Wavelets from Square-Integrable Representations

This paper was published in 1996 and has 122 citations in Litlas.

The continuous wavelet decompositions that arise from square-integrable representations of certain Lie groups on $L^2 (\mathbb{R}^n )$ are investigated. The groups are formed as the semidirect product of $\mathbb{R}^n $ with an n-dimensional subgroup H of $GL_n (\mathbb{R})$. There is a natural “translation and dilation” representation of such groups on $L^2 (\mathbb{R}^n )$. The basic formulas of Duflo and Moore, which lead to the resolution of the identity via a square-integrable representation, are given an elementary proof for this special case. Several two-dimensional examples are described. A method for discrete decompositions via frames is given using the representations under study.