힐버트 공간에 대한 프레임의 분류
CHARACTERIZATION OF FRAMES FOR HILBERT SPACES
- 주제(키워드) frames , 힐버트 공간 , 프레임 분류
- 발행기관 아주대학교 일반대학원
- 지도교수 하영화
- 발행년도 2008
- 학위수여년월 2008. 2
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 본문언어 영어
초록/요약
In this thesis we study two themes. One is a characterization of frames containing a Riesz basis in an infinite-dimensional Hilbert space, and the other is of finite frames generated by normal operators on $\mathbb{C}^d$. In our first study, we give a characterization of frames containing a Riesz basis which is one of topics in modern abstract frame theory. Riesz frames and near-Riesz basis behave in many respects like Riesz bases and they are frames containing Riesz bases. More generally, a frame with subframe property also has a Riesz basis as its subsequence. But, every frame containing a Riesz basis need not be a Riesz frame, a near-Riesz basis, nor a frame with subframe property. So we have a condition for a frame containing a Riesz basis to be a Riesz frame. The losses in the network are modeled as erasures of transmitted frame coefficients. Equal-norm tight frames have been shown to be useful for robust data transmission. Also the general class of equal-norm tight frames is a generalized harmonic frame, that is, a frame generated by a unitary operator on $\mathbb{C}^d$ and moreover, we note that the generalized harmonic frames have a maximal erasure. In the second study, more generally, we show that the frames generated by normal operators on $\mathbb{C}^d$ also have maximal erasures. On the other hand, in multiple access communication systems it is widely used for a pair of orthogonal frames. So we also characterize pairs of orthogonal frames by normal operators and we apply the results to orthogonal generalized harmonic frames. Moreover, we give a condition when a finite frame generalized by a normal operator has an alternate dual frame generalized by a normal operator.
more목차
Chapter 1 Introduction = 1
Chapter 2 Preliminaries = 6
Chapter 3 A characterization of frames in an infinite Hilbert space = 15
3.1 Frames versus Riesz bases = 15
3.2 Cosine angle between two closed subspaces of a Hilbert space = 20
3.3 Characterization of frames containing a Riesz basis = 22
Chapter 4 A chracteriztion of finite frames in C^(d) = 30
4.1 Finite frames in C^(d) = 30
4.2 Orthogonal frames = 44
4.3 Frames and orthogonal frames generated by normal operators = 46
4.4 Orthogonal generalized harmonic frames = 50
4.5 Dual frames of a frame generated bt a normal operator = 53
Bilbliography = 61
