Some Study About The Wavelet And Frame

Frame theory derives from the signal processing. In 1952, Duffin and Shaffer introduced the concept of frame for Hilbert spaces in order to study some deep problem in nonharmonic Fourier series. When wavelet theory is booming Daubechies, Grossmann and Meyer connected continuous wavelet transforms with frames theory and introduced wavelet frames. Today frames theory have been widely used in wavelet analysis, signal analysis, image processing, numerical analysis, Banach spaces theory, etc.This paper mainly talks about the relationship among frames, Riesz bases and orthonormal bases in Hilbert spaces. The paper designs approximate Hilbert transform pairs of wavelet and discuss bi-frame decomposition in Sobolev spaces. The results that are quoted in this paper are mostly classical conclusions or the newist conclusions which show the research level and the developing direction. On the basis of the results, this paper generalizes some results and gives some new results.This paper is composes of four parts: The chapter 1 is an introduction which summarizes the emergence,development of wavelet analysis and frames theory.The chapter 2 presents the basic properties of frame at first. The mainly study the relationship among frames, Riesz bases and orthonormal bases in Hilbert spaces. The results indicate that exact tight frame is orthonormal basis and exact frame is Riesz basis. Moreover the paper gives proper examples to illuminate that linear independent frame is not always exact frame and orthonormal basis is not always frame.The chapter 3 introduces the definition and the property of Hilbert transform. Then This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases. But one could start with a known wavelet and take its Hilbert transform to obtain the second wavelet, in that case the second wavelet would not be of finite support .This paper design a finitely supported wavelet to approximate the infinitely supported wavelet .This approach is analogous to the Daubechies construction of compactly supported wavelets with vanishing moments but where the approximate Hilbert transform will use a fiat delay filter.The first part in the chapter 4 definition and MRA of Sobolev spaces.The second part shows the bi-frame decomposition of functions in Sobolev space in a stable ways when the generators of the affine frames satisfy certain mild regularity and vanishing moment conditions. The paper provides three kinds of structure of function's bi-frame decomposition in Sobolev spaces: homogeneous frame decomposition, nonhomogeneous frame decomposition, and finite frame decomposition. This result shows the close property of Q_jH~a and Topology property of Q_j which is a translation in document [35], and extend these results to R_jH~a and R_j. The third part proves those result.