Classification and geometric aspects of vector valued Fourier transforms |
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Authors: | In Sook Park |
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Institution: | 1. BK Office 1215 Division of Electrical Engineering, Department of Electrical Engineering & Computer Science, Korea;2. Advanced Institute of Science and Technology, 373‐1 Kuseong‐dong, Yuseong‐gu, Taejeon 305‐701, Republic of Korea |
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Abstract: | 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 |?????p‖, satisfies ‖T |?????p‖ ≤ ‖T |?????p‖, where ?? is the direct product of ?2, ?3, ?4, … It is also shown that if ?? is not of bounded order then Cnp ‖T |?????p‖ ≤ ‖T |?????p‖, where ?? is the circle group, n is a onnegative integer and Cp = . From these inequalities, for any locally compact abelian group ?? ‖T |?????2‖ ≤ ‖T |?????2‖, and moreover if ?? is not of bounded order then ‖T |?????2‖ = ‖T |?????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) |
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Keywords: | Banach space operator Fourier transform vector valued function locally compact abelian group dual group |
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