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Strict local inclusion results between spaces of Fourier transforms

Bloom, W. (1982) Strict local inclusion results between spaces of Fourier transforms. Pacific Journal of Mathematics, 99 (2). pp. 265-270.

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Abstract

Let G denote a noncompact Hausdorff locally compact abelian group, Γ its character group, and write (Ls,lt)∧ for the space of Fourier transforms of functions in the amalgam (Ls,lt). We show that for 1 ≦ p < q ≦∞ the local inclusion (L1,lp)∧loc ⊂(L∞,lq)∧ is strict, that is, given any nonvoid open subset Ω of Γ there exists f ∈ (L∞,lq) such that f −ĝ does not vanish on Ω for any g ∈ (L1,lp). If in addition G is assumed to be second countable then we show there exists such an f independent of the choice of Ω. Of special interest is the case, included in the above results, where the amalgams (L1,lq), (L∞,lp) are replaced by Lp(G), Lq(G) respectively.

Publication Type: Journal Article
Murdoch Affiliation: School of Mathematical and Physical Sciences
Publisher: MSP
Copyright: 1982 Pacific Journal of Mathematics
URI: http://researchrepository.murdoch.edu.au/id/eprint/31409
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