Abstract
We study the convergence of series of the Fourier-Vilenkin coefficients of functions represented as multiplicative convolutions. In the trigonometric case similar results are obtained by C. Onneweer, M. Izumi, and S. Izumi. Moreover, we consider certain analogs of I. Hirshman and W. Rudin transforms of Fourier coefficients. Some results are proved to be unimprovable in a certain sense.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transforms: Theory and Applications (Nauka, Moscow, 1987) [in Russian].
F. Schipp, W. Wade, and P. Simon, Walsh Series (Akad. Kiado, Budapest, 1990).
M. T. Chen, “The Absolute Convergence of Fourier Series,” Duke Math. J. 9(4), 803–810 (1942).
M. Izumi and S.-I. Izumi, “Absolute Convergence of Fourier Series of Convolution Functions,” J. Approx. Theory 1(1), 103–109 (1968).
C.W. Onneweer, “On Absolutely Convergent Fourier Series,” Arkiv Mat. 12(1), 51–58 (1974).
C.W. Onneweer, “Absolute Convergence of Fourier Series on Certain Groups,” Duke Math. J. 39(4), 599–610 (1972).
C. W. Onneweer, “Absolute Convergence of Fourier Series on Certain Groups,”. II, Duke Math. J. 41(3), 679–688 (1974).
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].
A. Zygmund, Trigonometric Series, Vol. 1 (Cambridge University Press, Cambridge, 1959; Mir, Moscow, 1965).
A. Zygmund, Trigonometric Series, Vol. 2 (Cambridge University Press, Cambridge, 1959; Mir, Moscow, 1965).
S. S. Volosivets, “Convergence of Fourier Series with Respect to Multiplicative Systems and the p-Fluctuation Continuity Modulus,” Sib. Matem. Zhurn. 47(2), 241–258 (2006).
J. Pal and P. Simon, “On a Generalization of the Concept of Derivative,” Acta Math. Hung. 29(1–2), 155–164 (1977).
G. Hardy, J. Littlewood, and G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934; Inostу Litу, Moscow, 1948).
L. Leindler, “Further Sharpening of Inequalities of Hardy and Littlewood,” Acta Sci. Math. (Szeged) 54(3–4), 285–289 (1990).
S. Kaczmarz and H. Steinhaus, Theory of Orthogonal Series (Chelsea, New York, 1951; Fizmatgiz, Moscow, 1958).
L. Leindler “Über Strukturbedingungen für Fourierreihen,” Math. Z. 88(5), 418–431 (1965).
N. K. Bari and S. B. Stechkin, “The Best Approximations and Differential Properties of Two Conjugate Functions,” Trudy Mosk. Matem. Obshchestva 5, 483–522 (1956).
O. Szasz, “Über die Fourierschen Reihen gewisser Funktionenklassen,” Math. Ann. 100, 530–536 (1928).
O. Szasz, “Fourier Series and Mean Moduli of Continuity,” Trans. Amer. Math. Soc. 42(3), 366–395 (1937).
I. I. Hirshman, “Multiplier Transformations,” Duke Math. J. 26(2), 222–242 (1959).
W. Rudin, “Some Theorems on Fourier Coefficients,” Proc. Amer. Math. Soc. 10(6), 855–859 (1959).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the blessed memory of Petr Lavrent’evich Ul’yanov
Original Russian Text © S.S. Volosivets, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 11, pp. 27–39.
About this article
Cite this article
Volosivets, S.S. Convergence of series of Fourier coefficients for multiplicative convolutions. Russ Math. 52, 23–34 (2008). https://doi.org/10.3103/S1066369X08110030
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X08110030