Mathematical Notes

, Volume 105, Issue 3–4, pp 543–549 | Cite as

A Sobolev Orthogonal System of Functions Generated by a Walsh System

  • M. G. Magomed-KasumovEmail author


Properties of functions from the Sobolev orthogonal system \(\mathfrak{W}_{r}\) generated by the Walsh system are studied. In particular, recurrence relations for functions from \(\mathfrak{W}_{1}\) are obtained. The uniform convergence of Fourier series in the system \(\mathfrak{W}_{r}\) to functions f from the S obolev spaces \(W_{{L^p}}^r\), p ≥ 1, r = 1, 2,… is proved.


Sobolev orthogonality Walsh system uniform convergence recurrence relation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. I. Golubov, A. V. Efimov, and V.A. Skvortsov, Walsh Series and Transforms. Theory and Applications (Nauka, Moscow, 1987) [in Russian].zbMATHGoogle Scholar
  2. 2.
    I. I. Sharapudinov, “Sobolev-orthogonal systems of functions associated with an orthogonal system,” Izv. Ross. Akad. Nauk Ser. Mat. 82 (1), 225–258 (2018) [Izv. Math. 82 (1), 212–244 (2018)].MathSciNetzbMATHGoogle Scholar
  3. 3.
    I. I. Sharapudinov, “Approximation properties of Fourier series of Sobolev orthogonal polynomials with Jacobi weight and discrete masses,” Mat. Zametki 101 (4), 611–629 (2017) [Math. Notes 101 (4), 718–734 (2017)].MathSciNetzbMATHGoogle Scholar
  4. 4.
    I. I. Sharapudinov, “Special series in Laguerre polynomials and their approximation properties,” Sibirsk. Mat. Zh. 58 (2), 440–467 (2017) [Sib. Math. J. 58 (2), 338–362 (2017)].MathSciNetzbMATHGoogle Scholar
  5. 5.
    I. I. Sharapudinov, M. G. Magomed-Kasumov, and S. R. Magomedov, “Sobolev orthogonal polynomials associated with the Chebyshev polynomials of the first kind,” Daghestan Electronic Mathematical Reports, No. 4, 1–14 (2015).Google Scholar
  6. 6.
    I. I. Sharapudinov and T. I. Sharapudinov, “Polynomials orthogonal in the Sobolev sense, generated by Chebyshev polynomials orthogonal on a mesh,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 67–79 (2017) [Russian Math. (Iz. VUZ) 61 (8), 59–70 (2017)].Google Scholar
  7. 7.
    I. I. Sharapudinov and Z. D. Gadzhieva, “Sobolev orthogonal polynomials generated by Meixner polynomials,” Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 16 (3), 310–321 (2016).MathSciNetzbMATHGoogle Scholar
  8. 8.
    R. M. Gadzhimirzaev, “The Fourier series of the Meixner polynomials orthogonal with respect to the Sobolev-type inner product,” Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 16 (4), 388–395 (2016).MathSciNetzbMATHGoogle Scholar
  9. 9.
    G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus (Fizmatlit, Moscow, 2001), Vol. 2. [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Vladikavkaz Scientific Center of Russian Academy of SciencesVladikavkazRussia
  2. 2.Daghestan Scientific Center of Russian Academy of SciencesMakhachkalaRussia

Personalised recommendations