Wavelet Estimation Of Multivariate Functions Based On Multichannel Deconvolution Model

Multichannel deconvolution model is widely applied to signal processing,image restoration and so on.Since wavelet basis has the property of time-frequency localization and characterization for function space,it performs well in estimating the unknown function of deconvolution model.This dissertation applies the wavelet method to study the multichannel deconvolution model with Gaussian white noise.For the multivariate multichannel deconvolution model,we study the mean Lp(1≤p<∞)-consistency of wavelet estimator.We construct the linear wavelet estimator by using Fourier analysis,weighted summation and wavelet projection method.Without assumption of smoothness of the unknown function,we prove that the wavelet estimator satisfies the mean Lp(1≤p<∞)-consistency for two kinds of blurring functions.This provides theoretical support for studying the convergence rate of wavelet estimator.Based on the above work,we study the convergence rate of wavelet estimator for Lp(1≤p<∞)-risk.For the super-smooth case,we give the Lp(1≤p<∞)-risk convergence rate of the linear wavelet estimator.For the regular-smooth case,we construct a nonlinear wavelet estimator by the hard threshold method,and give its Lp(1≤p<∞)risk convergence rates.Here,the wavelet estimators are adaptive,and their convergence rates depend on the space dimension m.Moreover,the convergence rate of the regularsmooth case is better than that of the super-smooth case.Since the derivative function plays significant roles in application,this dissertation studies the Lp(1≤p<∞)-risk of wavelet estimators of derivative function for univariate multichannel deconvolution model.We construct adaptive wavelet estimators in both the regular-smooth and super-smooth cases,and give their Lp(1≤p<∞)-risk convergence rates.It turns out that the convergence rates depend on the derivative order d in the regular-smooth case,while it does not depend on d in the super-smooth case.The above work can provide theoretical preparation for the estimation of partial derivative function of multivariate function.