Estimates for polynomials orthogonal with respect to some Gegenbauer–Sobolev type inner product

Abstract

In this paper we obtain some estimates in [-1, 1] for orthogonal polynomials with respect to an inner product of Sobolev-type f, g) fg d/zo + where p(2c + 2) (1 x2) dx + M[6(x + 1) + 6(x-1)] d#0 2(2+1)p2(a + 1) d#l N[6(x + l) + 6(x-1)], M,N > O and a>-I Finally, the asymptotic behavior of such polynomials in [-1, 1] is analyzed.

A Cohen type inequality for Gegenbauer-Sobolev expansions

“…Thus, applying the Gram-Schmidt orthogonalization process to the canonical basis {x n } n≥0 , we get a sequence of orthonormal polynomials with respect to the above inner product. We denote it by { B (α) n } n≥0 (see [2,3,9]). They are called Gegenbauer-Sobolev type polynomials.…”

“…We summarize some properties of Gegenbauer-Sobolev type polynomials that we will need in the sequel (see [2,3,9]). Throughout the article, positive constants are denoted by c, c 1 , .…”

Об Одной Экстремальной Задаче Для Алгебраических Полиномов В Симметричном Дискретном Пространстве Гегенбауэра - Соболева

“…. Формула (3.9) из [18] в наших обозначениях (в статье [18] были использованы другие обозначе-ния параметров) имеет вид…”

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

“…where (c) n is the shifted factorial defined by and have proved the following representation (see also [10]):…”

“…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.…”