Abstract
Let G be a locally compact abelian group, G its dual. For any ideal I of L'(G) let cosp(I) denote the cospectrum of I; this is defined as the closed set in G formed by the common zeros of the Fourier transforms of all functions in I. Consider two closed ideals I, I2 with disjoint cospectra. Then the algebraic sum I, + I2, which is again an ideal, has empty cospectrum. But II + 12 may not be closed in L'(G); this can be shown by examples. In case it is closed, we have, by Wiener's theorem, I, + I2 = L'(G). This suggests the following definition. Let two (non-empty) disjoint subsets A1, A2 of G be given. We say that the pair (A1, A2) has the closure property with respect to L'(G) if all pairs of closed ideals I, I2 c L'(G) such that
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.