POLYNOMIALS AND FUNCTIONS WITH FINITE SPECTRA ON LOCALLY COMPACT ABELIAN-GROUPS

In this paper we define polynomials on a locally compact Abelian group G and prove the equivalence of our definition with that of Domar. We explore the properties of polynomials and determine their spectre. We also characterise the primary ideals of certain Beurling algebras L(w)(1)(Z) on the group of integers Z. This allows us to classify those elements of L(w)(infinity)(G) that have finite spectrum. If phi is a uniformly continuous function with bounded differences then there is a Beurling algebra naturally associated with phi. We give a condition on the spectrum of phi relative to this algebra which ensures that phi is bounded. Finally we give spectral conditions on a bounded function on R that ensure that its indefinite integral is bounded.