Abstract
It is shown that for every l∞-function f and for every ɛ, ɛ>0, there exists a function g such that mes {t=g} <ɛ, while the partial sums of the Fourier and Fourier-Walsh series of the function g are uniformly bounded by the number C log (ε−1)∥f∥∞. In the proof we make use of the characterization of the dyadic space H1, ∞ in terms of atomic decompositions (it is, apparently, new).
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 67–75, 1986.
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Kislyakov, S.V. A correction theorem and the dyadic space H1, ∞ . J Math Sci 42, 1584–1590 (1988). https://doi.org/10.1007/BF01665044
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DOI: https://doi.org/10.1007/BF01665044