Skip to main content
SpringerLink
Log in

The Weighted L 1-integrability of functions and the parseval equality with respect to multiplicative systems

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

In this paper we prove necessary and sufficient conditions for the weighted L 1-integrability of functions defined on [0, 1) in terms of Fourier coefficients with respect to a multiplicative system of bounded type. These results are counterparts of trigonometric ones obtained by M. and S. Izumi and M. M. Robertson.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transforms (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  2. C.W. Onneweer and D. Waterman, “Uniform Convergence of Fourier Series on Groups,” Michigan J. Math. 18(3), 265–273 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Leindler, “A New Class of Numerical Sequences and Its Application to Sine and Cosine Series,” Anal. Math. 28(4), 279–286 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Tikhonov, “Trigonometric Series with General Monotone Coefficients,” J. Math. Anal. Appl. 326(1), 721–735 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Heywood, “Integrability Theorems for Trigonometric Series,” Quart. J.Math. 13(2), 172–180 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. P. Boas, Integrability Theorems for Trigonometric Transforms (Springer, N.Y., 1967).

    MATH  Google Scholar 

  7. M. M. Robertson, “Integrability Theorems for Trigonometric Series and Transforms,” Math. Zeitschrift 91(1), 20–29 (1966).

    Article  MATH  Google Scholar 

  8. M. Izumi and S. Izumi, “Integrability Theorems for Fourier Series and Parseval Equation,” J. Math. Anal. Appl. 18(2), 252–261 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. S. Volosivets, “Certain Conditions in the Theory of Series with Respect toMultiplicative Systems,” Anal. Math. 33(3), 227–246 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  10. G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A. I. Rubinshtein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups (Elm, Baku, 1981) [in Russian].

    MATH  Google Scholar 

  11. J. A. Gosselin, “Convergence A. E. of Vilenkin-Fourier Series,” Trans. Amer. Math. Soc. 185, 345–370 (1973).

    Article  MathSciNet  Google Scholar 

  12. M. Dyachenko and S. Tikhonov, “Integrability and Continuity of Functions Represented by Trigonometric Series: Coefficients Criteria,” Stud. Math. 193(3), 285–306 (2009).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saratov State University, ul. Astrakhanskya 83, Saratov, 410012, Russia

    S. S. Volosivets

Authors
  1. S. S. Volosivets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. S. Volosivets.

Additional information

Original Russian Text © S.S. Volosivets, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 8, pp. 15–26.

About this article

Cite this article

Volosivets, S.S. The Weighted L 1-integrability of functions and the parseval equality with respect to multiplicative systems. Russ Math. 56, 11–21 (2012). https://doi.org/10.3103/S1066369X12080026

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X12080026

Keywords and phrases

Search

Navigation