Abstract
Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.
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This research has been supported by MTM2016-77054-C2-1-P (Ministerio de Economía, Industria y Competitividad, Spain).
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Sánchez Pérez, E.A. Maharam-type kernel representation for operators with a trigonometric domination. Aequat. Math. 91, 1073–1091 (2017). https://doi.org/10.1007/s00010-017-0507-6
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DOI: https://doi.org/10.1007/s00010-017-0507-6