Abstract
We study the problem of strong summability of Fourier series in orthonormal systems of polynomial-type functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of systems under consideration.
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References
G. H. Hardy and J. E. Littlewood, “Sur la serie de Fourier d’une fonction a carre summable,” Comput Revs., 153, 1307–1309 (1913).
G. H. Hardy and J. E. Littlewood, “On the strong summability of Fourier series,” Proc. London Math. Soc., 26, 233–286 (1926).
T. Carleman, “A theorem concerning Fourier series,” Proc. London Math. Soc., 21, 483–492 (1923).
G. H. Hardy and J. E. Littlewood, “The strong summability of Fourier series,” Fund. Math., 25, 162–189 (1935).
J. Marcinkiewicz, “Sur la summability forte des series de Fourier,” J. London Math. Soc., 14, 162–168 (1939).
A. Zygmund, “On the convergence and summability of power series on the circle of convergence,” Proc. London Math. Soc., 47, 326–350 (1941).
K. Tandori, “Uber die starke summation von Fourierreihen,” Acta Sci. Math., 16, 65–73 (1955).
K. Tandori, “Berichtigung Arbeit “Uber die starke summation von Fourierreihen,” Acta Sci. Math., 29, No. 3–4 (1968).
O. D. Gabisoniya, “On the points of strong summability of Fourier series,” Mat. Zametki, 14, No. 5, 615–628 (1973).
G. Hardy, J. Littlewood, and G. Polya, Inequalities [Russian translation], Inostr. Lit., Moscow (1948).
I. Ya. Novikov and V. A. Podin, “Characterization of the points of p-strong summability of trigonometric Fourier series,” Izv Vyssh. Uchebn. Zaved., 9, 58–62 (1988).
K. Tandori, “Uber die Cesaroche summierbarkeit der orthogonalen Polynomreihen. II,” Acta. Math. Sci. Hung., 5, No. 3–4, 236–253 (1954).
G. Alexits, Problems of Convergence of Orthogonal Series [Russian translation], Inostr. Lit., Moscow (1963).
G. Alexits and D. Kralik, “Uber Approximation im starken Sinne,” Acta Scient. Math. Szeged., 26, No. 1–2., 93–101 (1965).
A. Zygmund, Trigonometric Series [Russian translation], Mir, Moscow 1965.
N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow 1961.
A. Ya. Ismailov, “On the estimation of derivatives of polynomials of many variables,” Doki Akad. Nauk SSSR, 4, 239–243 (1956).
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Stepanets, A.I., Lasuriya, R.A. Strong summability of orthogonal expansions of summable functions. I. Ukr Math J 48, 294–313 (1996). https://doi.org/10.1007/BF02372053
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DOI: https://doi.org/10.1007/BF02372053