Abstract : There were three major research explorations. (1) Wavelets: Necessary and sufficient conditions on the wavelet, scaling function and projection kernel for given rates of convergence of wavelet expansions in the supremum and L (P) (Rd) norms have been given. (2) Image compression is developed using quasi-interpolant multivariate box splines and multi-resolution analysis has been developed. (3) Shallow Water Theory: A mathematical justification for the "shallow water theory for time-dependent two-dimensional flows of an inviscid, irrotational, incompressible fluid moving under the influence of gravity has been developed.