The Stability Of Frames And Banach Frames

In 1952, frames for Hilbert spaces were introduced by Duffin and Scha,effer as part of their research in nonharmonic Fourier series. In 1984, Grossmann found a fundamental new application of frames to wavelet and Gabor transform, which was the starting point for a growing interest in frames.Frames are sequences that have basis-like properties but which need not be bases. In particular, they allow element of a Hilbert space to be written as linear combinations of the frame elements in a stable manner. Because the frame elements may be linear dependent and therefore the uniqueness of representation characteristic of bases may be lost. This redundancy has important application in, for example, signal and image processing. Banach frame is a new important branch of frame theory, they were extended to Banach spaces by Grochnig in 1991.This paper is concerned with the stability of frames in Hilbert spaces and Banach frames in Banach spaces, which consists of four chapters. The first chapter are introduction and main content. In this part, we introduce the origin of frame, progress and application, otherwise introduce the main content of this paper. The second chapter are preliminaries. In this part, we give some fundamental definitions and several lemmas which will be used to prove our main results. In the third chapter, we study the stability of frames in Hilbert spaces. It is well known that Riesz bases are frames, but the converse is not true. In this part, we first give an example to support the above consequent, then we give a sample pattern of frames perturbations, lastly, we explore the stability of frames. This is inspired by corresponding classical perturbation results for bases, including the Paley-Wiener basis stability criteria and the perturbation theorem of Kato, and the result of Christensen in [3]. In the fourth chapter, we study the stability of Banach frames in Banach spaces. Firstly, we give an example to show that two results of Christensen have no necessary relation, then let X,Y be Banach spaces, {x_n}_n∈N be a frame of X, and define two operators U₁ and U₂ between X and Y, where U₁ is a linear homeomorphism, U₂ is a linear bounded operator, if the sequence {y_n}_n∈N in X satisfies a certain condition, we prove that {U₁x_n + U₂y_n}_n∈N is a frame of Y. At last, we construct a linear homeomorphism to affect the frame structure and explore the stability of Banach frames.