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

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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 α r,k (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 α k (x) (k = 0, 1,...). The polynomials l α r,k (x) are represented as expressions containing the Laguerre polynomials L α−r n (x). An explicit form of the polynomials l α r,k+r (x) is established as an expansion in the powers xr+l, l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l α r,k (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

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Authors and Affiliations

  1. Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, Dagestan, 367025, Russia

    I. I. Sharapudinov & M. G. Magomed-Kasumov

  2. Dagestan State Pedagogical University, Makhachkala, Dagestan, 367003, Russia

    I. I. Sharapudinov

  3. Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, North Ossetia, 362027, Russia

    M. G. Magomed-Kasumov

Authors
  1. I. I. Sharapudinov
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  2. M. G. Magomed-Kasumov
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Correspondence to I. I. Sharapudinov.

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Original Russian Text © I.I. Sharapudinov, M.G. Magomed-Kasumov, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 1, pp. 51–68.

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Sharapudinov, I.I., Magomed-Kasumov, M.G. On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials. Diff Equat 54, 49–66 (2018). https://doi.org/10.1134/S0012266118010068

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  • DOI: https://doi.org/10.1134/S0012266118010068

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