Some Problems On F_a-frame Theory

The study of structured frames is one of core issues in the frame theory such as that of Gabor and wavelet frames.So far,wavelet and Gabor analysis in L~2(R)have seen great achievements in the theory and applications.But the study of structured frames in L~2(R₊)has been less reported.It is because in contrast to R,R₊is not a group under addition.This results in the fact that L~2(R₊)admits no nontrivial shift invariant system,and thus admits no Gabor or wavelet frame.In practice,the time variable cannot be negative.L~2(R₊)models the casual signal space.Observe that R₊is a group under multiplication.Multiplication-based F_a-inner product and its associated F_a-frame have been proposed in recent years.This dissertation addresses F_a-frames for L~2(R₊),F_a-frames in the setting of subspaces of L~2(R₊),duality relations for F_a-frame theory in L~2(R₊)and F_a-equivalence and unitary F_a-equivalence.Chapter 1 is an introduction including concepts,notations,backgrounds and the main results of this dissertation.Chapter 2 focuses F_a-frames in the setting of subspaces of L~2(R₊),where a>1.We establish the connection between subspace F_a-frames and orthogonal projections;characterize all bounded linear operators on L~2(Z×T_a)that transform F_a-frames into other F_a-frames for a subspace V of L~2(R₊);and obtain some sufficient conditions for constructing F_a-frames for V from an arbitrarily given one.Chapter 3 focuses on F_a-equivalence and unitary F_a-equivalence between F_a-frames.We introduce the notions of F_a-equivalence and unitary F_a-equivalence between F_a-frames;present a characterization of the F_a-equivalence and unitary F_a-equivalence.This characterization looks like that of equivalence and unitary equivalence between frames,but the proof is nontrivial due to the particularity of F_a-frames.Chapter 4 addresses duality relations for F_a-frame theory in L~2(R₊).We introduce the notion of F_a-R-dual of a given sequence in L~2(R₊),and obtain some duality principles.Specifically,we prove that a sequence in L~2(R₊)is an F_a-frame(F_a-Bessel sequence,F_a-Riesz basis,F_a-frame sequence)if and only if its F_a-R-dual is an F_a-Riesz sequence(F_a-Bessel sequence,F_a-Riesz basis,F_a-frame sequence),and that two sequences in L~2(R₊)form a pair of F_a-dual frames if and only if their F_a-R-duals are biorthonormal.This dissertation is a theoretical study and its application needs further dis-cussion.