This chapter offers a detailed survey on intrinsically localized frames and
the corresponding matrix representation of operators. We re-investigate the
properties of localized frames and the associated Banach spaces in full detail.
We investigate the representation of operators using localized frames in a
Galerkin-type scheme. We show how the boundedness and the invertibility of
matrices and operators are linked and give some sufficient and necessary
conditions for the boundedness of operators between the associated Banach
spaces.Comment: 32 page