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Fourier type 2 operators with respect to locally compact abelian groups
Author(s) -
Hinrichs Aicke,
Piotrowski Mariusz
Publication year - 2004
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.200310212
Subject(s) - mathematics , abelian group , fourier transform , type (biology) , pure mathematics , bounded function , fourier inversion theorem , locally compact space , operator (biology) , fourier analysis , mathematical analysis , fractional fourier transform , chemistry , ecology , biochemistry , repressor , gene , transcription factor , biology
A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that∥ T $\hat f$ ∥ 2 ≤ c ∥ f ∥ 2 holds for all X ‐valued functions f ∈ L X 2 ( G ) where $\hat f$ is the Fourier transform of f . We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of ℝ n . (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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