On Fourier series of a discrete Jacobi-Sobolev inner product

22 pages, no figures.-- MSC2000 code: 42C05.MR#: MR1920116 (2003e:42007)Zbl#: Zbl 1019.42014Let $\mu$ be the Jacobi measure supported on the interval $[-1,1]$ and introduce the discrete Sobolev-type inner product \langle f,g\rangle= \int _{-1} f(x) g(x) d\mu(x)+ \sum _{k=1} \sum N_k}_{i=0} M_{k,i} f (i)}(a_k) g (i)}(a_k), where $a_k$, $1\le k\le K$, are real numbers such that $_k 1$ and $M_{k,i}> 0$ for all $k$, $i$. This paper is a continuation of [{\it F. Marcellán}, {\it B. P. Osilenker} and {\it 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 $(-1,1)\bigcup _{k=1}\{a_k\}$, 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.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 by an INTAS Grant 2000/272.Publicad

Divergent Cesàro Means of Jacobi-Sobolev Expansions

Let µ be the Jacobi measure supported on the interval [­1; 1]. Let introduce the Sobolev-type inner product …. In this paper we prove that, for certain indices δ, there are functions whose Ces_aro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product are divergent almost everywhere on [-1; 1].Let µ be the Jacobi measure supported on the interval [­1; 1]. Let introduce the Sobolev-type inner product …. In this paper we prove that, for certain indices δ, there are functions whose Ces_aro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product are divergent almost everywhere on [-1; 1]

Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

21 pages, no figures.-- MSC2000 codes: 42C05, 33C47.MR#: MR1971776 (2004a:42035)Zbl#: Zbl 1014.42019^aLet μ be a finite positive Borel measure supported in [−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 the mass points $a_k$ belong to [−1,1], $M_{k,i}\geq 0$, $i = 0,\dots,N_k-1$, and $M_{k,N_k} >0$. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants Mk,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in R\[-1,1].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 00-272.Publicad

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

In this paper we consider sequences of polynomials orthogonal with respect to the discrete Sobolev inner product (f.g)_s ∫_°^∞ f(x)g(x) x^(α ) e dx+F(c)ΑG(c)^t, α> 1 where f and g are polynomials with real coefficients A∈ R^2.2 and the vectors F(c), G(c) are A=(■(M&0@0&N)), F(c)=(f(c),f'(c) ) G(c)=(g(c),g'(c)) with M,N ∈ R and the mass point c is located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of classical Laguerre polynomials and we analyze the behavior of the coefficients of the corresponding five term recurrence relation when the degree of the polynomials is large enough. Also, the outer relative asymptotics of the Laguerre Sobolev type with re spect to the Laguerre polynomials is analyzed