# Generalized Absolute Convergence of Series with Respect to Multiplicative Systems of Functions of Generalized Bounded Variation

A. Zygmund proved that a 2π-periodic function with bounded variation and from any Lipschitz class Lip(α) has absolutely convergent Fourier series. This result was extended to many classes of functions of generalized bounded variation (for example, functions of bounded Jordan-Wiener p-variation, functions of bounded Λ-variation introduced by D. Waterman et al) and to different spaces defined with the help of moduli of continuity. We study the convergence of series \sum\limits^\infty_{k=1}\gamma_k|\hat{f}(k)|^\beta, where \{\gamma_k\}^\infty_{k=1} is a sequence from appropriate Gogoladze-Meskhia class, while \{\hat{f}(k)\}_{k=0}^\infty are Fourier coefficients of f ∈ L 1 [0,1) with respect to a multiplicative system. The sufficient conditions for convergence of these series are obtained under assertion of boundedness of generalized fluctuation determined by a number p > 1 and sequence Λ and in terms of uniform or integral moduli of continuity. By using fluctuation (i.e. the oscillations of a function are considered only with respect to the restricted class of partitions and its intervals) instead of variation we obtain more general assertions. The results of the present paper give an analogue or generalize some results of R.G.Vyas concerning trigonometric or Walsh series.

1. Golubov B. I., Efimov A. V., Skvortsov V. A. Walsh series and transforms. Theory and applications. Dordrecht, Kluwer Academic Publ., 1991. 380 p.

2. Waterman D. On the summability of Fourier series of functions of Λ-bounded variation. Studia Math., 1976, vol. 55, no. 1, pp. 87–95.

3. Shiba M. On absolute convergence of Fourier series of functions of class ΛBV(p) . Sci. Rep. Fukushima Univ., 1980, vol. 30, pp. 7–10.

4. Kita H., Yoneda K. A generalization of bounded variation. Acta Math. Hung., 1990, vol. 56, no. 3-4, pp. 229–238. DOI: doi.org/10.1007/BF01903837.

5. Gogoladze L., Meskhia R. On the absolute convergence of trigonometric Fourier series. Proc. Razmadze Math. Inst., 2006, vol. 141, pp. 29–40.

6. Uljanov P. L. O ryadakh po systeme Haara s monotonnymi koefficientami [On Haar series with monotone coefficients]. Izvestiya AN SSSR. Ser. matem., 1964, vol. 28, no. 4, pp. 925–950 (in Russian).

7. Moricz F. Absolute covergence of Walsh-Fourier series and related results. Analysis Math., 2010, vol. 36, no. 4, pp. 275–286. DOI: doi.org/10.1007/s10476-010-0402-z.

8. Golubov B. I, Volosivets S. S. Generalized absolute convergence of single and double Fourier series with respect to multiplicative systems. Analysis Math., 2012, vol. 38, no. 2, pp. 105–122. DOI: doi.org/10.1007/s10476-012-0202-8.

9. Vyas R. G. Generalized absolute convergence of trigonometric Fourier series. Modern Mathematical Methods and High Performance Computing in Science and Technology, Springer Proceedings in Mathematics and Statistics, 2016, vol. 171, pp. 231–237. DOI doi.org/10.1007/978-981-10 1454-3_19.

10. Vyas R. G. Absolute convergence of Walsh–Fourier series. Annales Univ. Sci. Budapest. Sect. Math., 2013, vol. 56, pp. 71–77.