Some Properties Of Generalized Frames

Frames were first introduced in 1952 by Duffin and Schaeffer. Subsequently, with the development of frames and their applications in signal and image processing, the theory of frame has been under tremendous growth in the past three decades. Today, frames have been widely used in image processing, signal processing, data compression and many other fields.In 2006, Professor Wenchang Sun introduced generalized frames in a Hilbert space, showed that generalized frames were natural generalizations of frames. In this thesis, we will discuss some properties of generalized frames. The thesis consists of four chapters.Chapter 1 and chapter 2 give introduction of this thesis and basic knowledge to be used throughout the thesis, respectively. Chapter 3 investigates the properties of analysis operator, pre-frame operator and frame operator for generalized frame which is restricted in some index set.Final chapter is the main work of the thesis. Given a generalized frame F={∧j}j€J for U with respect to{Vj}j∈J, we introduce the linear subspace UFP of U:UFP={f ∈ U|{∧jf}j∈J ∈ lp({Vj}j∈J)}, where 1≤ p< 2. Our focus is to study the general theory of these spaces. We investigate different aspects of these spaces in relation to reconstruc-tions, generalized p-frames, realizations and dilations. In particular, we show that for closed linear subspaces of U, only finite dimensional one can be UFP space for some gener-alized frame F. We also prove that under a mild condition on the generalized frame F, the generalized frame expansion of any element in UFP converges in both the Hilbert space norm and the‖·‖F,P-norm which is induced by the lp({Vj}j∈J)-norm.