Recurrence relations and outer relative asymptotics of orthogonal polynomials with respect to a discrete Sobolev type inner product

Abstract

We investigate algebraic and analytic properties of sequences of polynomials orthogonal with respect to the Sobolev type inner product polynomials. When the values M k,i are nonnegative real numbers, we can deduce the coefficients of the recurrence relation in terms of the connection coefficients for the sequences of polynomials orthogonal with respect to the Sobolev type inner product and those orthogonal with respect to the measure μ. The matrix of a symmetric multiplication operator in terms of the above sequence of Sobolev type orthogonal polynomials is obtained from the Jacobi matrix associated with the measure μ. Finally, we focus our attention on some outer relative asymptotics of such polynomials, which are deduced by using the above connection formulas.

Generalized Sobolev orthogonal polynomials, matrix moment problems and integrable systems

“…The auxiliary polynomial (18) is the keystone for the following result in concordance with [10] and slightly generalizing [21].…”

Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications

“…In the setting of orthogonal polynomial theory these kernels have been especially used by Freud and Nevai [4,21,22] and, more recently, the remarkable Lubinsky's works [9,10] have caused heightened interest in this topic. Also, other interesting and related results corresponding to Fourier-Sobolev expansions may be found in [11][12][13][14][15][17][18][19]25].…”

Asymptotics for Laguerre–Sobolev type orthogonal polynomials modified within their oscillatory regime

“…The study of asymptotic properties of the sequences of orthogonal polynomials with respect to particular cases of the inner product (1) has been done by considering separately the cases 'mass points inside' or 'mass points outside' of suppµ 0 , respectively, being suppµ 0 a bounded interval of R or, more recently, an unbounded interval of the real line (see, for instance [7-10, 12, 19, 26]). The first results in the literature about asymptotic properties of orthogonal polynomials with respect to a Sobolev-type inner product like (1) appear in [27], where the authors considered d = 0, N 0 = 1, a (0) 11 = a (0) 12 = a (0) 21 = 0, a (0) 22 = λ, with λ > 0. Therein, such asymptotic properties when there is only one mass point supporting the derivatives either inside or outside [−1, 1] and µ is a measure in the Nevai class M(0, 1) are studied.…”