利用多小波构造整个Sobolev空间的不规则采样逼近及研究其精度的提升

The sampling approximation for Sobolev space H^s(R^d) is an important problem in wavelet analysis, where s>d/2. Its related theory has been applied to various research domains such as differential equation and signal analysis. The existing sampling approximation, however, just holds for the subspaces of H^s(R^d). The sampling approximation holding for the whole H^s(R^d) has not been established. In this project, based on dual multiwavelets of (H^s(R^d),H^-s(R^d)), we shall establish the irregular sampling approximation for the whole H^s(R^d). Moreover, we shall investigate how to lift approximating accuracy. Therefore, we focus on the following contents: (1)Constructing the dual multiwavelets of (H^s(R^d),H^-s(R^d)). (2)For any function in H^s(R^d), we shall estimate its scale-truncated error and shift-perturbed error of the multiwavelet series, used to express the function by the multiwavelets in (H^s(R^d),H^-s(R^d)). (3)Based on a special pair of dual multiwavelets, we shall establish the sampling approximation holding for the whole H^s(R^d). (4)Without requiring the function belong to the approximating space as well as the sampling set condition hold, we shall lift the approximating accuracy by iterative method. When mentioning the applicable scope of sampling approximation as well as lifting approximating accuracy, our sampling approximation to be established in this project will be a breakthrough for the existing sampling approximation theory.

Sobolev空间H^s(R^d)的采样逼近是小波分析的一个重要课题，其理论在微分方程、信号分析等领域中有广泛应用，其中s>d/2。然而，现有的采样逼近大多只适用于其子空间，适用于整个H^s(R^d)的采样逼近尚未建立。本项目将基于(H^s(R^d),H^-s(R^d))的对偶多小波，构造适用于整个H^s(R^d)的不规则采样逼近，并研究其精度提升。主要内容包括：(1)构造(H^s(R^d),H^-s(R^d))的对偶多小波；(2)对于H^s(R^d)的任意函数，估计它在(H^s (R^d ),H^(-s) (R^d))多小波展开下的级数截断误差及平移扰动误差；(3)基于特殊对偶多小波，构造适用于整个H^s(R^d)的不规则采样逼近；(4)在函数不必属于逼近空间且不要求采样集条件成立的情况下，研究逼近精度的迭代提升。本项目的采样逼近，将是现有采样逼近理论，在适用范围以及精度提升方面的突破。

Sobolev空间是一类重要的函数空间，其上函数理论广泛应用于微分方程、逼近论以及信号分析等学科领域。在本项目开展之前，函数的采样逼近理论大多仅适用于该空间上的某个子空间。本项目主要的研究内容有：建立对偶Sobolev (H^s(R), H^(-s)(R))空间上对偶小波框架级数的收敛性理论；基于特殊的对偶小波框架理论，建立适用于整个Sobolev空间的小波框架采样逼近；建立小波框架级数关于平移扰动的误差估计；把相应采样逼近结论应用于相位恢复以及Riesz变换重构等问题。本项目取得的成果有：1. 不依赖于函数傅里叶变换点态衰减以及函数的导数性质下，建立对偶Sobolev (H^s(R), H^(-s)(R))空间上对偶小波框架级数的收敛速率估计，该收敛速率是指数阶的，并且仅依赖于伸缩因子以及H^s(R)上框架的消失矩。2. 基于Delta分布，构造了适用于整个Sobolev空间的小波框架采样逼近，该逼近所需的数据是多尺度采样；特别地，项目采用了迭代采样方法，提升了逼近精度。3. 基于小波多尺度采样逼近，建立了Sobolev空间上函数的Riesz变换采样重构。我们首先建立小波框架级数关于平移自适应扰动的误差估计。Riesz变换是奇异积分，本项目恢复Riesz变换时基于逼近系统的自适应扰动，消除了数值奇异。4. 基于Sobolev空间的采样逼近理论，建立了该空间上实信号的相位恢复。