Order Hermite-Fejer Interpolation for Stieltjes Polynomials

Let omega(lambda) (x) := (1-x(2))(lambda-1/2) and P-lambda,P-n be the ultraspherical polynomials with respect to omega(lambda) (x). Then, we denote the Stieltjes polynomials E-lambda,E-n+1 with respect to omega(lambda) (x) satisfying integral(1)(-1) omega(lambda) (x)P-lambda,P-n(x)E-lambda,E-n+1(x)x(m)dx(= 0,0 <= m < n + 1; not equal 0, m = n + 1). In this paper, we consider the higher-order Hermite-Fejer interpolation operator Hn+1,(m) based on the zeros of E-lambda,(n+1) and the higher order extended Hermite-Fejer interpolation operator H-2n+1,H-m based on the zeros of E-lambda,E-n+1 P-lambda,P-n. When m is even, we show that Lebesgue constants of these interpolation operators are O (n(max{(1-lambda)m-2,0}))(0 < lambda < 1) and O(n(max{(1-2 lambda)m-2,0}))(0 < lambda < 1/2), respectively; that is, parallel to H-2n+1,H-m parallel to = O(n(max{(1-2 lambda)m-2,0}))(0 < lambda < 1) and parallel to H-n+1,H-m parallel to = O(n(max{(1-lambda)m-2,0}))(0 < lambda < 1/2). In the case of the Hermite-Fejer interpolation polynomials H-2n+1,H-m [.] for 1/2 < lambda < 1, we can prove the weighted uniform convergence. In addition, when m is odd, we will show that these interpolations diverge for a certain continuous function on [-1, 1], proving that Lebesgue constants of these interpolation operators are similar or greater than log n.