  Matrix type operators with the off-diagonal decay of polynomial or
sub-exponential types are revisited with weaker assumptions concerning row or
column estimates, still giving the continuity results for the frame type
operators. Such results are extended from Banach to Fr\'{e}chet spaces.
Moreover, the localization of Fr\'{e}chet frames is used for the frame
expansions of tempered distributions and a class of Beurling
ultradistributions.
