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Classification and geometric aspects of vector valued Fourier transforms
Author(s) -
Park In Sook
Publication year - 2008
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.200310625
Subject(s) - mathematics , abelian group , convexity , bounded function , locally compact space , pure mathematics , fourier transform , norm (philosophy) , order (exchange) , banach space , type (biology) , bounded operator , integer (computer science) , combinatorics , discrete mathematics , mathematical analysis , biology , ecology , computer science , finance , political science , financial economics , law , economics , programming language
It is shown that for any locally compact abelian group and 1 ≤ p ≤ 2, the Fourier type p norm with respect to of a bounded linear operator T between Banach spaces, denoted by ‖ T |ℱ p ‖, satisfies ‖ T |ℱ p ‖ ≤ ‖ T |ℱ p ‖, where is the direct product of ℤ 2 , ℤ 3 , ℤ 4 , … It is also shown that if is not of bounded order then C n p ‖ T |ℱ p ‖ ≤ ‖ T |ℱ p ‖, where is the circle group, n is a onnegative integer and C p = . From these inequalities, for any locally compact abelian group ‖ T |ℱ 2 ‖ ≤ ‖ T |ℱ 2 ‖, and moreover if is not of bounded order then ‖ T |ℱ 2 ‖ = ‖ T |ℱ 2 ‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)