Directional time frequency analysis via continuous frame

Grafakos and Sansing \cite{GS} have shown how to obtain directionally sensitive time-frequency decompositions in $L^2({\text{mr}}^n)$ based on Gabor systems in $\text{ltr};$ the key tool is the “ridge idea,” which lifts a function of one variable to a function of several variables. We generalize their result by showing that similar results hold starting with general frames for $L^2(\text{mr}),$ both in the setting of discrete frames and continuous frames. This allows to apply the theory for several other classes of frames, e.g., wavelet frames and shift-invariant systems. We will consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretize the representations using $ϵ$-nets.