Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Volosivets S. S., Likhacheva T. V. Several Questions of Approximation by Polynomials with Respect to Multiplicative Systems in Weighted Lp Spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2015, vol. 15, iss. 3, pp. 251-257. DOI: 10.18500/1816-9791-2015-15-3-251-258, EDN: UKIVDH

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 171)

Several Questions of Approximation by Polynomials with Respect to Multiplicative Systems in Weighted Lp Spaces

Volosivets Sergei Sergeevich, Saratov State University
Likhacheva Tatyana Vladimirovna, Saratov State University

In this paper we study approximation by Vilenkin polynomials in weighted Lp spaces. We prove the Butzer – Scherer type result on equivalence between the rate of best approximation of a function f and the growth of generalized derivatives and approximating properties of the best approximation polynomial tn(f). Some applications to the approximation by linear means of the Fourier – Vilenkin series are given.

  1. Голубов Б. И., Ефимов А. В., Скворцов В. А. Ряды и преобразования Уолша. Теория и применения. М. : Наука, 1987. 344 с.
  2. Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function // Trans. Amer. Math. Soc. 1972. Vol. 165. P. 207–226.
  3. Young W. S. Weighted norm inequalities for Vilenkin–Fourier series // Trans. Amer. Math. Soc. 1993. Vol. 340, № 1. P. 273–291.
  4. Волосивец С. С. Приближение полиномами по мультипликативным системам в весовых пространствах Lp // Сиб. матем. журн. 2015. Т. 56, № 1. С. 82–93.
  5. Ky N. X. On approximation by trigonometric polynomials in Lp u-spaces // Studia Sci. Math. Hungar. 1993. Vol. 28. P. 183–188.
  6. Ky N. X. Moduli of mean smoothness and approximation with Ap weights // Annales Univ. Sci. Budapest. Eӧtvӧs Sect. Math. 1997. Vol. 40. P. 37–48.
  7. Kokilashvili V., Yildirir Y. E. On the approximation in weighted Lebesgue spaces // Proc. A. Razmadze Math. Inst. 2007. Vol. 143. P. 103–113.
  8. Тиман А. Ф. Теория приближения функций действительного переменного. М. : Физматгиз, 1960. 624 с.
  9. Butzer P. L., Scherer K. On the fundamental approximation theorems of D. Jackson, S. N. Bernstein and theorems of M. Zamansky and S. B. Steˇ ckin // Aequationes Math. 1969. Vol. 3. P. 170–185.
  10. Butzer P. L., Wagner H. J. On dyadic analysis based on the pointwise dyadic derivative // Analysis Math. 1975. Vol. 1, № 3. P. 171–196.
  11. Iofina T. V., Volosivets S. S. On the degree of approximation by means of Fourier–Vilenkin series in H¨older and Lp norm // East J. Approximations. 2009. Vol. 15, № 3. P. 143–158.