Beurling Spectra of Functions on Locally Compact Abelian Groups

الأصل · EN

Let G be a locally compact abelian topological group. For locally bounded measurable functions φ: G→ C we discuss notions of spectra for φ relative to subalgebras of L¹(G). In particular we study polynomials on G and determine their spectra. We also characterize the primary ideals of certain Beurling algebras Lw¹(Z) on the group of integers Z. This allows us to classify those elements of Lw¹(G) that have finite spectrum. If φ is a uniformly continuous function whose differences are bounded, there is a Beurling algebra naturally associated with φ. We give a condition on the spectrum of φ relative to this algebra which ensures that φ is bounded. Finally we give spectral conditions on a bounded function on R that ensure that its indefinite integral is bounded.