On Herz'S Extension Theorem

We present a self-contained proof of the following famous extension theorem due to Carl Herz. A closed subgroup H of a locally compact group G is a set of p-synthesis in G if and only if, for every u is an element of A(p)(H) boolean AND C-00(H) and for every epsilon > 0, there is v is an element of A(p)(G) boolean AND C-00(G), an extension of u, such that parallel to v parallel to A(p)(G) < parallel to u parallel to A(p)(H) + epsilon.