A time-frequency analysis of the properties of orthogonal transforms

This paper deals with the problem of selecting an optimal base for representing a given signal. This problem, which is important in coding and filtering applications, has often been treated from a statistical point of view: the signal is assumed to be part of an ensemble with known statistics and the optimal base is the one which is best on average for this ensemble. This paper takes a different point of view: the signal of interest is assumed to be part of a signal space of locally band-limited functions. The paper shows that the optimal base for the signal space of locally band-limited signals consists of warped polynomials, which were introduced earlier by the author. It also shows that the local bandwidth point of view easily explains why, e.g., wavelet-approximation often outperforms approximation by cosines