The Zeros of Orthogonal Polynomials and Markov–Bernstein Inequalities for Jacobi-Exponential Weights on (−1,1)

Let Ux=∏i=1rx−tipi, 0<p<∞, −1=tr<tr−1<⋯<t2<t1=1, r≥2, pi>−1/p, i=1,2,…,r, and W=e−Qx where Q:−1,1⟶0,∞. We give the estimates of the zeros of orthogonal polynomials for the Jacobi-Exponential weight WU on −1,1. In addition, Markov–Bernstein inequalities for the weight WU are also obtained.

INTEGRAL ERROR REPRESENTATION OF HERMITE INTERPOLATING POLYNOMIAL AND RELATED INEQUALITIES FOR QUADRATURE FORMULAE

We consider integral error representation related to the Hermite interpolating polynomial and derive some new estimations of the remainder in quadrature formulae of Hermite type, using Holder’s inequality and some inequalities for the Čebyšev functional. As a special case, generalizations for the zeros of orthogonal polynomials are considered.