  The family of harmonic Hilbert spaces is a natural enlargement of those
classical $L^{2}$-Sobolev space on $\R^d$ which consist of continuous
functions. In the present paper we demonstrate that the use of basic results
from the theory of Wiener amalgam spaces allows to establish fundamental
properties of harmonic Hilbert spaces even if they are defined over an
arbitrary locally compact abelian group $\G$. Even for $\G = \R^{d}$ this new
approach improves previously known results. In this paper we present results on
minimal norm interpolators over lattices and show that the infinite minimal
norm interpolations are the limits of finite minimal norm interpolations. In
addition, the new approach paves the way for the study of stability problems
and error analysis for norm interpolations in harmonic Hilbert and Banach
spaces on locally compact abelian groups.
