The Finite Frame Analysis And Frame Multiresolution Analysis

Wavelet analysis is a new tool of time-frequency analysis besides Fourier analysis and Gabor analysis. Multiresolution Analysis (MRA) formulated by Mallat and Meyer made it come to true that wavelets is applied in signal processing, image processing and numerical analysis. As an improvement of Multiresolution Analysis, Frame Multiresolution Analysis (FMRA) also shows a great value in theory and applications. Frame and Frame Multiresolution Analysis become two key questions in wavelets theory recently.1. We discuss the effect on the frame lower and upper bounds when the vectors of a frame of C~N are added by some vectors in C~N or some vectors of a frame are erased.2. We give a method of construction of tight frames.3. We show the existence of Frame Multiresolution Analysis wavelets for certain matrices of non-integer entry.4. We give a sufficient condition and a necessary condition for the spectrum of an FMRA, and then develop a different characterization of the spectrums of the central spaces of FMRAs to be contained in an MRA.