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The Wiener Spectrum in Spectral Synthesis
Author(s) -
Benedetto John J.
Publication year - 1975
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm197554291
Subject(s) - mathematics , ideal (ethics) , abelian group , spectrum (functional analysis) , infinity , fourier transform , pure mathematics , locally compact space , space (punctuation) , arithmetic , mathematical analysis , physics , philosophy , epistemology , quantum mechanics , linguistics
Let E be a closed subset of a locally compact Abelian group Г and let k ( E ) be the space of absolutely convergent Fourier transforms on Г that vanish on E . (1) k ( E ) is characterized as an ideal of arithmetic means defined by the behavior of pseudo‐measures at infinity; and so spectral synthesis holds for an ideal I if I and its corresponding ideal of arithmetic means coincide. (2) We then prove that it is possible to synthesize pseudomeasures at infinity in a manner that runs parallel to Beurling's synthesis of weighted spaces. These two results are closely related by the notion of Wiener's spectrum.