Tight Wavelet Frame

In this dissertation, we consider the following five problems:Framelet lift-ing:Construction of tight framelet packets; Norm and pointwise convergence of tight wavelet frame expansions:Construction of periodic tight wavelet frames; Characterizations of function spaces using tight wavelet frames.In Chapter two, we give some fundamental knowledge of frames including wavelet frames and Gabor frames, and two general lifting schemes of them.In Chapter three, we first prove the high-dimension case of the Extension Principles (Unitary Extension Principle and Oblique Extension Principle) by dif-ferent method from that given by Ron and Shen. And then give the multireso-lution structure and the fast algorithm of tight wavelet frames generated by the Extension Principles. At last, tight wavelet frame packages and lifting schemes of such the tight wavelet frame are also given.In Chapter four, we prove that if the mother framelets belong to L1(Rd), then the periodization of those framelets may generate a tight wavelet frame for L2([0,1]d).In Chapter five, we prove that the tight wavelet frame expansion not only converges to f in terms of Lp-norm,1< p<∞, but also converges pointwisely to f.In Chapter six, we give characterizations of function spaces using tight wavelet frames, such as LP(R).1< p<∞, Hard space H1(R), Sobolev spaces, Lipschitz space∧a,0< a< 1, and Zygmund class∧*.