Asymptotic behaviour of orthogonal polynomials relative to measures with mass points

Abstract

General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ + Mδc, in terms of those of the measures dμ and (x − c)2dμ. In particular, these relations allow us to show that Nevai's class M(0, 1) is closed under adding a mass point, as well as obtain several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product

“…Comparing the results of Lemmas 3 and 4, we see that we must estimate Bn(1) and /3~(1). Note first that according to (8)and (19) Note that B,(1) < 0 and /3~(1) > 0 for sufficiently large n. This means that there exists at least one zero of Bn(x) to the right (and, by symmetry, at least one zero to the left) of the interval [-1, 1]. But, by the lemma preceding Theorem 2.1 in [2], /3n(x) has at least n -2 distinct zeros of odd multiplicity in (-1, 1).…”

“…We consider only the case N > 0, since the growth of polynomials orthogonal with respect to weight functions belonging to the Badkov class for the case in which the point masses were on the interval [-1, 1], was analyzed in [6,8,9].…”

Estimates for jacobi—sobolev type orthogonal polynomials

“…In such a way we can give a complete answer in order to estimate the behaviour on [−1, 1] of such polynomials. Notice that some of the above results, when dµ 0 = wdx + M δ c where w is a generalized Jacobi weight and µ k = 0 (k = 1, ..., m), have been obtained in [5].…”

Orthogonal polynomials in several variables for measures with mass points

“…In the case of one variable (d = 1), such an estimate has been carried out in [6]. We shall give one result on the strong asymptotics of the Christoffel function with respect to dν on the simplex T d .…”