Study Of Bessel Multipliers And The Related Problems In Banach Spaces

The concept of frame in Hilbert spaces was introduced by Duffin and Schaeffer when they studied nonharmonic Fourier series in 1952. Compared with a basis, a frame pos-sesses similar property:it can represent every element in a Hilbert space H, in contrast to a basis, the way of representation is not unique. Just the nonuniqueness of repre-sentation makes frame theory apply in pure mathematics, applied mathematics, physics, engineering technology and so on. Especially in the fields such as signal processing, image processing and communication, frames have been widely used. At present, the study of frame develops rapidly and the content of frame is rich. The frame in a Banach space is one of the rich contents.This thesis discusses Bessel multipliers and the related problems which associateds with the frames in a Banach space. And it consists of five chapters.Chapter 1 lists background knowledge of frames and Bessel multipliers, and intro-duces the main contents and structure of the thesis.Chapter 2 reviews the basic concepts, properties and some results of Xd frames and p order frames.Chapter 3 is one of the main contents. It mainly introduces the concepts of Banach frames and atomic decompositions and some basic conclusions, and then gives character-izations of Banach frames, atomic decompositions, Xd frames, duals, synthesis-pseudo-duals and relevant sequences.Chapter 4 is another main content. Firstly, we construct (p, q) order Bessel multipliers in a Banach space and study their properties. Secondly, we discuss the sufficient conditions of (p, q) order Bessel multipliers which become a (r, p, g)-kernel operator. Finally, the continuous dependence of the parameters for multipliers is described.The final part is Chapter 5. It gives the basic concept of (Xd, Xd*) Bessel multipliers in a Banach space and then studys its related properties, and describes the continuous dependence of the parameters for multipliers.