Convergence of subsequences of partial cubic sums of Fourier series in mean and almost everywhere
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Under certain assumptions on the regularity of a function Φ necessary and sufficient conditions are found for Φ under which the integrability of Φ(¦f¦) implies, for every function f measurable onTd, the existence of a subsequence of cubic sums of the Fourier series of f that converges to f in mean or almost everywhere.
KeywordsFourier Fourier Series
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- 1.S. V. Konyagin, “On the convergence of a subsequence of partial sums of multiple trigonometric Fourier series,” Trudy MIAN, 190, 102–116, Nauka, Moscow (1989).Google Scholar
- 2.R. Edwards, Fourier Series in Modern Applications, Vol. 2 [Russian translation], Mir, Moscow (1985).Google Scholar
- 3.R. D. Getsadze, “On convergence in measure of multiple Fourier series,” Soobshch. Akad. Nauk Gruz. SSR,122, No. 2, 269–271 (1986).Google Scholar
- 4.P. Stein, “On a theorem of M. Riesz,” J. Lond. Math. Soc.,8, 242–247 (1933).Google Scholar
- 5.J. Bourgain, “On the behavior of the constant in the Littlewood-Paley inequality,” Lect. Notes in Math.,1376, 202–208 (1989).Google Scholar
- 6.B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1981).Google Scholar
- 7.S. V. Konyagin, “On the divergence of subsequences of rectangular partial sums of dual trigonometric Fourier series almost everywhere,” Dokl. Rassh. Zased. IPM im. I. N. Vekua,5, No. 2, 71–74 (1990).Google Scholar
- 8.N. I. Akhiezer, Lectures on Theory of Approximation [in Russian], Nauka, Moscow (1965).Google Scholar
- 9.S. M. Nikol'skii, “Approximation in mean of functions by means of trigonometric polynomials,” Izv. Akad. Nauk SSSR. Ser. Mat.,10, No. 1, 207–256 (1946).Google Scholar