Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential-Type Weights

Let wρ(x):=|x|ρexp(−Q(x)), ρ>−1/2, where Q∈C2:(−∞,∞)→[0,∞) is an even function. In 2008 we have a relation of the orthonormal polynomial pn(wρ2;x) with respect to the weight wρ2(x); pn′(x)=An(x)pn−1(x)−Bn(x)pn(x)−2ρnpn(x)/x, where An(x) and Bn(x) are some integrating functions for orthonormal polynomials pn(wρ2;x). In this paper, we get estimates of the higher derivatives of An(x) and Bn(x), which are important for estimates of the higher derivatives of pn(wρ2;x)

An estimate for derivative of the de la Vallee Poussin mean

In this paper, we discuss derivative of the de la Vallee Poussin mean for exponential weights on real line. When we lead an inequality, an estimate for the Christoffel function plays an important role.Comment: 18 page

Some Properties of Orthogonal Polynomials for Laguerre-Type Weights

Let , let be a continuous, nonnegative, and increasing function, and let be the orthonormal polynomials with the weight . For the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted -norms ( ) of .</p

Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

Let ( ) := (1− 2 ) −1/2 and , be the ultraspherical polynomials with respect to ( ). Then, we denote the Stieltjes polynomials . In this paper, we consider the higher-order Hermite-Fejér interpolation operator +1, based on the zeros of , +1 and the higher order extended Hermite-Fejér interpolation operator H 2 +1, based on the zeros of , +1 , . When m is even, we show that Lebesgue constants of these interpolation operators are ( max{(1− ) −2,0} )(0 &lt; &lt; 1) and ( max{(1−2 ) −2,0} )(0 &lt; &lt; 1/2), respectively; that is, ‖H 2 +1, ‖ = ( max{(1−2 ) −2,0} )(0 &lt; &lt; 1) and ‖ +1, ‖ = ( max{(1− ) −2,0} )(0 &lt; &lt; 1/2). In the case of the Hermite-Fejér interpolation polynomials H 2 +1, [⋅] for 1/2 ≤ &lt; 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