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

Authors: EDMUNDO JOSÉ HUERTAS CEJUDO, FRANCISCO MARCELLÁN ESPANOL, MARÍA FRANCISCA PÉREZ VALERO, YAMILET QUINTANA

Abstract: Let consider the Sobolev type inner product \langle f, g\rangle_S = \int_0^{\infty} f(x)g(x)d \mu (x) + Mf(c)g(c) + Nf^{\prime}(c) g^{\prime}(c), where d\mu (x) = x^{\alpha} e^{-x}dx, \alpha > -1, is the Laguerre measure, c < 0, and M, N \geq 0. In this paper we get a Cohen-type inequality for Fourier expansions in terms of the orthonormal polynomials associated with the above Sobolev inner product. Then, as an immediate consequence, we deduce the divergence of Fourier expansions and Cesàro means of order \delta in terms of this kind of Laguerre--Sobolev polynomials.

Keywords: Sobolev-type orthogonal polynomials, Cohen-type inequality, Fourier--Sobolev expansions

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