Consistency of D-dimensional Wavelet Estimators

To investigate estimator’s convergence rate over Lebesgue measurable space L _p (R ^d ), wavelet method was used and consistency under L _p norm was given. The main result is that if wavelet scaling function φ has compact support and satisfies condition S, then for any f∈ L _p (R ^d ),1≤ p<∞, we have $\begin{array}{c}\underset{n\to \infty }{\mathit{lim}}\end{array}$ E‖ f- $\begin{array}{c}{\widehat{f}}_n\end{array}$ ‖ _p =0.