Wavelet transform on the torus: A group theoretical approach

We construct a continuous wavelet transform (CWT) on the torus T2T2 following a group-theoretical approach based on the conformal group SO(2,2)SO(2,2). The Euclidean limit reproduces wavelets on the plane R2R2 with two dilations, which can be defined through the natural tensor product representation of usual wavelets on RR. Restricting ourselves to a single dilation imposes severe conditions for the mother wavelet that can be overcome by adding extra modular group SL(2,Z)SL(2,Z) transformations, thus leading to the concept of modular wavelets. We define modular-admissible functions and prove frame conditions.