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

Abstract

Let be a finite positive Borel measure supported on [−1, 1] and introduce the discrete Sobolev-type inner productwhere the mass points a k belong to [−1, 1], and M k,i > 0(i = 0, 1, . . . , N k ). In this paper, we obtain generalized trace formula and asymptotics of the averaged Turan determinant for the Sobolev-type orthogonal polynomials. Asymptotics of the recurrence coefficients for symmetric Gegenbauer-Sobolev orthogonal polynomials is obtained. Trace formula and asymptotics of Turan's determinant for Gegenbauer-Sobolev orthogonal polynomials are also given.

On multipliers for Fourier series in Sobolev orthogonal polynomials

“…1. For sufficiently large n there exists a pair of real zeros lying outside [−1, 1] (all the zeros of P That the recurrence coefficients in (6.4) have bounded variation was shown in [36]. The proof of (6.3) for recurrence relation (6.2) is similar.…”

On linear summability methods of fourier series in polynomials orthogonal in a discrete Sobolev space

“…with C 1 and C 2 constants independent of n. For a nine-termed recurrence relation the asymptotics (4.4) is specified in [15], the proof of (4.3) is similar. The polynomials B (α)…”

“…n (x) (as well as the classical Gegenbauer polynomials P (α) n (x)) are uniformly bounded on every compact set from (−1, 1) [15]. Put R α = f : To every function f ∈ R α (α > −1), there corresponds the Fourier-Gegenbauer-Sobolev series (M > 0, N > 0):…”