Extensions of the Heisenberg Group by Dilations and Frames

Abstract

Two standard tools for signal analysis are the short-time Fourier transform and the continuous wavelet transform. These tools arise as matrix coefficients of square integrable representations of the Heisenberg and affine groups, respectively, and discrete frame decompositions of L 2 arise from approximations of corresponding reproducing formulae. Here we study two groups, the so-called affine Weyl–Heisenberg and upper triangular groups, which contain both affine and Heisenberg subgroups. Generalized notions of square-integrable group representations allow us to fashion frames for L 2 and other function spaces. Such frames combine advantages of the short-time Fourier transform and wavelet transform and can be tailored to analyze specific types of signals.