Integrals, Singular

This thesis presents a comprehensive analysis of a relatively new transform for discrete time signals, called the Discrete Wavelet Transform (DWT). We find how this transform is connected with the already existing theory of perfect reconstruction filter banks and the recently introduced theory of multiresolution analysis. We use the conditions obtained from these two theories in order to understand the construction of wavelet filters, which also generate continuous functions that prove to constitute an orthonormal basis for the L$\sp2$ space. We also investigate the connection of this transform to the sampled wavelet series of nonorthogonal functions with good time-frequency localization properties. Finally, we see the way that the DWT maps a discrete signal in the phase plane and the applications that such representations incorporate.