High Dimensional Wavelet Function Approximation

Approximation theory of functions is a Mathematical subject that has a long history, rich contents and strong practicality. It has close connection with the development of modern computational mathematics. In the classical age of the development of approximation theory, it takes constructive theory of function with single variable as research key point. The main direction of modern approximation theory of functions research has gradually turned to multivariate approximation and the construction of different kinds of approximation tools. Now, modern approximation theory of functions has become one of the most active branches in the theory of functions. A lot of modern mathematics branches, including basic mathematics like algebra, topology, functional analysis, as well as such applied mathematics branches as mathematical equation of physics, probability and statistics and computational mathematics, all of which have countless ties with the theory of approximation. Approximation theory of functions was basically one single branch of classical analysis in the past, now it intercrosses many mathematical branches and closely contact with the real life, which lead to a big improvement of its comprehensive feature. The appearance of wavelet analysis brings a new developing direction to approximation theory of functions, which is called approximation of wavelet function. Lately, wavelet theory has been successfully applied to these fields such as musical speech synthesis, image compression, signal processing, quantum physics. They are mainly using two most important properties of wavelet function, one is the good capacity of time-frequency location owned by all sorts of wavelet and wavelet basis; the other is that wavelet basis can be used to compose unconditional basis of many frequently-used spaces. The special properties of wavelet contains orthogonality, compactness, decay characteristics, smoothness, time-frequency window area and vanishing moment.Based on the optimal recovery in one-dimensional functions, we have first proved approximation results of one-dimensional smooth functions. Then function class results are extended to Besov space with wavelet as transitional tool and we get approximation results of multi-dimensional wavelet using multi-resolution analysis to do further research about the above approximation results. Lastly, we have got the best approximation order about upper bound estimate using Dirichlet kernel as approximation tool to reconstruct wavelet function.