Wavelet approximation of deterministic and random signals: convergence properties and rates

The multiresolution decomposition of deterministic and random signals and the resulting approximation at increasingly finer resolution is examined. Specifically, an nth-order expansion is developed for the error in the wavelet approximation at resolution 2^-l of deterministic and random signals. The deterministic signals are assumed to have n continuous derivatives, while the random signals are only assumed to have a correlation function with continuous nth-order derivatives off the diagonal-a very mild assumption. For deterministic signals square integrable over the entire real line, for stationary random signals over finite intervals, and for nonstationary random signals with finite mean energy over the entire real line, the smoothness of the scale function can 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 intervals and for deterministic signals which are only locally square integrable