For almost-periodic Besicovitch functions whose spectrum has a limit point only at infinity, we establish criteria for the absolute Cesàro summability of their Fourier series of order greater than –1.
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L. V. Grepachevskaya, “Absolute summability of orthogonal series,” Mat. Sb., 65(107), No. 3, 370–389 (1964).
L. V. Grepachevskaya, “On absolute summability by Cesàro, Riesz, and Zygmund methods,” Dokl. Akad. Nauk SSSR, 155, No. 3, 173–179 (1964).
L. Leindler, “Über die absolute Summierbarkeit der Orthogonalreihen,” Acta Sci. Math. (Szeged), 22, 243–268 (1961).
K. Tandori, “Über die orthogonalen Funktionen IX.(Absolute Summation),” Acta Sci. Math. (Szeged), 292–299 (1960).
M. F. Timan, “On absolute convergence and summability of Fourier series,” Soobshch. Akad. Nauk Gruz. SSR, 26, No. 6, 641–646 (1961).
A. F. Timan, Approximation Theory of Functions of Real Variables [in Russian], Fizmatgiz, Moscow (1960).
S. B. Stechkin, “On absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 2, 37–40 (1955).
B. M. Levitan, Almost-Periodic Functions [in Russian], Gostekhteorizdat, Moscow (1953).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1989).
Yu. Kh. Khasanov, “On absolute convergence of Fourier series of functions almost periodic in the Besicovitch sense,” Dokl Akad. Nauk Resp. Tadzh., 39, No. 9-10, 42–47 (1996).
G. Sunouchi, “On the absolute summability of Fourier series,” J. Math. Soc. Jpn., 1, No. 2, 57–65 (1949).
A. Zygmund, Trigonometric Series, Vol. 1, Cambridge University, Cambridge (1959).
Yu. Kh. Khasanov, “On absolute convergence of Fourier series of almost-periodic functions,” in: Abstracts of the Conference “Constructive Theory of Functions,” St. Petersburg (1992), pp. 66–68.
S. A. Baron, Introduction to the Theory of Summable Series [in Russian], Valgus, Tallin (1977).
G. Alexits, Convergence Problems of Orthogonal Series [Russian translation], Inostrannaya Literatura, Moscow (1963).
Yu. Kh. Khasanov, “Modulus of smoothness as an apparatus for investigation of absolute convergence of Fourier series of almost-periodic functions,” in: Differential and Integral Equations and Their Applications [in Russian], Dushanbe (1996), pp. 67–72.
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Timan, M.F., Khasanov, Y. On the absolute summability of Fourier series of almost-periodic besicovitch functions. Ukr Math J 61, 1499–1510 (2009). https://doi.org/10.1007/s11253-010-0292-z
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DOI: https://doi.org/10.1007/s11253-010-0292-z