Signal processing bases and the Jacobi group

The Jacobi group GJ can be used to produce bases for signal processing. Bases produced by GJ are functions on $H\times C$ where $H$ is the upper half complex plane. This thesis studies the conditions under which these bases are complete. A basis is complete provided that the only signal which is orthogonal to every basis element is the zero signal.; Completeness for continuously parametrized bases in Hilbert spaces is shown in the first part of this thesis, and in the second part, completeness is shown for bases for image processing. Methods from real analysis and analytic number theory are used in the proofs of most theorems.; A summary of results follows. If f is a holomorphic function on $H\times C$ which is square-integrable with respect to the Petersson inner product, then the action of GJ on f produces a complete basis in the space of square-integrable holomorphic functions on $H\times C$. Complete bases for L2($R$) and L2($R2$) are produced from the action of many different families of elements of GJ. Some of these bases are classical windowed Fourier and wavelet bases, and some of these bases are new. One of these new bases is tested by using it to reconstruct Chebyshev polynomials. Finally, the subgroup SL(2, $R$) of GJ is used to create two different complete bases for image processing.