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On Fourier series of a discrete Jacobi–Sobolev inner product. (English) Zbl 1019.42014
Summary: Let $$\mu$$ be the Jacobi measure supported on the interval $$[-1,1]$$ and introduce the discrete Sobolev-type inner product $\langle f,g\rangle= \int^1_{-1} f(x) g(x) d\mu(x)+ \sum^K_{k=1} \sum^{N_k}_{i=0} M_{k,i} f^{(i)}(a_k) g^{(i)}(a_k),$ where $$a_k$$, $$1\leq k\leq K$$, are real numbers such that $$|a_k|> 1$$ and $$M_{k,i}> 0$$ for all $$k$$, $$i$$. This paper is a continuation of [F. Marcellán, B. P. Osilenker and I. A. Rocha, “On Fourier series of Jacobi-Sobolev orthogonal polynomials”, J. Inequal. Appl. 7, 673-699 (2002; Zbl 1016.42014)] and our main purpose is to study the behaviour of the Fourier series associated with such a Sobolev inner product. For an appropriate function $$f$$, we prove here that the Fourier-Sobolev series converges to $$f$$ on $$(-1,1)\bigcup^K_{k=1}\{a_k\}$$, and the derivatives of the series converge to $$f^{(i)}(a_k)$$ for all $$i$$ and $$k$$. Roughly speaking, the term appropriate means here the same as we need for a function $$f$$ in order to have convergence for its Fourier series associated with the standard inner product given by the measure $$\mu$$. No additional conditions are needed.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 33C47 Other special orthogonal polynomials and functions
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##### References:
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