Convergence properties of wavelet estimators with multiple sampling rates

Adaptive sampling schemes with multiple sampling rates have the potential to significantly improve the efficiency and effectiveness of methods for signal analysis. For example, in the case of equipment which transmits data continuously, multi-rate methods can reduce the cost of transmission. For equipment which transmits data only periodically they can reduce the costs of both storage and transmission. When multiple sampling rates are used in connection with wavelet estimators, the most natural algorithms for rate-switching are arguably those based on threshold-crossings by wavelet coefficients. In this paper we study the performance of such algorithms, and show that even simple threshold-crossing rules can achieve near-optimal convergence rates. A new mathematical model is suggested for assessing performance, combining the simplicity and familiarity of global approaches with an account of the local variation to which multi-rate sampling responds.