On generalized bounded variation functions on Vilenkin groups andapplications
Keywords:
Vilenkin Group, Functions of bounded variation, Summability by Cesaro methodAbstract
In the present paper, certain classes of functions of weighted bounded
oscillation defined on bounded Vilenkin groups. For such
classes, we employ the summability methods of the theory of double
Vilenkin-Fourier series.
References
G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinshtein, Multiplikativnye sistemy funktsii
i garmonicheskii analiz na nulmernykh gruppakh. (Russian) [[Multiplicative systems of functions
and harmonic analysis on zero-dimensional groups]] Elm, Baku, 1981. 180 pp.
M Avdispahic. Concepts of generalized variation on vilenkin groups and convergence of Fourier
Vilenkin series. In Colloq. Math. Soc. JÆnos Bolyai, volume 49, pages 145 163, 1985.
M Avdispahic and N Memic. On the Lebesgue test for convergence of Fourier series on unbounded
Vilenkin groups. Acta Math. Hungar, 129(4):381392, 2010.
A. Bakhvalov. On local behaviour of multi-dimensional harmonic variation. Izvestiya: Mathematics,
(4):641, 2006.
A. Bakhvalov. Convergence and localization of multiple Fourier series for classes of bounded λ
variation. Moscow University Mathematics Bulletin, 63:8591, 2008.
A. Bakhvalov. On the local behaviour of the multidimensional λ-variation. Sbornik: Mathematics,
(11):1563, 2010.
A. Bakhvalov. Continuity in λ-variation and summation of multiple Fourier series by Ces ro meth
ods. Mathematical Notes, 90:469484, 2011.
E. I. Berezhnoi. Spaces of functions of generalized bounded variation. II. Problems of the uniform
convergence of Fourier series. Sibirsk. Mat. Zh., 42(3):515 532, i, 2001.
E. I. Berezhnoi. Two-sided estimates of the K-functional for spaces of functions of generalized
bounded variation. Funct. Anal. Appl., 56(1):1926, 2022. Translation of Funktsional. Anal. i
Prilozhen. 56 (2022), no. 1, 2636.
E. I. Berezhnoi. Two-sided estimates of Lebesgue constants for spaces of functions of generalized
bounded variation and Hlder spaces. J. Math. Anal. Appl., 537(2):Paper No. 128347, 2024.
Z. Chanturiya. The modulus of variation of a function and its application in the theory of Fourier
series. In Dokl. Akad. Nauk SSSR, volume 214, pages 63 66, 1974.
M. I. Dyachenko and D. Waterman. Convergence of double Fourier series and W-classes. Trans.
Amer. Math. Soc., 357(1):397407, 2005.
U Goginava. On the uniform convergence of multiple trigonometric Fourier series. East J. Approx,
(3):253266, 1999.
U Goginava. CesÆro means of negative order of double Fourier series and generalized bounded
variation. Siberian Mathematical Journal, 54(6):10051013, 2013.
U. Goginava and A. Sahakian. On the convergence of double Fourier series of functions of bounded
partial generalized variation. East J. Approx., 16(2):153165, 2010.
U. Goginava and A. Sahakian. On the convergence of CesÆro means of negative order of double
trigonometric Fourier series of functions of bounded partial generalized variation. Acta Scientiarum
Mathematicarum, 77:451471, 2011.
U. Goginava and A. Sahakian. Convergence of double Fourier series and generalized λ-variation.
Georgian Mathematical Journal, 19(3):497509, 2012.
U. Goginava and A. Sahakian. On the convergence of multiple Fourier series of functions of bounded
partial generalized variation. Anal. Math., 39(1):4556, 2013.
U. Goginava and A Sahakian. Summability of multiple Fourier series of functions of bounded gen
eralized variation. Proceedings of the Steklov Institute of Mathematics, 280(1):144155, 2013.
U. Goginava and A. Sahakian. Convergence and summability of multiple Fourier series and gener
alized variation. Bull. TICMI, 18(1):3654, 2014.
U. Goginava and A. Sahakian. On the convergence and summability of double Walsh-Fourier series
of functions of bounded generalized variation. Izv. Nats. Akad. Nauk Armenii Mat., 49(6):5165,
G. H. Hardy, On double fourier series, and especially those which represent the double zeta-function
with real and incommensurable parameters. Quart. J. Math, 37(1):5379, 1906.
C. Jordan. Sur la series de fourier. CR Acad. Sci., Paris, 92:228230, 1881.
C. W Onneweer. Uniform convergence of fourier series on groups. ii. The Rocky Mountain Journal
of Mathematics, 1(4):623631, 1971.
C. W. Onneweer and D. Waterman. Uniform convergence of Fourier series on groups. i. Michigan
Mathematical Journal, 18(3):265 273, 1971.
A. I Sablin. λ-variation and Fourier series. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika,
(10):6668, 1987.
A Sahakian. On convergence of double Fourier series of functions of bounded harmonic variation.
In Proc. Armenian Acad. Sci. Math. Ser., volume 21, pages 517529, 1986.
F. Schipp, W. R. Wade, and P. Simon. Walsh series. Adam Hilger, Ltd., Bristol, 1990. An introduc
tion to dyadic harmonic analysis, With the collaboration of J. PÆl.
Gv. Shavardenidze. On the convergence of Ces ro means of negative order of Vilenkin-Fourier series.
Studia Scientiarum Mathematicarum Hungarica, 56(1):2244, 2019.
T. Tepnadze. On the approximation properties of Ces ro means of negative order of double Vilenkin
Fourier series. Ukran. Mat. Zh., 72(3):391406, 2020.
D. Waterman. On convergence of Fourier series of functions of generalized bounded variation. Studia
math, 44(2):107117, 1972.
N. Wiener. The quadratic variation of a function and its Fourier coe cients. Journal of Mathematics
and Physics, 3(2):7294, 1924.
L. C Young. Sur un generalization de la notion de variation de Winer et sur la convergence de series
de fourier. CR Acad. Sci. Paris, 204:470472, 1937
