Behaviour approximated on subgroups

The recovery of behaviour from its approximation over substructures is fraught with pathology. Here the extent is considered to which the behaviour of a continuous function on a locally compact Abelian group can be approximated by its behaviour on proper closed subgroups. Known results are summarised when the behaviour concerns integrability and the group is the circle; then boundedness and other limiting behaviour ‘at infinity’ are considered for more general groups. It is shown that if a continuous function is bounded on each proper closed subgroup of a connected locally compact Abelian group then it is bounded on the whole group. As befits this Festschrift, the techniques are predominantly topological. In passing we reflect on criteria for the difficult problem of identifying ‘substructures’ in Computer Science.