Wavelet Frame And The Construction Of Wavelet Tight Frame

Fame 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, Grossman 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 theory of wavelet frame for L² (R) and wavelet tight frames construction. The results that are quoted in this paper are mostly classical conclusions or the newest 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 composed of four parts: The chapter 1 is an introduction which summarizes the emergence, development of wavelet analysis, frames theory and simply introduces the dual frame theory.The chapter 2 presents the basic properties of frame. The frame is a special Bessel sequence, which extends the Parseval equality of orthonormal basis in Hilbert space to the two-sided inequality in general sequence. For every V/eH , we is the noise data. This shows theframe decomposition have some resistance to the noise. With the redundancy of frame increasing, the resistance will become stronger. Therefore it is necessary to discuss the frame properties.The first part in the chapter 3 studies the judgment and property of wavelet frame in L²(R). The wavelet frame is a special sequence which is a frame for L²(R). The sequence is acquired by a special function acted by two transforms: dilation operator D_an ? with real parameter a>1, n∈Z and translation operator T_mb with real parameterb>0, m ∈ Z. It is very difficult to say a sequence is a wavelet frame or not. A function...