On fourier series of Jacobi-Sobolev orthogonal polynomials

Abstract

Let J.l be the Jacobi measure on the interval [-I, I] and introduce the discrete Sobolev-type inner productwhere c E (1,00) and M, N are non negative constants such that M + N > O. The main purpose of this paper is to study the behaviour of the Fourier series in terms of the polynomials associated to the Sobolev inner product. For an appropriate function f, we prove here that the Fourier-Sobolev series converges to f on the interval (-I, I) as well as to f( c) and the derivative of the series converges to f' (c). The term appropriate means here, in general, the same as we need for a function f(x) in order to have convergence for the series of f(x) associated to the standard inner product given by the measure J.l. No additional conditions are needed.

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].…”

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

“…The polynomials q n (x) and their particular cases are studied in [8][9][10][11][12] (see the bibliography therein). They do not possess the properties characteristic for orthogonal polynomials such as the three-term recurrent relation, their zeros can lie outside the orthogonality segment, and so on (see Section 4 for more detail).…”

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

“…ii nonnegative constants. The pointwise convergence of the Fourier series associated to such an inner product was studied when µ 0 is the Jacobi measure (see also [20,21]). On the other hand, the asymptotics for orthogonal polynomials with respect to the Sobolev-type inner product (1) with µ 0 ∈ M(0, 1), c k belong to suppµ \ [−1, 1], and A k are complex diagonal matrices such that a (k) 1+N k ,1+N k = 0, was solved in [2].…”