Constructing Tight Frames In Finite Dimensional Hilbert Spaces

In order to study some problems of nonharmonic Fourier series, Duffin and Schaeffer firstly introduced the notion of frames for Hilbert spaces in 1952. Frames are redundant sets of vectors in a Hilbert space. What frame is different from orthonormal basis is that frames may have many different representations for a given vector. It is very important in the actual application for this nature. Recently, frames for finite dimensional Hilbert spaces have got more and more attentions from some scholars. On the one hand, frames have been used in signal processing because of their strong stability. On the other hand, finite dimensional space and infinite dimensional space have substaintial distinction. Therefore, we need to study the frame in finite dimensional Hilbert spaces more deeply. Compared to the general frame, tight frame reconstruction formula is simple. Tight frame can avoid the big trouble to calculate the inverse operator of a frame operator. In actual application, so we need to construct some tight frames. In this paper, the structure of our general arrangement is as follows:In chapter 1, we make an introduction about the generation and development of frame theory.In chapter 2, firstly, we review simply some basic concepts and important properties of the frames. Secondly, we introduce some of the research results on the existing construction of tight frame. Lastly, we list the main works of this paper.In chapter 3, firstly, we get the necessary and sufficient condition for the tight frame which is made up of a number of vectors in R2; Secondly, we get any sequence can become a tight frame in R2 by adding arbitrary number vectors, and we can write down the added vector specifically; Thirdly, we give a necessary and sufficient condition for an (r, k)-surgery on unit-norm tight frames in R2. Finally, we generalize the condition under which a unit-norm tight frame can carry on the ((a-1) · (k+u), k)-surgery.In chapter 4, firstly, we study some properties of tight frames inR". Secondly, we get a necessary condition for an (r, k)-surgery on unit-norm tight frames in R ". Finally, an example is given to prove the sufficient condition of the above theorem about an (r, k)-surgery on unit-norm tight frames inR2 is not true in R3.