Wavelet Analysis and Its Applications.

Time-frequency localization is one of the most essential features of the wavelet transform. It was shown that while high order Daubechies and Battle-Lemarie wavelets give poor time-frequency localizations, the Chui-Wang spline-wavelets provide asymptotically optimal time-frequency windows. On the other hand, we also showed that by using the scale 3 instead of 2, symmetry can be achieved by orthonormal wavelets with compact support. Multivariate wavelets, particularly those with matrix dilation, were studied, and the theory of oversampling frames was extended to this setting. Interpolating wavelets have distributional duals that lead to the notion of functional wavelet transform. Other extensions required a study of the stability issue and algorithmic construction in multivariate splines. Applications to systems theory lead to the study of Hankel approximation and localization of neural networks. AN