Summability of general orthonormal Fourier series
2015 ◽
Vol 52
(4)
◽
pp. 511-536
Keyword(s):
S. Banach in [1] proved that for any function f ∈ L2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0. D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0. In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φn(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.
1980 ◽
Vol s3-41
(2)
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pp. 217-253
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1980 ◽
Vol 87
(3)
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pp. 383-392
2011 ◽
Vol 43
(3)
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pp. 688-711
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1987 ◽
Vol 10
(3)
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pp. 443-452
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1996 ◽
Vol 60
(3)
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pp. 405-420