Abstract
We give some necessary and sufficient conditiosn for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray, and Č.V. Stanojević and extend some results of F. Móricz proved for Walsh–Fourier series.
REFERENCES
Č. V. Stanojević and V. B. Stanojevic, “Generalizations of the Sidon–Telyakovskiĭ theorem,” Proc. Am. Math. Soc. 101, 679–684 (1987). https://doi.org/10.1090/s0002-9939-1987-0911032-2
S. A. Telyakovskii, “Concerning a sufficient condition of Sidon for the integrability of trigonometric series,” Math. Notes Acad. Sci. USSR 14, 742–748 (1973). https://doi.org/10.1007/BF01147448
S. Yu. Sheng, “The extension of the theorems of Č. V. Stanojević and V. B. Stanojević,” Proc. Am. Math. Soc. 110, 895–904 (1990). https://doi.org/10.1090/s0002-9939-1990-1031672-2
W. O. Bray and Č. V. Stanojević, “Tauberian L 1-convergence classes of Fourier series. I,” Trans. Am. Math. Soc. 275, 59–69 (1983). https://doi.org/10.1090/s0002-9947-1983-0678336-3
F. Móricz, “Sidon-type inequalities,” Publ. Inst. Math. (Beograd) 48 (62), 101–109 (1990).
B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transforms: Theory and Applications, Mathematics and Its Applications, Vol. 64 (Nauka, Moscow, 1987; Springer, Dordrecht, 1991). https://doi.org/10.1007/978-94-011-3288-6
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).
N. Ya. Vilenkin, “On a class of complete orthonormal systems,” Transl., Ser. 2, Am. Math. Soc. 28, 1–35 (1963). https://doi.org/10.1090/trans2/028/01
J. Pal and P. Simon, “On a generalization of the concept of derivative,” Acta Math. Acad. Sci. Hung. 29, 155–164 (1977).
S. S. Volosivets, “Approximation of the function of bounded p-fluctuation by polynomials on multiplicative systems,” Anal. Math. 21 (1), 61–77 (1995).
F. Schipp, W. R. Wade, and P. Simon, Walsh Series. An Introduction to Dyadic Analysis (Akad. Kiado, Budapest, 1990).
S. S. Volosivets and T. V. Likhacheva, “Sidon-type inequalities and strong approximation by Fourier sums in multiplicative systems,” Sib. Math. J. 57, 486–497 (2016). https://doi.org/10.1134/s0037446616030095
F. Móricz, “On L 1-convergence of Walsh–Fourier series. II,” Acta Math. Hung. 58, 203–210 (1991). https://doi.org/10.1007/bf01903561
F. Móricz, “On L 1-convergence of Walsh–Fourier series. I,” Rend. Circolo Matematico Palermo 38, 411–418 (1989). https://doi.org/10.1007/bf02850023
Č. V. Stanojević, “Classes of L 1-convergence of Fourier and Fourier–Stieltjes series,” Proc. Am. Math. Soc. 82, 209–215 (1981). https://doi.org/10.1090/s0002-9939-1981-0609653-4
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Translated by E. Oborin
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Agafonova, N.Y., Volosivets, S.S. Integrability of Series with Respect to Multiplicative Systems and Generalized Derivatives. Russ Math. 68, 1–10 (2024). https://doi.org/10.3103/S1066369X24700142
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DOI: https://doi.org/10.3103/S1066369X24700142