Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

SOME PROPERTIES OF BESSEL SEQUENCES IN HILBERT SPACES

XIU-GE ZHU1, XIAO-JING ZHANG2 , GUO-CHANG WU3

1Computer Center, Henan University, Kaifeng 475004, P. R. China2Basic Department, Zhengzhou Huaxin College, Zhengzhou 451150, P. R. China

3Department of Mathematics and Information Science,Henan University of Economics and Law, Zhengzhou 450002, P. R. China

E-MAIL: waveletOlll@163.com.archang-Olll@163.com

Abstract:Frame theory plays an important role in signal process­

ing, image processing, data compression and sampling the­ory. In this short paper, some properties of Bessel sequencesin Hilbert spaces are obtained, which generalize the exist­ing results of frame to the case of Bessel sequences.

Keywords:Hilbert space; Frame; Bessel sequence

1. Introduction

Frames were first introduced in 1952 by Duffin and Schaef­fer ([1]) to address some very deep problems in nonharmonicFourier series. Outside of signal processing, frames did notseem to generate much interest until the ground breaking workof Daubechies, Grossmann and Meyer ([2]). Since then, thetheory of frames have been widely studied (see [3-5]) and havebeen intensively applied in wavelet and frequency analysis the­ories. One of the most important properties of a frame is thatevery element in the space can be represented as a series interms of elements in a frame. The series representations bymeans of frames are, however, not unique because frames areusually overcomplete. The redundancy and flexibility offeredby frames has spurred their application in a variety of areasthroughout mathematics and engineering. Traditionally, frameshave been used in signal processing, image processing, datacompression, and sampling theory. Recently, frames are alsoused to mitigate the effect of losses in packet-based commu­nication systems and hence to improve the robustness of datatransmission([6], [7]), and to design high-rate constellationwith full diversity in multiple-antenna code design [8]. Also,frames are used in wireless communications ([9]) and a - 6quantization([10]).

978·1-4673·1535·7/121$31.00 ©2012 IEEE

With the in-depth study of frame theory and its applica­tions, various generalizations of frames have been proposedrecently, such as pseudo-frames ([11]), oblique frames ([12]),outer frames ([13]) and so on. In particular, Sun introduced ageneralization of frames ([14]) and showed that this includesother cases of generalizations of frame concept.

We need to recall the definition and some properties offrames in Hilbert spaces.

Let 1-£ be a Hilbert space and J be a countable index set. Aframe for 1-£ is a sequence {/j : j E J} such that there are twopositive constants A and B satisfying

AII/I12 < E 1(I, Ij) 1

2:::;BII/I1

2 (1)jEJ

for all I E 1-£. The constants A and B are called lower andupper frame bounds, respectively. Those sequences which sat­isfy only the upper inequality are called Bessel sequences. IfA = B, then this frame is called a A-tight frame, and ifA = B = 1, then it is called a Parseval frame.

Define frame operator as following:

S : 1-£~ 1-£, SI = E(/, Ij)/jo (2)jEJ

then, the frame operator S is a self-adjoint positive invertibleoperator in 1-£, which leads to the frame reconstruction formula:

I = E(/, Ij)S-l Ij = E(/, s:' Ij)/j, \;f I E 1-£, (3)jEJ jEJ

here, the collection {h == S-l/j : j E J} is also a frame forH, which is called the canonical dual frame of {/j : j E J}.

In general, the frame {gj : j E J} for 1-£ is called an alternatedual frame of {/j : j E J} if \;f I E 1-£,

I = E(/,gj)/jo (4)jEJ

317

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

(8)

( E bj(f'9j)(f,fj)) - E bj(f,gj)fjI12jEJ jEJ

= (E (1 - bj)(f,gj)(f, fj))jEJ

-llj~(l-bj )(f,gj )1i 11

2

.

In this short paper, we will generalize the existing results offrame to the case of Bessel sequences and get some propertiesof Bessel sequences in Hilbert spaces, which include all exist­ing results as corollaries.

(5)E l(f,fj)1 2-II E (f,fj) f jI12

jEK jEK

= E l(f,fj)1 2-II E (f,fj) f jI12jEKC jEKC

where KC = J\K.

Theorem 1.1 If {fj : j E J} is a Parsevalframe for 1-£,then \;fK c J and \;f f E 1-£,

In [15], the authors verified a longstanding conjecture of sig- for every bounded sequence {bj : j E J} and \;f f E 1-£,nal processing community: a signal can be reconstructed with­out information about the phase. While working on efficientalgorithms for signal reconstruction, the authors of [16] estab­lished a remarkable Parseval frame equality below.

Recently, Theorem 1.1 was generalized to alternate dualframes ([17]). The following form was given in [17]. 2. Main result and its proof

Theorem 1.2 If{fj : j E J} is a frame for 1-£ and Let {Xj}jEJ and {Yj}jEJ be two Bessel sequences in Hilbert{gj : j E J} is an alternate dualframe of {fj : j E J}, then spaces H. The operator 0 is defined by\;fK c J and \;f f E 1-£,

0: H~ H, Ox = E(x,Yj)Xj, (9)jEJ

E aj(x, Yj)(Ox, 0-l Xj )jEJ

-(0-1(Eaj(x, Yj)Xj), E aj (x, Yj)Xj)jEJ jEJ

= E(I-aj)(x,Yj)(x,Xj)-jEJ

( 0 - 1 (E(1 - aj)(x, Yj)Xj) ,E(I- aj)(x, Yj)Xj)jEJ jEJ

(10)

then it is easy to verify that 0 is well defined, linear andbounded. Now, the main result is stated as follows.

(6)

(7)

Theorem 2.1 Suppose that {Xj}jEJ and {Yj}jEJ are twoBessel sequences in Hilbert spaces H. If the operator 0 definedby (9) is invertible, then for any bounded number sequence

In [18], X.G. Zhu and G.e. Wu generalized Theorem 1.2 to {aj}jEJ and x E H,the more general forms which did not involve the real parts ofthe complex numbers. They obtained the following results.

Theorem 1.3 Let {fj : j E J} be a frame for 1-£ and{gj : j E J} be an alternate dual frame of{fj : j E J}, thenfor \;fK c J and \;f f E 1-£,

Proof. For any bounded number sequence {aj}jEJ' it isTheorem 1.4 Let {Ii : j E J} be a frame for 1£ and not hard to check that Eaj(x,yj)Xj and E(l-aj)(x,Yj)Xj

{gj : j E J} be an alternate dualframe of{fj : j E J}, then jEJ jEJ

318

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

are convergent. Since the operator 0 is invertible, 0-1 is con- hence, by (11) and (12),tinuous. Therefore for \Ix E H, we have

E aj(x, Yj)(Ox, 0-lXj )+jEJ

( 0 - 1 (E(l - aj)(x, Yj)Xj) ,E(l- aj)(x, Yj)Xj)jEJ jEJ

= (x, Ox) - (x, E aj(x, Yj)Xj) +jEJ

( 0 - 1(Eaj(x, Yj)Xj), E aj (x, Yj)Xj)jEJ jEJ

= (x,E(x,Yj)Xj - Eaj(x,Yj)Xj) +jEJ jEJ

( 0 - 1(Eaj(x, Yj)Xj), E aj (x, Yj)Xj)jEJ jEJ

Note that Remark. If {Xj }jEJ is a frame for H and {Yj}jEJ is thedual frame of {Xj}jEJ , then 0 = I, where I denotes identityoperator on H, consequently we have that above Theorem 1.3and Theorem 1.4 hold.

3. Conclusions

Acknowledgements

Frames have been used in signal processing, image process­ing, data compression and sampling theory. In this note, theexisting results of frame are generalized to the case of Besselsequences and some properties of Bessel sequences in Hilbertspaces are obtained, which include all existing results as corol­laries.

The work was supported by the National Natural ScienceFoundation of China (No.61071189) and the Natural ScienceFoundation for the Education Department of Henan Province

(12) of China (No. 2010AII0002).

( X - 0-1 (Eaj(x,Yj)Xj) ,Ox - Eaj(x,Yj)Xj)jEJ jEJ

+ (x,Eaj(x,Yj)Xj)jEJ

= (x, Ox) - ( 0-1 (~aj(x, Yj)Xj) ,ox)

+ (o-l(Eaj(x,Yj)Xj), Eaj(x,Yj)Xj) ,jEJ jEJ

319

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

References

[1] R. J. Duffin, A. C. Schaeffer, A class of nonharmonicFourier series, Trans. Amer. Math. Soc. 72(1952) 341­366.

[2] I. Daubechies, A. Grossman, Y. Meyer, Painlessnonorthogonal expansions, J. Math. Phys. 27(1986) 1271­1283.

[3] I. Daubechies, Ten Lectures on Wavelets, in: CBMS-NSFRegional Conference Series in Applied Mathematics, vol.61, SIAM, Philadelphia, 1992.

[4] O. Christensen, An introduction to frames and Rieszbases, Birkhduser, Boston, 2003.

[5] P. G. Casazza, The art of frame theory, Taiwanese J. Math.4(2000) 129-201.

[6] P. G. Casazza and J. Kovaevic, Equal-norm tight frameswith erasures, Adv. Comput. Math. 18 (2003) 387-430.

[7] V.K. Goyal, J. Kovaevic and J. A. Kelner, Quantizedframes expansions with erasures, Appl. Comput. Harmon.Anal. 10(2001) 203-233.

[8] B. Hassibi, B. Hochwald, A. Shokrollahi, and W.Sweldens, Representation theory for high-rate multiple­antenna code design, IEEE Trans. Inform. Th. 47 (2001)2355-2367.

[9] T. Strohmer, Jr. R. W. Heath, Grassmannian frames withapplications to coding and communication, Appl. Com­put. Harmon. Anal. 14(2003) 257-275.

[10] J. J. Benedetto, A. Powell, O. Yilmaz, Sigma-Delta(E d) quantization and finite frames, IEEE Trans. In­form. Theory 52(2006) 1990-2005.

[11] S. D. Li andH. Ogawa, Pseudo-frames for subspaces withapplications, J. Fourier Anal. Appl. 10 (2004) 409-431.

[12] O. Christensen and Y. C. Eldar, Oblique dual framesand shift-invariant spaces, Appl. Comp. Harm. Anal. 17(2004) 48-68.

[13] A. Aldroubi, C. Cabrelli and U. Molter, Wavelets on ir­regular grids with arbitrary dilation matrices and frameatomics for L2(Rd ) , Appl. Comp. Harm. Anal. 17(2004)119-140.

[14] W. C. Sun, G-frames and g-Reisz bases, J. Math. Anal.Appl. 322(2006) 437-452.

[15] R. Balan, P. G. Casazza, D. Edidin, On signal recon­struction without phase, Appl. Comput. Harmon. Anal.20(2006) 345-356.

[16] R. Balan, P. G. Casazza, D. Edidin, G. Kutyniok, Anew identity for Parseval frames, Proc. Amer. Math. Soc.135(2007) 1007-1015.

[17] P. Gavruta, On some identities and inequalities for framesin Hilbert spaces, J. Math. Anal. Appl. 321(2006) 469­478.

[18] X. G. Zhu and G. C. Wu, A note on some equalities forframes in Hilbert spaces, Applied Mathematics Letters23(2010) 788-790.

320