The Study Of MRA E Tight Frame Wavelets And Related Questions

, ψ∈L2(R2) and ψjk(x) =21/2 ψ(Ejx-k),where j ∈ Z, k ∈ Z2, E = M or E = D. ψ is called an E-tight frame wavelet if is a tight frame for L2(R2). In this thesis, a class of " generalized " E-filters and E—pseudo scaling functions are also defined. Now, suppose that ψ is an E-tight frame wavelet. If there exist an E- pseudo scaling function φ and a " generalized " E-filter H ∈ Fφ such thatfor some unimodular, 2π - Z2 periodic function p0, where, then ψ is called an MRA E-tight frame wavelet.The main contents of this thesis consist of three parts. First, one method that constructs MRA E-tight frame wavelets by making use of " generalized " low pass E-filters is given, and it is proved that all of MRA E-tight frame wavelets can be constructed by this method.An MRA E-tight frame wavelet is an E-tight frame wavelet, but the opposite implication is not true. So a sufficient and necessary condition for an E-tight frame wavelet to be an MRA E-tight frame wavelet is the second content of this thesis. Accurately, an E-tight frame wavelet ψ is from MRA if and only if the dimension of Fψ(ξ) is either zero Finally, we also character the MRA E-tight frame wavelet multipliers as well as the E-pseudo scaling function multipliers and the " generalized " low pass E-filter multipliers. As an application of these multipliers, the connectivity of MRA E-tight frame wavelets is studied.