Frames and frame multiresolution analyses

Abstract

This thesis is devoted to a study of an abstract theory of frames and frame multiresolution analysis. First, we give two equivalent conditons for a frame to be a Riesz basis of a separable Hilbert space by a careful examination of the ``projection method`` which approximates the coefficients of a frame expansion, and obtain formulas of Riesz bounds in terms of the eigenvalues of the Gram matrices of finite subsets of a frame. We then generalize bi-orthogonal (non-orthogonal) MRA to frame MRA in which the family of integer translates of a scaling function forms a frame for the initial ladder space $V_0$. We probe the internal structure of frame MRA``s and establish the existence of a dual scaling function, and show that, unlike bi-orthogonal MRA, there exists a frame MRA that has no ``wavelet.`` We prove the existence of a dual wavelet under the assumption of the existence of a wavelet and present easy sufficient conditions for the existence of a wavelet. Finally, we give a new proof of an equivalent condition for the translates of a function in $L^2(R)$ to be a frame of its closed linear span, and present a proof that, among all complex numbers, Duffin-Schaeffer``s choice in the Neumann series expansion of the inverse of a frame operator has the best possible convergence rate.