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Differential Equations

, Volume 54, Issue 1, pp 49–66 | Cite as

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

  • I. I. Sharapudinov
  • M. G. Magomed-Kasumov
Ordinary Differential Equations
  • 11 Downloads

Abstract

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l r,k α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product
$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$
, and generated by the classical orthogonal Laguerre polynomials L k α (x) (k = 0, 1,...). The polynomials l r,k α (x) are represented as expressions containing the Laguerre polynomials L n α−r (x). An explicit form of the polynomials l r,k+r α (x) is established as an expansion in the powers x r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l r,k α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dagestan Scientific CenterRussian Academy of SciencesMakhachkala, DagestanRussia
  2. 2.Dagestan State Pedagogical UniversityMakhachkala, DagestanRussia
  3. 3.Vladikavkaz Scientific CenterRussian Academy of SciencesVladikavkaz, North OssetiaRussia

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