An invariance property of spectral synthesis
Karel de Leeuw, Carl Herz · Illinois Journal of Mathematics · 1965
In this paper we study the behavior under group representations of certain spaces of functions that arise in harmonic analysis.We indicate in this section the essential nature of the results we obtain.Let G1 and G2 be locally compact abelian groups, h: G1--G2 a faithful representation, that is, a one-one continuous mapping that is an algebra homomorphism.For example, take G1 to be the real line, G the torus, and h an injection of G as a dense one-parameter subgroup of G or G1 arbitrary, G the Bohr compactification of G, and h the canonical imbedding of G1 ia G.Let $1 be a compact subset of G and $2 its image in G under h.If A (S) is the space of functions in S that are restrictions of Fourier trans- forms in G, it is easy to demonstrate the inclusion (1.1) {f o h f e A(S.)} c A(SI).That equality holds in (1.1) is the main point of Theorem 1 below.Let 0k be the dual group of G, the adjoint of h G --G.If B'(S) is the space of bounded continuous functions on having spectrum contained in S, it is easy to establish the inclusion (1.2) {4):e B'(S)} c B'(S2).That equality holds in (1.2) is the main point of Theorem 2 below.From the fact that equality holds in (1.1) and (1.2) it is possible to con- clude that either both $1 and S are sets of spectral synthesis, or neither is a set of spectral synthesis.A slightly more general result is Theorem 4 below.A consequence of equality in (1.2) is the following, which we prove as Theorem 5. A bounded function on a discrete abelian group G, that is dis- continuous in some locally compact topology on G, cannot have in its spectrum only characters continuous in that topology.2. In this section we define the spaces of functions with which we shall be concerned.Let G be a locally compact abelian group, its dual, C() the Banach space of bounded continuous complex-valued functions on , M() the Banach algebra of finite measures on ( under convolution.We denote by A(G) the Banach algebra of functions on G that are Fourier transforms of measures in M().A (G) is isometrically isomorphic to M(() under the Fourier transform mapping --.