The current project aims to pursue further developments in three areas situated in the intersection of functional analysis and time-frequency analysis. We aim to do deal with decay properties of wavelet transforms, sequence space representations of major locally convex spaces of smooth functions and multidimensional quasiasymptotics. The intention is to employ a mixture of classical analysis and abstract functional analytic tools in order to obtain new results in each of these branches. Our specific objectives are to: 1) Obtain optimal continuity results for the wavelet transform in the context of specified rapid decay in time and scale. Developing such a framework will enable us to provide new tools for the study of pointwise regularity and asymptotic behavior of functions. In particular, we aim to quantify the right wavelet decay properties characterizing when a function is real analytic. 2) Establish, in a systematic way, isomorphisms between a large spectrum of subspaces of the Denjoy-Carleman classes and sequence spaces. These representations will provide great insight into the topological invariants of the considered subspaces. Our considerations will be closely connected to Gabor frame theory. 3) Provide structural theorems for the quasiasymptotic behavior of generalized functions in dimension 2 or higher. New insights into this long standing open problem will further enhance our understanding of the asymptotic behavior, enabling greater applicability.