Behaviour approximated on subgroups (Q1030197)

The author considers the question to what extent the behaviour of a function over a group can be inferred from that of its restrictions to appropriate substructures. First, the circle and Cantor groups are treated and subsequently, connected locally compact abelian groups (LCA groups, for short) are dealt with. By definition, a \textit{\(d\)-subgroup} of a connected locally compact abelian group is a subgroup of the form \((r_1\mathbb Z\times\cdots r_n\mathbb Z)\times C\), where \(r_j\in \mathbb R\), \(1\leq j\leq n\) and \(C\) is a compact subgroup. Amongst others, the following results are proved: (1) Suppose that \(G\) is a connected LCA group and \(f\) is a continuous complex-valued function on \(G\) whose restriction to each \(d\)-subgroup of \(G\) is bounded. Then \(f\) is bounded on \(G\); (2) Suppose that \(G\) is a connected LCA group, \(f\) is continuous on \(G\) and for each \(d\)-subgroup \(H\) of \(G\) there exists \(c_H\in \mathbb C\) such that \((f-c_H)|_{H}\) vanishes at \(\infty\). Then for some \(c\in \mathbb C\), \((f-c)\) vanishes at \(\infty\) on \(G\). The author also makes a brief discussion on the difficult problem of identifying substructures in Computer Science.