Advertisement

Mathematical Notes

, Volume 106, Issue 3–4, pp 616–638 | Cite as

The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent

  • I. I. SharapudinovEmail author
Article
  • 14 Downloads

Abstract

The problem of the basis property of ultraspherical Jacobi polynomials in a Lebesgue space with variable exponent is studied. We obtain sufficient conditions on the variable exponent p(x) > 1 that guarantee the uniform boundedness of the sequence S n α,α (f), n = 0,1,..., of Fourier sums with respect to the ultraspherical Jacobi polynomials P k α,α (x) in the weighted Lebesgue space L μ p( x) ([-1, 1]) with weight μ = μ(x) = (1 - x2)α, where α >-1/2. The case α = -1/2 is studied separately. It is shown that, for the uniform boundedness of the sequence Sn-1/2, -1/2 (f), n = 0,1,..., of Fourier—Chebyshev sums in the space L μ p( x) ([-1,1]) with μ(x) = (1 - x2)-1/2, it suffices and, in a certain sense, necessary that the variable exponent p satisfy the Dini-Lipschitz condition of the form
$$\left| {p(x) - p(y)} \right| \leq \frac{d}{{ - \ln \left| {x - y} \right|}},\;\;\;\text{where}\;\left| {x - y} \right| \leq \frac{1}{2},\;\;x,y \in [ - 1,1],\;\;d > 0,$$
and the condition p(x) > 1 for all x ∈ [-1,1].

Keywords

the basis property of ultraspherical polynomials Fourier—Jacobi sums Fourier— Chebyshev sums convergence in a weighted Lebesgue space with variable exponent Dini—Lipschitz condition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding

This work was supported by the Russian Foundation for Basic Research under grant 16-01-00486.

References

  1. 1.
    J. Newman and W. Rudin, “Mean convergence of orthogonal series,” Mean convergence of orthogonal series, 219–222 1952.zbMATHGoogle Scholar
  2. 2.
    B. Muckenhoupt, “Mean convergence of Jacobi series,” Mean convergence of Jacobi series, 23, 306–310 1969.MathSciNetzbMATHGoogle Scholar
  3. 3.
    H. Pollard, “The mean convergence of orthogonal series,” The mean convergence of orthogonal series, 387–403 1947.zbMATHGoogle Scholar
  4. 4.
    H. Pollard, “The mean convergence of orthogonal series. II,” The mean convergence of orthogonal series. II, 355–367 1948.zbMATHGoogle Scholar
  5. 5.
    H. Pollard, “The mean convergence of orthogonal series. III,” The mean convergence of orthogonal series. III, 189–191 1949.zbMATHGoogle Scholar
  6. 6.
    I. I. Sharapudinov, “The basis property of the Legendre polynomials in the variable exponent Lebesgue space Lp(x)(-1,1),” The basis property of the Legendre polynomials in the variable exponent Lebesgue space Lp(x)(-1,1), 200(1), 137–160 2009).[Sb. Math. 200(1), 133–156 (2009)].Google Scholar
  7. 7.
    I. I. Sharapudinov, “Topology of the space JW([0, 1])”, Mat. Zametki 26 (4), 613–632 1979).[Math. Notes 26(4), 796–806 (1979)].Google Scholar
  8. 8.
    I. I. Sharapudinov, Some Questions of Approximation Theory in Lebesgue Spaces with Variable Exponent, in Itogi Nauki. Yug Rossii, Mathematical monograph (YuMI VNTs RAN and RSO-A, Vladikavkaz, 2012). Vol. 5 [in Russian].Google Scholar
  9. 9.
    I. I. Sharapudinov, “Some questions of the theory of the approximation in the spaces Lp(x),” Anal. Math. 33 (2), 135–153 (2007).Google Scholar
  10. 10.
    I. I. Sharapudinov, “Approximation of functions in Lp(x)2r by trigonometric polynomials,” Approximation of functions in Lp(x)2r by trigonometric polynomials, 77 (2), 197–224 2013).[Izv. Math. 77 (2), 407–434 (2013)].Google Scholar
  11. 11.
    I. I. Sharapudinov, “On the basis property of the Haar system lp(t)([0,1]) and the principle of localization in the mean,” On the basis property of the Haar system lp(t)([0,1]) and the principle of localization in the mean, 130 (172) (2(6)), 275–283 1986).[Math. USSR-Sb. 58 (1), 279–287 (1987)].Google Scholar
  12. 12.
    I. I. Sharapudinov, “Uniform boundedness in Lp, (p = p(x)) of some families of convolution operators,” Uniform boundedness in Lp, (p = p(x)) of some families of convolution operators, 59 (2), 291–302 1996).[Math. Notes 59 (2), 205–212 (1996)].Google Scholar
  13. 13.
    L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, in Lecture Notes in Math. (Springer, Heidelberg, 2011). Vol. 2017.Google Scholar
  14. 14.
    D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis (Springer, Heidelberg, 2013).CrossRefGoogle Scholar
  15. 15.
    A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959; Mir, Moscow, 1965). Vol. 1.Google Scholar
  16. 16.
    G. Szegö, Orthogonal Polynomials in Colloquium Publ. (Amer. Math. Soc, Providence, RI, 1959; Fizmatgiz, Moscow, 1962).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Daghestan Scientific Center of Russian Academy of SciencesMakhachkalaRussia
  2. 2.Vladikavkaz Scientific Center of Russian Academy of SciencesVladikavkazRussia
  3. 3.Daghestan State Pedagogical UniversityMakhachkalaRussia

Personalised recommendations