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On the Fourier‐Coefficients of Vector‐Valued Functions
Author(s) -
König Hermann
Publication year - 1991
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19911520118
Subject(s) - mathematics , fourier transform , lorentz space , fourier series , sequence (biology) , scalar (mathematics) , banach space , lorentz transformation , regular polygon , mathematical analysis , space (punctuation) , function space , pure mathematics , vector valued function , type (biology) , differentiable function , fourier analysis , physics , geometry , ecology , linguistics , philosophy , classical mechanics , biology , genetics
We study the decay of the Fourier‐coefficients of vector‐valued functions F : T → X, X a Banach space. Differentiable functions f generally have absolutely summable Fourier‐coefficients, f (n) <, iff X is K ‐convex. More precise statements on the decay of f (n) for regular functions f can be given if X has Fourier‐type p . If f belongs to the Besov space, the sequence (||f(n)||) belongs to the Lorentz sequence space l t,v with 1/ t = λ + 1/max (u′, p′). This result is the best possible in the vector‐valued case and generalizes the well‐known scalar results.