Stability of alternate dual frames

Transcription

1 Stability of alternate dual frames Ali Akbar Arefijamaal Abstrat. The stability of frames under perturbations, whih is important in appliations, is studied by many authors. It is worthwhile to onsider alternate duals instead of anonial duals in the frame deomposition. In this paper, we apply a new haraterization of alternate dual frames to disuss a stability problem for them. Mathematis Subjet Classifiation 010). Primary 4C15; Seondary 4C99. Keywords. Frames; dual frames; alternate duals; perturbation of frames. 1. Introdution and preliminaries Let H be a separable Hilbert spae. A sequene {f i } H is alled a frame for H if there are onstants A, B > 0 satisfying A f f, f i B f, f H). 1.1) If the right-hand side of 1.1) holds, it is said to be a Bessel sequene. If {f i } is a frame, the frame operator is defined by S : H H, Sf = f, f i f i. This series onverging unonditionally and S is a bounded, invertible, and self-adjoint operator. This leads to the frame deomposition f = S 1 Sf = f, S 1 f i f i, f H). 1.) A sequene {g i } H is alled a dual frame for {f i} if f = f, g i f i, 1.3)

2 Every frame at least has a dual. In fat, if {f i } is a frame, then 1.) follows that {S 1 f i }, whih is a frame with bounds B 1 and A 1, is a dual for {f i } ; it is alled the anonial dual, a dual whih is not the anonial dual is alled an alternate dual, or simply a dual. To see a general text in frames theory see [8]. Reently, various generalization of frames have been proposed. For example, ontinuous frames [1,, 13], g-frames [17], fusion frames [7], von Neumann-Shatten frames [16] and so on. Due to the wide variety of appliations, there has been a great influx of researhers into the subjet in different approahes, f. [3, 4, 5, 14]. Let {f i } be a frame in H and {g i} in some sense be losed to {f i }, does it follow that {g i} is also a frame? This question, whih plays an essential role in appliations, is referred as the stability of frames. The stability of frames was introdued by Favier and Zalik [11]. Although, it is developed by many other authors [6, 15] in the last deade, but most of known results on this topi, are stated about anonial dual, see [1] for frames and [17, 18] for g-frames. Not only omputing the anonial dual is highly nontrivial but also the struture of a frame is not shared by its anonial dual, in general [10]. Hene, we try to hoose a suitable dual instead of a anonial dual. In this artile, we first review some haraterizations of dual frames. Then we disuss a perturbation property for alternate duals.. Main Results Let {f i } be a frame in H with the frame operator S. For eah f H due to Lemma of [8] the sequene { f, S 1 f i } has minimal l -norm among all sequenes { f, g i } where {g i} is an arbitrary alternate dual. However, finding alternate duals is ompliated and disorderly, see [8, 9]. In order to derive our haraterizations of duals we must define the synthesis operator. If {f i } is a Bessel sequene for H, then the operator T : l H defined by T { i } := i f i. is a bounded linear operator, alled the synthesis operator or the pre-frame operator. It is surjetive and bounded if and only if {f i } is a frame. Moreover, S = T T, where T : H l ; f { f, f i } is the adjoint of T [8]. The next lemma follows immediately from the definition. Lemma.1. Let {f i } and {g i} be two frames with the synthesis operators T and U, respetively. Then the following are equivalent: 1. {f i } is a dual for {g i}.. T U = I. 3. UT = I. 4. T U) = T U.

3 The following theorem desribe the alternate dual in terms of the left inverse of the synthesis operator. Theorem.. [8] Let {f i } be a frame for H. The dual frames for {f i} are preisely as {Φδ i }, where Φ : l H is a bounded left inverse of T. The following haraterization of alternate duals shows that the differene between an alternate dual and the anonial dual an be onsidered as a bounded operator. Theorem.3. Let {f i } be a frame for H with the bounds A and B and the synthesis operator T. Then there exists an one to one orrespondene between duals of {f i } and operators Ψ BH, l ) suh that T Ψ = 0. Proof. First assume that {g i } is a dual of {f i}. Denote the synthesis operator of {g i } by U. Define Ψ : H l by Ψf = U f T T T ) 1 f. Clearly, Ψ is a bounded operator and by using lemma.1 we have T Ψf = T U f T T T T ) 1 f = 0. Conversely, let Ψ BH, l ) with T Ψ = 0, also let {e i } be an orthonormal basis of l. Take g i = S 1 f i + Ψ e i ), where S is the frame operator of {f i }. Then f, g i f, S 1 f i + f, Ψ e i A 1 + Ψ + ) A 1 Ψ f, for all f H. This follows that {g i } is a Bessel sequene. Moreover, UT = f, f i g i = f, f i S 1 f i + f, f i Ψ e i ) = f + Ψ T f = f. Therefore, {f i } and {g i} are dual of eah other by Lemma.1. It is not diffiult to see that every two Bessel sequenes {f i } and {g i} satisfying in 1.3) must be frames. Now we are ready to state our main result about the stability of alternate duals. Theorem.4. Let {f i } and {f i } be two frames for H with the bounds A 1, B 1 and A, B, respetively. Also let {g i } be a fix alternate dual for {f i }. If {f i f i } is a Bessel sequene with suffiiently small bound ɛ > 0, then there exists an alternate dual {g i } for {f i } suh that {g i g i } is also Bessel and its bound is a multiple of ɛ.

4 Proof. Assume that T 1 and T are the synthesis operators of, {f i } and {f i }, respetively. Also denote S 1 = T 1 T1 and S = T T as their frame operators. Due to the proof of Theorem.3 there exits a Ψ BH, l ) suh that T 1 Ψ = 0 and g i = S1 1 f i + Ψ e i ),.1) where {e i } is an orthonormal basis of H. Take Obviously that {h i } A 1 h i = S 1 f i + Ψ e i )..) is a Bessel sequene with the bound A 1 + Ψ + Ψ. Denote the synthesis operator of {h i} by T 3, then f T T3 f = f f, h i f i i Therefore, T T 3 = T Ψf = T Ψf T 1 Ψf T 1 T Ψ f ɛ Ψ f, is invertible for suffiiently small ɛ > 0. In partiular, I T T 3 ɛ Ψ..3) Put {g i } = {T T3 ) 1 h i }. We are going to show that is the desired frame. On the other hand, S 1 S = T 1 T 1 T 1 T + T 1 T T T = T 1 T T 1 + T ) ɛ B 1 + B ). Moreover, by using.1) and.) for all finite salars i we obtain i g i h i ) = i S1 1 f i S 1 f i) i S1 1 f i S 1 f i) + i S 1 f i S 1 f i) S 1 1 S 1 T 1 i + S 1 T 1 i T i S 1 S S 1 S 1 1 T 1 + S 1 ɛ A B1 + ) ) B 1 B i. A 1 If take W = T 3 T ) 1, then we obtain from.3), W 1 1 I W 1 < 1 1 ɛ Ψ, T 1 T ) i ) 1

5 and Consequently, i g i g i) = I W = W I W 1 < ɛ Ψ 1 ɛ Ψ. i g i W g i + W g i W h i ) I W i g i + W i g i h i ) ɛ Ψ B + B 1 + B 1 B + 1 ) i ) 1 A 1 A A 1 ɛ Ψ where B is the upper bound of {g i }. This ompletes the proof. Aknowledgments The author would like to thank the anonymous reviewer for many helpful suggestions that helped improve the presentation of this paper. Referenes [1] S. T. Ali, J. P. Antoine, and J. P. Gazeau, Continuous frames in Hilbert spaes, Ann. Physis 1993) [] A. Askari-Hemmat, M. A. Dehghan, and M. Radjabalipour, Generalized frames and their redundany, Pro. Amer. Math. So. 194) 001) [3] J. Benedetto, A. Powell, O. Yilmaz, Sigm-Delta quantization and finite frames, IEEE Trans. Inform. Theory 5 006) [4] H. Bolskel, F. Hlawatsh, H.G. Feihtinger, Frame-theoreti analysis of oversampled filter banks, IEEE Trans. Signal Proess ) [5] E. J. Candès, D.L. Donoho, New tight frames of urvelets and optimal representations of objets with pieewise C singularities, Comm. Pure. Appl. Math ) [6] O. Christensen, Perturbation of operators and appliations to frame theory, J. Fourier Anal. Appl., ) [7] P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed proessing, Appl. Comput. Harmon. Anal. 51) 008) [8] O. Christensen, Frames and bases. An introdutory ourse, Birkhauser Boston, 008. [9] O. Christensen and R. Y. Kim, On dual Gabor frame pairs generated by polynomials, J. Fourier Anal. Appl ) [10] I. Daubehies, The wavelet transform, time-frequeny loalization and signal analysis, IEEE Trans. Inform. Theory 365) 1990) [11] S. Favier and R. Zalik, On the stability of frames and Riesz bases, Appl. Comput. Harmon. Anal. 1995) [1] H. G. Feihtinger and W. Sun, Stability of Gabor frames with arbitrary sampling points, Ata Math. Hungar. 1133) 006)

6 [13] J. P. Gabardo and D. Han, Frames assoiated with measurable spae, Adv. Comput. Math. 183) 003) [14] R. W. Heath, A. J. Paulraj, Linear dispersion odes for MIMO systems based on frame theory, IEEE Trans. Signal Proess ) [15] Y. Y. Koo and J. K. Lim, Perturbation of frame sequenes and its appliations to shift-invariant spaes, Linear Algebra Appl ) [16] Gh. Sadeghi, A. Arefijamaal, von Neumann-Shatten frames in separable Banah spaes, Mediterr. J. Math. 93) 01), [17] W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl ) [18] Y. C. Zhu, Charaterizations of g-frames and g-bases in Hilbert spaes, Ata Math. Sinia 4 008) Ali Akbar Arefijamaal Department of Mathematis and Computer Sienes, Hakim Sabzevari University Sabzevar, Iran. Arefijamaal@hsu.a.ir, Arefijamaal@gmail.om