Abstract
We consider Fourier coefficients for functions of bounded variation with respect to general orthonormal systems (GONS). We show that in general case the sequence of coefficients may have an arbitrarily prescribed order of vanishing. In this connection we seek for a class of GONS such that Fourier coefficients of a function from V(0, 1) satisfy the same inequality as in the case of classical systems (trigonometric, Walsh, and Haar ones). In the present paper we study this problem and related issues.
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References
A. M. Olevskii, “Orthogonal Series in Terms of Complete Series,” Matem. Sborn. 58(2), 707–748 (1962).
P. L. Ul’yanov, “Series with Respect to the Haar System,” Matem. Sborn. 63(3), 356–391 (1964).
V. Tsagareishvili, “On the Fourier Coefficients for General Orthonormal Systems,” Proc. A. RazmadzeMath. Inst. 124, 131–150 (2000).
N. J. Fine, “On theWalsh Functions,” Trans. Amer.Math. Soc. 65, 372–414 (1949).
G. Alexits, Convergence Problems of Orthogonal Series (New York-Oxford-London-Paris, Pergamon Press, 1961; In. Lit., Moscow, 1963).
B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transforms (Nauka, Moscow, 1987) [in Russian].
N. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961) [in Russian].
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Original Russian Text © L.D. Gogoladze and V.Sh. Tsagareishvili, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 8, pp. 14–23.
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Gogoladze, L.D., Tsagareishvili, V.S. The Fourier coefficients of functions with bounded variation. Russ Math. 57, 10–19 (2013). https://doi.org/10.3103/S1066369X13080021
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DOI: https://doi.org/10.3103/S1066369X13080021