On hilbert, Fourier, and Wavelet transforms

Several new families of wavelets, which are derived from earlier work by the author on Teichmuller theory, are introduced and studied. These wavelets are associated with the modular group rather than the affine group of the real line as for classical wavelets. In further contrast to the usual setting, there are explicit and essentially algebraic formulas relating wavelet and Fourier expansions. Furthermore, rather than relying on equally spaced multiscale sampling methods, the wavelet expansions introduced here rely instead on an explicit, nonuniformly spaced, multiscale sampling method that depends upon the Farey sequence of classical number theory. These ingredients combine to give novel methods for calculating wavelet, Fourier, and Hilbert transforms with potential applications to myriad problems in electrical engineering. By employing a new characteristic length associated to 2 × 2 matrices, finiteness and convergence properties of the wavelet expansions are derived. For functions with more than 5/2 derivatives in L2, one of our wavelet families forms a frame and the Hilbert transform is calculable in hyperbolic coordinates. © 2002 Wiley Periodicals, Inc.