Some Studies Of Wavelet Transform

Wavelet transformation is composed as the main part in wavelet analysis, and it is an ideal time-frequency analysis tool. It had solved many difficult problems about Fourier transformation. It includes continuous wavelet transformation and discrete wavelet transformation, so it is called mathematical microscope feature. Now it becomes a hot problem for scholars both at home and abroad. At the same time, Wavelet has great potential in noise reduction, edge detection, signal or image processing and so on.Based the definition of traditional integrable wavelet transformation, we propose a new definition about the normalized wavelet transformation, and some properties for wavelet transformation are researched. At the same time, based on the strong reconstruction formulas of the normalized windowed Fourier transformation Twwinf, some properties and reconstruction formulas for wavelet transformation are researched. For arbitrary f in L2(R), a function sequence{fn}n=1∞is defined by the normalized integrable wavelet transformation Twwavf, and it is proved that{fn}n=1∞converges to f in norm. Thus the strong reconstruction formulas of Twwav f is established. There are three chapters in this thesis as follows.In Chapter1, we first recall the history of the development of wavelet theory and the motivation, especially introduce the development of wavelet transformation in detail. It is main results relating to the thesis are introduced.In Chapter2, a function sequence {Fn}n∞=1is defined by the normalized windowed Fourier transformation Twin f, and it is proved that{Fn}n=1∞converges to f in norm. Thus the strong reconstruction formulas of Twwm f is established.In Chapter3, we mainly study some properties and reconstruction formulas for wavelet transformation are researched. For arbitrary f in L2(R),a function sequence {fn}n=1∞is defined by the normalized integrable wavelet transformation Twwav f, and it is proved that{fn}n=1∞converges to f in norm. Thus the strong reconstruction formulas of Twwavv f is established.