Localization of Fr\'echet frames and expansion of generalized functions

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.