Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Fadeev A. A., Volosivets S. S. Weighted Integrability of Sums of Series with Respect to Multiplicative Systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 2, pp. 129-136. DOI: 10.18500/1816-9791-2014-14-2-129-136, EDN: SHHICR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
09.06.2014
Full text:
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Language: 
Russian
Heading: 
Mathematics
UDC: 
517.518
DOI: 
10.18500/1816-9791-2014-14-2-129-136
EDN: 
SHHICR

Weighted Integrability of Sums of Series with Respect to Multiplicative Systems

Autors: 
Fadeev Aleksandr Andreevich, Saratov State University
Volosivets Sergei Sergeevich, Saratov State University
Abstract: 

A necessary and sufficient condition for Lp-integrability with power weight of a function f represented by the series with respect to multiplicative systems with generalized monotone coefficients is obtained. The integrability of the majorant of partial sums of a representing series is also described by the same conditions. In addition we study the integrability of difference quotient (f(x) − f(0))/x.

Key words: 
weighted Lp-integrability
multiplicative system
power weight
generalized monotone sequences
References: 
  1. Golubov B. I., Efimov A. V., Skvortsov V. A. Walsh series and transforms. Theory and applications. Dordrecht, Kluwer, 1991.
  2. Tikhonov S. Trigonometric series with general monotone coefficients. J. Math. Anal. Appl., 2007, vol. 326, no. 1, pp. 721–735.
  3. Leindler L. A new class of numerical sequences and its application to sine and cosine series. Analysis Math., 2002, vol. 28, no. 4, pp. 279–286.
  4. Konyushkov A. A. Nailuchshee priblizhenie trigonometricheskimi polinomami i koefficienty Fur’e [The best approximation by trigonometrical polynomials and Fourier coefficients]. Mat. sbornik, 1958, vol. 44, no. 1, pp. 53–84 (in Russian).
  5. Boas R. P. Integrability theorems for trigonometric transforms. Berlin, Springer, 1967.
  6. Dyachenko M., Tikhonov S. Integrability and continuity of functions represented by trigonometric series: coefficients criteria. Studia Math., 2009, vol. 193, no. 3, pp. 285–306.
  7. Moricz F. On Walsh series with coefficients tending monotonically to zero. Acta Math. Hung., 1983, vol. 38, no. 1–4, pp. 183–189.
  8. Timan M. F., Rubinstein A.I. On embedding of function classes defined on zero-dimensional groups. Izvestiya vuzov. Matematika. [Soviet Math.], 1980, no. 6, pp. 66–76 (in Russian).
  9. Volosivets S. S. On certain conditions in the theory of series with respect to multiplicative systems. Analysis Math., 2007, vol. 33, no. 3, pp. 227–246 (in Russian).
  10. Volosivets S. S., Fadeev R. N. Estimates of best approximations in integral metrics and Fourier coefficients with respect to multiplicative systems. Analysis Math., 2011, vol. 37, no. 3, pp. 215–238.
  11. Askey R., Wainger S. Integrability theorems for Fourier series. Duke Math. J., 1966, vol. 33, no. 2, pp. 223–228.
  12. Vukolova T. M., Dyachenko M. I. On the properties of trigonometric series sums with monotone coefficients. Vestnik Mosk. Universiteta. Ser. Matem. mech., 1994, no. 3, pp. 22–31 (in Russian).
  13. Agaev G. N., Vilenkin N. Ya., Dzhafarli G. M., Rubinstein A. I. Mul’tiplikativnye sistemy funkcii i garmonicheskii analiz na nul’-mernyh gruppah [Multiplicative systems of functions and harmonic analysis on zero-dimensional groups]. Baku, Elm, 1980 (in Russian).
  14. Hardy G. H., Littlewood J. E., Polya G. Inequalities. Cambridge, Cambridge University Press, 1934.
  15. Leindler L. Inequalities of Hardy – Littlewood type. Analysis Math., 1976, vol. 2, no. 2, pp. 117–123.
  16. Leindler L. Generalization of inequalities of Hardy andf Littlewood. Acta Sci. Math. (Szeged), 1970, vol. 31,no. 1–2, pp. 279–285.
  17. 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, no. 2, pp. 143–158.
Received: 
12.11.2014
Accepted: 
08.04.2014
Published: 
30.05.2014
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