Study On The Construction And Properties Of Tight Frames In Finite-dimensional Hilbert Spaces

The concept of frames for Hilbert spaces was first introduced in 1952 by Duffin and Schaeffer in the context of nonharmonic Fourier series.Some properties of frame in Hilbert spaces are similar to those of basis.Frames are redundant sets of vectors.In signal processing,a signal has more flexibility linear representation by using frame elements,and the coefficient of linear representation is easy to get by calculating the inner product of the signal and each frame element.However,when the signal dimension is large,we need to calculate a large number of additions and multiplications,and this makes a frame decomposition intractable in applications with limited computing budget.Therefore,it is meaningful to study the sparsity of frames.Among all the types of frames,using the equal-norm tight frames to code is optimal in modern communication networks.It does not depend on the location of the loss of a packet in data transmission,which minimized the error operator of operator norm in some sense.Therefore,it is significant to construct these types of frames.In particular,equal-norm tight frames in finite-dimensional Hilbert spaces have attracted a lot of attention in recent years due to their numerous applications.In this paper,based on the concept of prime tight frames or divisible tight frames,which were introduced by Jakob L,we analyze the Prescribed Norms Spectral Tetris algorithm(PNST)for construction of equal-norm tight frames and provide a necessary and sufficient condition that the equal-norm tight frames are prime equal-norm tight frames.Secondly,we focus on the equal-norm tight frames which constructed by the PNST algorithm in finite-dimensional real spaces and give the sparsity for such frames with respect to the standard unit vector basis.Finally,we present a method for constructing equal-norm tight frames in finite-dimensional Hilbert spaces.This method for constructing tight frames in finite-dimensional real spaces are different from the PNST algorithm by Casazza.We using the existing equal-norm tight frames for Hn to construct the equal-norm tight frames for Hn+1.In addition,we give two concrete constructions which can be used to produce a unit norm frame with n+1 elements for a n-dimensional real space.In fact,we also provide a feasible scheme for constructing Parseval frames and equiangular tight frames in finite-dimensional Hilbert spaces.