CURRICULUM VITAE

Name: ZAKHAROVA Anastasia Date and place of birth: July 31, 1983, Moscow Nationality: Russian Address: 32 rue Tronchet, 69006 Lyon, France Professional address: Le2i - CNRS UMR 5158 IUT - 12 rue de la Fonderie, 71200 Le Creusot Telephone number: (+33) 6 48 12 78 04 E-mail: [email protected], [email protected] Webpage: http://azakharova.free.fr

Contents 1 Curriculum vitae

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2 List of publications

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3 Conference presentations

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4 Research statement

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5 Teaching activities

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6 Reference persons

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7 Appendix 1: Compressed sensing

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8 Appendix 2: Multiscale decompositions, applications to image compression 14 9 Appendix 3: Integral and Generalized Frames

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Curriculum vitae

Present position 09/2010 - 08/2011 Post-doctorate, University of Burgundy, Le2i laboratory (compressed sensing, application to the analysis of ECG signals and to 3D images) Professional experience 09/2009 - 08/2010

Temporary research and teaching assistant (ATER) Grenoble INP, Ensimag, France (multiscale decompositions, their applications to contour detection, image compression and reconstruction) 01/2009 - 08/2009 Post-doctorate Laboratory Jean Kuntzmann (Applied Mathematics), Grenoble INP, France (multiscale decompositions, their applications to contour detection, image compression and reconstruction)

Education 2005-2008 PhD studies, Department of Mechanics and Mathematics, Moscow State Lomonosov University Thesis ”Integral and Generalized Frames”, defended on October 17, 2008 2000-2005 Master studies, Department of Mechanics and Mathematics, Moscow State Lomonosov University Language and programming skills • Languages: English (fluent), French (fluent), German. • Programming: C/C++, Matlab.

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List of publications

1. A. Zakharova, Integral Riesz Systems and Their Properties, Vestnik Moskovskogo Universiteta, Seria 1. Matematika. Mekhanika, 2004, N 6 (English translation in ’Moscow University Mathematics Bulletin’, Allerton Press, New York, USA). 2. A. Zakharova, On the properties of Integral Frames, in the Proceedings of the International Workshop on Geometry and Analysis, Abrau-Durso, Russia, 2004, p. 104-115 (in Russian). 3. A. Zakharova, Integral Frames and Riesz Bases, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, 2005, N 76, p.667-676. 4. A. Zakharova, On the properties of generalized frames, Mathematical Notes, 2008, 83:2, p. 210-220. 5. B. Matei, S. Meignen and A. Zakharova, Smoothness of Nonlinear and Non-Separable Subdivision Schemes, accepted for publication in Asymtotic Analysis, 20 p. 6. B. Matei, S. Meignen and A. Zakharova, Interpolatory Nonlinear and Non-separable Multiscale Representation : Application to Image Compression, submitted to the proceedings of the Seventh Conference on Curves and Surfaces, 31 p. 7. B. Matei, S. Meignen and A. Zakharova, Smoothness Characterization and Stability of Nonlinear and Non Separable Multiscale Decomposition, submitted to Journal of Approximation Theory, in revision, 37 p. 8. A. Zakharova, O. Laligant and C. Stolz, ECG denoising using Dictionary Learning, submitted. The papers 3-8 are available on the web-page http://azakharova.free.fr.

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Conference presentations • Workshop: Signal Processing with Adaptive Sparse Structured Representations, June 2011 (accepted as poster contribution). • International congress ”Curves and surfaces”, Avignon, France, June 2010. Talk. • Mathematical and Numerical Models for the Cardiovascular System, summer course, Cortona, Italy, August 2008. Talk. • International Summer School ”New Trends and Directions in Harmonic Analysis”, Inzell, Germany, September 2007. Poster. • International Conference ”New Trends in Harmonic Analysis”, Strobl, Austria, June 2007. Poster. • Voronezh Winter Workshop on Modern Tools of Function Theory and Adjoining Problems, Voronezh, Russia, January 2007. Talk. • 13th Saratov Winter Workshop on Modern Problems of Function Theory and Their Applications, Saratov, Russia, January 2006. Talk. • International Conference and workshop ”Theory of functions, its applications and adjoining problems”, Kazan, Russia, June 2005. Talk. • International Conference and workshop ”Function Spaces, Approximation Theory, Nonlinear Analysis”, Moscow, Russia, May 2005. Poster. • International Workshop on Geometry and Analysis, Abrau-Durso, Russia, August 2004. Talk. • European Mathematical Congress, Stockholm, Sweden, July 2004. Poster. • 5th International Conference of Functional Analysis and Approximation Theory, Acquafredda di Maratea, Italy, June 2004. Talk. • 12th Saratov Winter Workshop on Modern Problems of Function Theory and Their Applications, Saratov, Russia, January 2004. Talk.

Participation in workshops (without contribution) • Semaine d’Etude Maths-Entreprises, Institut Henri Poincar´e, April 2011. • Vision and Machine Learning Research School, ENS Lyon, January 2011. • Apprentisage et Parcimonie (Learning and Sparsity), workshop, November 2010.

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Visits abroad 10/2007 - 11/2007

Mathematics Department of Technical University of Denmark, (collaboration with Prof. Ole Christensen)

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Research statement

The main axes of my research are: -compressed sensing and its applications to the analysis of ECG signals and to the acquisition of 3D images, in collaboration with Olivier Laligant (Professor at the University of Burgundy) and Christophe Stolz (Associate Professor at the University of Burgundy); -non-separable multiscale decompositions and their applications to image compression, in collaboration with Sylvain Meignen (Associate Professor at LJK, Grenoble INP) and Basarab Matei (Associate Professor at LAGA, University Paris 13); -wavelet (frame) theory, subject of my PhD thesis under the supervision of T.P. Lukashenko, Professor of Moscow State Lomonosov University. Below I give a short overview of the results. For more details, please refer to the Appendices 1-3.

Compressed sensing and its applications to the analysis of ECG signals and to the acquisition of 3D images In its modern formulation, the concept of compressed sensing appeared in the papers of Cand`es ([2], Appendix 1) and Donoho ([3], Appendix 1). This approach is based on the notion of sparsity - one says that a signal (an image) x is sparse in some dictionary D (which could be a basis or a redundant system) if the number of nonzero coefficients in the decomposition of x in D is much smaller than their total number (or compressible, if there exists a sparse signal which is close to x). Evidently, a sparse (or compressible) signal could be efficiently reconstructed from a limited number of coefficients. The methods of compressed sensing allow us to reconstruct efficiently by measuring only the part of the coefficients, i.e. the number of measurements is considerably reduced in comparison to the dimension of the original object. One of the problems we are now working on concerns the acquisition systems. These systems are designed using different properties of matter and radiation. For example, light polarization allows us to get the 3D information from a set of 2D images. A compressed sensing approach makes it possible to improve the acquisition procedure for an arbitrary object. Another application of compressed sensing we consider concerns the denoising of ECG signals. Since the denoised signals are classified afterwards, it is extremely important to preserve the time localization of the signal and the form of QRS complex which characterizes the signal. In order to satisfy these requirements, we used the techniques of dictionary learning that let us obtain a dictionary which is adapted to the type of considered signals. Using the signals from MIT-BIH database for simulation, we show that the proposed method outperforms the methods of sparse 2D separable transform [6], soft thresholding [4], and extended Kalman smoother filtering [5]. If the training set contains a sufficiently large variety of signals, then the dictionary could be learnt at the preprocessing step. For more details, see Appendix 1 or [8] in the publication list.

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Multiscale decompositions, application to image compression A multiscale decomposition of an image v could be written down as M v := (v 0 , d0 , d1 , d2 , ...) with v 0 := (vk0 ) ∈ Z d being the coarsest approximation of v and the sequences dj := (djk ) ∈ Z d , j ≥ 0 being the details characterizing fluctuations between two successive scales thus we get an hierarchic decomposition of an image. The advantage of such a representation is its compressibility that involves its application to image reconstruction and compression. The simplest case is that of wavelets, or the linear case, when the reconstruction process does not depend on image structure. However, linear method is subject to improvements, since it does not adapt to image. For one-dimensional signals, a nonlinear method was proposed by Harten ([5], Appendix 2) who suggested using a reconstruction operator that depends on given signal. It was extended to non-linear decompositions of images in the works of Cohen, Dyn, Oswald in a separable way: that is, to obtain a nonlinear representation of a two-dimensional image one takes a tensor product of two multiscale decompositions of dimension 1. The disadvantage of such an approach is that it privileges vertical or horizontal contours of an image. In order to obtain a nonlinear method adapted to all the directions of the contour we introduced a nonseparable multiscale decomposition defined by an integer invertible matrix M such that lim M −n = 0 instead of M = 2Id for the separable case. We studied the regularity and stability of such representations as well as the numerical properties of the detail coefficients (dj )j≥0 . Besides, we applied this approach to image compression and reconstruction and we showed that it outperforms separable methods. For more details, see Appendix 2 or [5],[6],[7] from the publication list.

Wavelet (frame) theory My research in this field was devoted to frame theory, the branch of wavelet theory that attracts researchers from all over the world since early 90s. Thanks to their redundance (that means, that a frame has more elements than a basis) frames are widely used in applications such as image reconstruction and compression, signal detection, etc.(see for example [7], [8], [9], [10], [11] in Appendix 3). Frames offer the following advantages: they give a representation which is invariant (it is of big importance for the applications, however it is not always true for bases), nonlinear and flexible (one can get several representations of the same object using only one frame). Finally, they provide an object representation for the cases when it is difficult to find a complete representation. Among the most well-known frames we can mention exponential frame, wavelet frame, frame of localized Fourier transform etc. One of an interesting problems is to find another frames. For example, Fourier transform, as well as Hilbert transform cannot be represented as frames though they possess many of frame properties such as Parseval equality. This implies the necessity to describe a class of systems that includes frames together with other systems with similar properties. To do this, we introduced the notion of generalized frames and we studied the properties of this class, including the relationship between generalized frames and linear bounded invertible operators. We showed that Fourier and

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Hilbert transforms belong to this class and we also provided some nontrivial examples of generalized frames. Besides, we carried out a deep study of integral frames. For more details, see Appendix 3 or [1]-[4] from the publication list.

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Teaching activities

2010-2011 Numerical analysis, 20h (exercises) (polynomial approximation of numerical functions, numerical integration, iterative methods of solving equations, numerical methods of solving differential equations) 2009-2010 Simulation and numerical calculus, 15h (lectures) + 15h (exercises) (polynomial approximation of numerical functions, numerical integration, iterative methods of solving equations, numerical methods of solving differential equations, direct and iterative methods of solving linear systems, illustrated by practical exercises in Matlab) 2009-2010 exercises in Analysis for engineers, 18h (measure theory, integrable functions, L1 (Rn ) space, Fourier transform in L1 (Rn ) and L2 (Rn ) spaces, distributions theory) 2009-2010 Numerical analysis,12h (lectures) + 12h (exercises) (polynomial interpolation, numerical integration, iterative methods of solving equations, numerical methods of solving differential equations, numerical methods of solving systems of linear equations, illustrated by examples from chemical engineering) 2006-2008 exercises in Analysis and Algebra,116h + 116h (sequences, plane and space geometry, linear algebra, differential calculus, integral calculus, Riemann integrals, improper integrals, multiple integration)

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Reference persons • Olivier Laligant, Professor at the University of Burgundy email: [email protected] • Christophe Stolz, Associate Professor at the University of Burgundy email: [email protected]

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Appendix 1: Compressed sensing

An important issue in the diagnosis of cardiovascular diseases is the analysis of the form of ECG signals. These signals are usually damaged with noise coming from different sources, therefore to identify a waveform or to fix an anomaly we need to remove noise. The most important features of the signal which should be preserved while denoising are the form of QRS complex and time localization of peaks since they contain the key information about the signal. Among the most popular approaches one could mention wavelet denoising first proposed by Donoho in [4]. We introduce a new algorithm of ECG denoising based on dictionary learning. That is, we first build an overcomplete dictionary adapted to different types of ECG signals and use it for denoising. We will see that a dictionary learned on a sufficiently big training set performs well on different ECG signals. In particular, the denoising method based on such a dictionary preserves the form of QRS complex that allows us to recognize an anomaly. We show that the proposed algorithm outperforms the algorithm of ECG denoising by sparse 2d-separable transform introduced in [6]. Let us consider a signal x ∈ Rm and a dictionary D = [d1 , . . . , dk ] ∈ Rm×k (the case k > m is allowed, meaning that the dictionary is overcomplete). If we define ∥α∥0 = |(α)|

(1)

(we use a conventional notation ∥ · ∥0 though it is not a norm), then the decomposition coefficients are found by solving the l1 -minimization problem (known also under the name of basis pursuit): min ∥α∥1 s.t. Dα = x, (2) ∑k where ∥α∥1 = i=1 |αi |. The real-valued case of 2 is equivalent to a linear programming task, so standard numerical methods could be applied to solve it, although more performing methods also exist. A common denoising method is to transform the signal using a fixed (complete or overcomplete) dictionary, to perform shrinkage using either hard or soft thresholding and finally to apply inverse transform. For example, Ghaffari et al. [6] used two-dimensional overcomplete DCT and DCT+Wavelet dictionaries for ECG denoising. It appears that the choice of the dictionary is crucial for the performance of a denoising method. However it is not always clear which dictionary to choose. Usually several dictionaries are compared ’manually’ and the one giving the sparsest representation is chosen. To avoid this, we will train our dictionary so that the representation of typical signals is as sparse as possible. Let D0 ∈ Rm×k be the initial dictionary, X = [x1 , . . . , xn ] ∈ Rm×n be the training set. The number of signals n in the training set is significantly larger than the signal dimension m. The size of the dictionary k is equal or bigger than m. Generally, k ≪ n. We are looking for a dictionary D which will provide us a good sparse approximation for the signals from the training set. To do this, we optimize the cost function n 1∑ (3) f (D) = l(xi , D), n i=1 11

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Figure 1: Performance of the denoising algorithm on the signal with natural noise. On the left a segment of signal with noise and another segment of the same signal, but without noise, are shown. On the right, the denoised versions are shown. As we see, the method based on dictionary learning outperforms the sparse 2D-separable method.

where l(x, D) is a loss function which is small if the representation of x is sparse in D. To find the dictionary we solve a joint optimization problem min

D∈C,α∈Rk

1 ∥x − Dα∥22 + λ∥α∥1 , 2

(4)

The proposed method of minimization (see [7]) is to alternate between two variables: while minimizing one, another is fixed. In order to study the performance of the algorithm we ran the simulations with signals with white noise added and also with signals damages by some unknown noise. In both cases, the dictionary was learnt on the signals from the database of Creusot - Montceaules-Mines hospital, see [1] for details. We show that proposed method outperforms the methods of soft thresholding [4], extended Kalman smoother filtering [5] et the one proposed by Ghaffari et al. [6].

Publications 1. A. Zakharova, O. Laligant et C. Stolz, ECG denoising based on dictionary learning, soumis, 2011.

R´ ef´ erences 2. E. Cand`es, Compressive sampling. Int. Congress of Mathematics, 3, pp. 1433-1452, 12

Madrid, Spain, 2006. 3. D.L. Donoho, Compressed sensing. IEEE Trans. on Information Theory, vol. 52(4), p. 1289–1306, 2006. 4. D.L. Donoho, Denoising by soft-thresholding. IEEE Trans. on Information Theory, vol. 41, p. 613–627, 1995. 5. R. Sameni, M.B. Shamsollahi, C. Jutten, and G.D. Clifford, A nonlinear Bayesian filtering framework for ECG denoising. IEEE Trans. on Biomedical Engineering, vol. 54, p. 2172–2185, 2007. 6. A. Ghaffari, H. Palangi, M. Babaie-Zadeh, C. Jutten, ECG denoising and compression by sparse 2D separable transform with overcomplete mixed dictionaries, 2010. 7. H. Lee, A. Battle, R. Raina and A.Y. Ng, Efficient sparse coding algorithms. Advances in Neural Information Processing Systems, vol. 19, p. 801–808, 2007. 8. J. Mairal, F. Bach, J. Ponce, G. Sapiro, Online learning for matrix factorization and sparse coding. Journal of Machine Learning Research, vol. 11, p. 19–60, 2010.

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Appendix 2: Multiscale decompositions, applications to image compression

We study a nonlinear multiscale decomposition defined by an inversible matrix M with integer entries, such that lim M −n = 0. The idea is similar to that of A. Harten for dyadic scales. That is, one associates to every function v a sequence M v := (v 0 , d0 , d1 , d2 , ...) with v 0 := (vk0 ) ∈ Z d being the coarsest approximation of v and the sequences dj := (djk ) ∈ Z d , j ≥ 0 are the detail sequences characterizing the fluctuations of function from one scale to the following. One defines a projection operator (which we suppose to be always linear) and a prediction operator which could be nonlinear in such a way that the obtained definition generalizes one given by Oswald [9]. In [1], we study the subdivision schemes associated to a matrix M . We study the conditions under which a scheme converges in Sobolev space and in the spaces of integrable functions Lp (Rd ) (the dyadic case was considered in [7],[8]). In order to obtain sufficient conditions, we use the finite differences which lead us to the notion of the polynomial reproduction of subdivision operator. Besides, we define the joint spectral radius of the subdivision operator and get the sufficient conditions of convergence also in terms of spectral radius. Furthermore, we show that a limit function could be defined with the help of box splines. In [2] we apply the obtained results to the image compression. We consider only the case of interpolatory schemes. Let us consider the decompositions given by quincunx matrix ) ( −1 1 M = 1 1 and hexagonal matrix ( M =

2 1 0 −2

)

respectively. We construct then ENO and WENO methods associated with nonlinear multiscale decompositions based on the quincunx or hexagonal matrix. In order to define these methods, we introduce a weight function which help us to determine the prediction interval. We verify that for both matrices the obtained subdivision rules satisfy the sufficient conditions of convergence and we show that the methods are stable. The methods are then applied to image reconstruction and compression. As a coding algorithm we use the EZW (Embedded Zerotree Wavelet) encoder adapted to our case. Comparing the results with those obtained with a linear method, we see that a nonlinear method outperforms significantly the linear method for the case of quincunx matrix. On the contrary, for the case of hexagonal matrix, the difference between the linear and nonlinear case is hardly visible. A possible reason is that in this case the image is too much compressed (every time we pass from one scale to another we divide the number of points by 4). 14

In [3] we aim to describe the regularity of a function v in terms of numerical properties of its decomposition details (dj )j≥0 and vice versa, if the function v lies in Besov space s Bp,q (R), with p, q ≤ 1, s being greater or equal to the degree of polynomials reproduced by the given subdivision rule. The conditions are given in terms of joint spectral radius. Furthermore, we study stability of a nonlinear multiscale decomposition in Besov spaces. Publications 1. B. Matei, S. Meignen and A. Zakharova, Smoothness of Nonlinear and Non-Separable Subdivision Schemes. Accepted to publication in ”Asymptotic Analysis”. 2. B. Matei, S. Meignen and A. Zakharova, Interpolatory Nonlinear and Non-separable Multiscale Representation : Application to Image Compression. Submitted. 3. B. Matei, S. Meignen and A. Zakharova, Smoothness Characterization and Stability of Nonlinear and Non Separable Multiscale Decomposition. In revision. References 4. A. Cohen, N. Dyn and B. Matei, Quasi-linear Subdivision Schemes with Applications to ENO Interpolation, Appl. Comput. Harmon. Anal., vol. 15,pp. 89-116, 2003. 5. A. Harten, Discrete Multiresolution Analysis and Generalized Wavelets, J. Appl. Num. Math. 12 (1993) 153-193. 6. B. Matei, Smoothness Characterization and Stability in Nonlinear Multiscale Framework: Theoretical Results, Asymptotic Analysis (2005), vol. 46 : 277-309. 7. A.S. Cavaretta, W. Dahmen and C.A. Michelli, Stationary Subdivision, Memoirs of Amer. Math. Soc.,Volume 93.,1991. 8. N. Dyn, Subdivision Schemes in computer aided geometric design, Advances in Numerical Analysis II., Subdivision algorithms and radial functions, W.A. Light (ed.), Oxford University Press, 1992, pp. 36-104. 9. S. Harizanov and P. Oswald, Stability of Nonlinear Subdivision and Multiscale Transforms, Constr. Approx., to appear.

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Appendix 3: Integral and Generalized Frames

The notion of frame was introduced by Duffin and Schaeffer in 1952 in order to study exponential systems. However, this notion was not frequently used until the beginning of 90s, when it appeared in the works of I.Daubechies, C. Chui, S.Mallat and others. From here on, frames were deeply studied mostly because of their redundancy which leads to possible applications in image analysis, such as compression, reconstruction etc. However, there exist some well-known systems of functions which share many properties with frames but they cannot be represented as a frame. As examples, we can consider the transforms of Fourier and Hilbert. This naturally lead us to the idea of generalization of frame notion in such a way that it describes also such transforms. In my PhD thesis we generalized the notion of the frame in the following way: Let {Hn }∞ n=1 be a nested sequence of closed subspaces of H, having the density property ∪∞ ω ω n=1 Hn = H). We say that a system {φ }ω∈Ω in H is called generalized if φ is a ω ω sequence {φωn }∞ n=1 of elements of H and φn is the orthogonal projection of φn+1 onto Hn . A generalized system {φω }ω∈Ω ⊂ H is a generalized frame, if there exist a, b : 0 < a 6 b < ∞ such that for any y ∈ Hn all the functions (y, φωn ) are µ-measurable and ∫ 2 (5) a ∥ y ∥Hn 6 |(y, φωn )|2 dµ(ω) 6 b ∥ y ∥2Hn . Ω

It is evident that frames are a special case of generalized frames and it is easy to show that the transforms of Fourier and Hilbert are ones as well. Besides, an example of a non-tight (a ̸= b) generalized frame is provided. The introduced notion is then studied in my thesis. In particular, we obtain necessary and sufficient conditions for a generalized frame to be a frame (a discrete frame), we study the convergence and measurability of the frame coefficients sequence and we get some results concerning frame decomposition ([4]). Finally, we established a connection between bounded linear operators in a Hilbert space and generalized frame. That is, for any bounded linear operator A : H → L2 (Ω), there exists a generalized frame {φω }ω∈Ω in H such that x ∈ H we have A(x) = (L2 ) lim (x, φωn ) (and vice versa, any operator A : x → (L2 ) lim (x, φωn ) is a bounded n→∞

linear operator if {φω }ω∈Ω is a generalized frame) ([4]).

n→∞

We also obtained some results concerning integral frames ([1]-[3]). Publications 1. A. Zakharova, Riesz Systems and Their Properties, Vestnik Moskovskogo Universiteta, Seria 1. Matematika. Mekhanika, 2004, N 6 (English version in ”Moscow University Mathematics Bulletin”, Allerton Press, New York, USA). 2. A. Zakharova, On the properties of Integral Frames, in the Proceedings of the International Workshop on Geometry and Analysis, Abrau-Durso, Russia, 2004, p. 104-115. 3. A. Zakharova, Integral Frames and Riesz Bases, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, 2005, N 76, p.667-676. 4. A. Zakharova, On the Properties of Generalized Frames, Mathematical Notes, 2008, 16

83:2, p. 210-220 (English version in ”Mathematical Notes”, 2008, 83:2, p. 190-200).

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