| Fourier analysis and wavelet analysis are widely applied to various fields of science and technology. How to approximate to non-periodic multivariate functions by Fourier series car periodic wavelet series is one of the most interesting and challenging problems. In 2003, N. Saito presented "polyharmonic local sine transform", which is a good idea to solve this problem. Along this path we investigate systematically nonlinear approximation of multivariate functions in this thesis.; For a target function on a general domain, we give the free partition and adaptive partition of the domain based on the smoothness of the target function at each point. Afterwards, on each subcube we discuss the approximation of the target function by a new approximation tool: a combination of polynomials and sine sums and obtain a precise approximation order.; On the other hand, combining the polyharmonic local sine transform (PHLST) and the periodic wavelet transform, we give a new algorithm in data compression which is called a polyharmonic wavelet transform (PHWT). This algorithm can compress data efficiently.; The other new approximation tool discussed by us is frames. Based on frames in the Hilbert space, we give an approximation formula by a linear combination of finitely many frame elements. We also research the convergence and Gibbs phenomenon of the periodic wavelet frame series. In the end of this thesis, passing a characterization of frequency bands of scaling functions, we give a general approach to the construction of wavelets. |
