Discrete Laguerre–Sobolev expansions: A Cohen type inequality

Abstract

C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre-Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán. 2000MSC: 42C05

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

“…Notice that Theorem 2 generalizes the asymptotic behavior of the diagonal Laguerre kernels given in [6] (where only the case 0 ≤ j, k ≤ 1 has been analyzed). The interested reader can find an analogous result of Theorem 2 when c = 0, 0 ≤ j, k ≤ 1, and c = 0, 0 ≤ j, k ≤ n − 1, in [3,24], respectively. Also, it is worthwhile to point out that with a different approach the authors of [8] obtained a lower bound for the Christoffel functions when c ≥ 0.…”

A Cohen type inequality for Laguerre--Sobolev expansions with a mass point outside their oscillatory regime

“…It is worth pointing out that Corollary 4.1 says that as for the results of [9,5,11], the divergence of Fourier expansions in terms of this kind of Laguerre-Sobolev-type orthonormal polynomials remains true. …”

“…For instance, the authors of [5,11] obtained Cohen-type inequalities for Laguerre orthonormal expansions with respect to Sobolev-type inner products with only one mass point at c = 0 , i.e. the mass is located in the boundary of the support of the measure.…”

Asymptotic behavior of varying discrete Jacobi–Sobolev orthogonal polynomials

“…From here, in [6] the authors found the asymptotic behavior of a family of orthogonal polynomials with respect to a varying Sobolev inner product similar to (1), involving the Laguerre weight w(x) = x α e −x , α > −1. We remark that the techniques used in [6] are not useful in this case, and now we need to use more powerful techniques based on those considered in [11]. More recently, in [12] the same authors have even improved these techniques in such a way that they have obtained relevant results for the orthogonal polynomials with respect to a non-varying discrete Sobolev inner product being µ 0 a general measure.…”