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

Abstract

ABSTRACT. For the Legendre-Sobolev orthonormal polynomials B,t(z) = B,,(z ; M, N) depending on the parameters M > 0, N >_ 0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials B,t(z) for M > 0, N > 0 decay at the rate of n -z/2 at the points 1 and -1. The values of B'(:t:I) are calculated.KEY WORDS: Legendre-Sobolev polynomials, orthogonality with respect to the Legendre-Sobolev inner product, Legendre polynomials.

Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product

“…It should be noted that the Gegenbauer-Sobolev orthonormal polynomials B ( ) n (x) have some properties other than the corresponding Gegenbauer orthonormal polynomials R ( ) n (x) (see, for example [2][3][4][5]9,10,15,17,18,20,21,[28][29][30][31]). We mention the following properties only.…”

Estimates for jacobi—sobolev type orthogonal polynomials

“…and n ∈ N. We want to find similar estimates for the new kernels. Let L n (x, y) be the kernels relative to the inner product (8). If we consider their expansion in terms of Jacobi orthonormal polynomials, we can deduce, (see [1, p.744]), L n (x, y) = K n (x, y) − M L n (y, 1)K n (x, 1) − N L (0,1) n (y, 1)K (0,1) n (x, 1)…”

“…The aim of this paper is to cover this lack in the literature. In fact, a first approach was given by Marcellán and Osilenker [8] when m = 1, dµ 0 = χ [−1,1] dx + M (δ 1 + δ −1 ) and dµ 1 = N (δ 1 + δ −1 ) using some previous work by Bavinck and Meijer ([3], [4]), (δ c denotes a Dirac measure supported at the point c).…”

OnW1,p-convergence of Fourier–Sobolev expansions

“…In fact, the solution of this problem can be very hard, or relatively easy, depending on either the sense of the convergence, or in terms of additional restrictions on f and the pair of weights (w 0 , w 1 ), (see for instance, [1][2][3][4][5]). …”