WAVELET APPROXIMATION OF DETERMINISTIC AND RANDOM SIGNALS - CONVERTENCE PROPERTIES AND RATES

The multiresolution decomposition of deterministic and random signals and the resulting approximation at increasingly finer resolution is ex amined. Specifically, an nth-order expansion is developed for the erro r in the wavelet approximation at resolution 2-l of deterministic and random signals. The deterministic signals are assumed to have n contin uous derivatives, while the random signals are only assumed to have a correlation function with continuous nth-order derivatives off the dia gonal-a very mild assumption. For deterministic signals square integra ble over the entire real line, for stationary random signals over fini te intervals, and for nonstationary random signals with finite mean en ergy over the entire real line, the smoothness of the scale function c an be matched with the signal smoothness to substantially improve the quality of the approximation. In sharp contrast, this is feasible only in special cases for nonstationary random signals over finite interva ls and for deterministic signals which are only locally square integra ble.