Covariant Functional Calculi from the Affine Groups

Abstract

Invoking the Clifford-Hermite Wavelets from Clifford analysis, we use the covariances of affine groups to construct a kind of functional calculi for several non-commuting bounded operators. Functional calculi are the intertwining transforms between the representations of affine groups in the space $L^2(\mathbb R^m)$ and in the space of bounded operators. It turns out that the Weyl calculus is the value of this new functional calculus at the identity of affine groups. Our approach is inspired by the mathematical ideas contained in the paper ``V. V. Kisil. Wavelets in Banach spaces. Acta Appl. Math. 1999, {\bf 59}(1): 79-109".