Frames and Allied Concepts in Banach Spaces

Abstract

The thesis entitled Frame and Allied concept in Banach Spaces , aims newline to study the notion of frame in Banach spaces in general. Fractional calculus newline has been used extensively in signal and image processing during the past ten newline years. Enhancing this study, fractional Gabor frames in L2(R) are defined newline and studied. In fact, fractional Weyl-Heisenberg packet (fractional WH packet) newline with respect to a fractional Gabor system is defined and a sufficient condition newline for a fractional WH packet to form a frame for L2(R) is obtained. Also, a newline necessary and sufficient condition for a fractional WH packet to form a frame newline for L2(R) is given. Finite sum of fractional Gabor frames is considered and a newline necessary and sufficient condition for the finite sum of fractional Gabor frames newline to be a fractional Gabor frame is obtained. Stability of a fractional Gabor newline Bessel sequence is discussed under perturbation and obtain conditions under newline which it becomes a fractional Gabor frame for L2(R). µ-exact and finitely newline exact Banach frames in Banach spaces are introduced, discussed its existence newline through examples and investigate relationship between these types of exact newline Banach frames. A necessary condition for the existence of a normalize µ-exact newline Banach frame in a Banach space is given. Also, as an application, investigated newline the existence of quasi-complementary subspaces in Banach spaces using exact newline Banach frames and obtain a sufficient condition for the existence of quasi newlinecomplementary subspaces. A necessary condition for an exact retro Banach newline frame, in terms of a bounded isometry, is given. Furthermore, a sufficient newline condition for a sequence to form a K-frame (controlled K-frames) is obtained. newline Finally, sufficient condition, in terms of an orthonormal basis of the Hilbert newline space, for a sequence to be a controlled K-frame for the space is given. newline