On Fourier series of Jacobi-Sobolev orthogonal polynomials

27 pages, no figures.-- MSC2000 code: 42C05.MR#: MR1931261 (2003g:42043)Zbl#: Zbl 1016.42014Let $\mu$ be the Jacobi measure on the interval $[-1,1]$ and introduce the discrete Sobolev-type inner product $$\langle f,g\rangle= \int^1_{-1} f(x) g(x) d\mu(x)+ Mf(c) g(c)+ Nf'(c) g'(c),$$ where $c\in (1,\infty)$ and $M$, $N$ are nonnegative constants such that $M+ N>0$. 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 $(-1,1)$ 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 $\mu$. No additional conditions are needed.The work of F. Marcellán was supported by a grant of Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain BFM-2000-0206-C04-01, and INTAS project INTAS 2000-272.Publicad