\(\mathcal{I}_H\)-regular Borel measures on locally compact abelian groups (Q2330056)

Let \(G\) be an LCA group, \(H\) a closed subgroup, \(\Gamma\) the dual group of \(G\) and \(\Lambda\) the annihilator group of \(H\) in \(\Gamma\). Let \(\pi_H\) be the canonical homeomorphism from \(G\) onto the factor group \(G/H\) and \(\tilde{x} := \pi_H(x)\). For a non-empty subset \(S\) in \(G\) let \(P(S)\) denote the linear space of all trigonometric \(S\)-polynomials and let \(P(x+H) =: P(\tilde{x})\), \(x \in G\). Moreover let \(\mu\) be a regular finite non-negative measure on the Borel \(\sigma\)-algebra \(B(\Gamma)\). The measure \(\mu\) is called \(J_H\)-regular if and only if \[ \bigcap_{x\in G} C_\alpha P(x + H) = \bigcap_{\tilde{x}\in G/H}C_\alpha P(\tilde{x}) = \{0\}\] and is called \(J_H\)-singular if \(C_\alpha P(\tilde{x}) = L^\alpha(\mu)\). Here \(C_\alpha P(\tilde{x})\) denotes the closure of \(P(\tilde{x})\) in \(L^\alpha(\mu)\). \par A characterization of \(J_H\)-regular measures is given in terms of be Radon-Nikodym derivatives of some measures defined by elements of the annihilator. Moreover the Wold type decomposition is obtained and relations to the Whittaker-Shannon-Kotel'nikov theorem are discussed.