PROJET AURORA: COMPLEX AND HARMONIC ANALYSIS RELATED TO GENERATING SYSTEMS: PHASE SPACE LOCALIZATION PROPERTIES, SAMPLING AND APPLICATIONS

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1 PROJET AURORA: COMPLEX AND HARMONIC ANALYSIS RELATED TO GENERATING SYSTEMS: PHASE SPACE LOCALIZATION PROPERTIES, SAMPLING AND APPLICATIONS The aim of the AURORA project CHARGE is to join the efforts of mathematicians in Norway and in France that have had a major impact in the last years in the applications of complex and harmonic analysis techniques to results on localization in the time-frequency plane (i.e. in phase space). The problems CHARGE members tacle stem both from applied analysis (PDEs, control theory, signal processing) as well as from recent developpements in complex and harmonic analysis. Let us be a bit more precise. The main unifying theme of CHARGE is phase-space localization. This is an important problem in mathematical analysis with applications to signal processing as well as theoretical physics. The aim here is to obtain good concentration properties both in the direct domain (usually tagged as time ) and simultaneously in the Fourier transform domain (frequency). Of course there are strong limitation to this which can have various mathematical formulations and are generically named uncertainty principles (UP) (see e.g. [FS] for a recent overview). There has been an important rebirth of interest in the UP in recent years as it appears in many parts of applied science. As a first example, let us mention the blooming subject of compressed sensing in which discrete versions of UPs play a central role (see e.g. [CRT]). In this work one typically assumes that a signal is well concentrated in one basis and one tryes to reconstruct the full signal by measuring only a few of its coefficients in another basis (so called compressed sampling ). An other example comes from unique continuation properties of solutions of PDE s (see for instance [EKPV] and ongoing work by those authors). Here one typically assumes that the difference between two solutions of a given PDE (say (i t )u = 0) is small at two distinct times and one concludes that the two solutions are almost the same at all times (or even coincide). As a last example, let us mention control theory of PDE s. The aim here is to obtain control of the solution of a PDE on a domain by fixing its values on a small part of that domain (or of its boundary). There are many approaches to such problems. It turns out that in the one by Lebeau-Robbiano, a key property takes the a form of the uncertainty principle (so called-annihilating pairs). While control has mainly been obtained for large subdomains (open sets or open subsets of the boundary), the work of members of CHARGE opens perspectives towards control over both smaller sets and less structured ones. The second unifying theme in CHARGE is the design of good generating systems. This means systems Φ = {φ λ } λ Λ that (i) are generating in a strong sense orthonormal or Riesz bases, frames 1 ; (ii) are easy to generate typically we want a finite set of building blocks w 1,..., w N to which one applies a few transformations, e.g. given by time-frequency shifts of the building blocks w j,k,l = w l (t α j )e iβ kt ) (iii) are well concentrated in time and frequency. 1 Recall that Φ is a frame if there exists frame bounds A, B such that the following generalized Parseval type inequality holds: A f 2 λ Λ f, φ λ 2 B f. 1

2 2 AURORA PROJECT CHARGE The first property is of essential importance for stability issues. It allows to reconstruct every signal s in terms of the generating system Φ (1) s = λ Λ c λ φ λ in such a way that that the coefficients c λ depend continuously on s and that s can easily be reconstructed from them (at least theoretically). The second property is essential in many areas and was one of the key ideas behind the theory of wavelets and of its more modern successors such as curvelets and shearlets. This property is essential and one wants to compute the coefficients c λ from s as well as to properly extract information on s when one knows c λ. The last property is often important in numerical applications. Let us mention two examples. The first one is the problem of sampling. Typically, a sampling formula will hold if the function has good concentration properties in frequency (like band-limitness). Shannon s sampling formula then provides us with a good generating system (of band limited functions). However, to practically implement such a formula, one needs to truncate the corresponding series (1). This only leads to a good approximation if the φ λ s and s are also concentrated. The UP thus limits the practicability of the sampling formula. An other instance of this comes from computarised tomography. Here a major line of research consists of designing good generating system in which the Radon transform is easy to express. This, with the help of the Fourier slice theorem is often done in the Fourier domain and the Fourier inversion brings this back into the time domain. Unfortunately this last step can usually not be done with closed form formulas so that numerical integration and truncation the Fourier integration is needed. The generating system we consider thus needs to be well concentrated in frequency and the UP tells us that it therefore is badly concentrated in time. But the functions to analyse are time limited so that there is a trade-off here. One of our aim is to design better generating systems to allow for better trade-offs between time and frequency concentration without losing the simplicity of generation. A last unifying theme of CHARGE is sampling theory, which we have already seen to be connected to the two other themes. Here a common feature is that we assume a signal to be in some model space (e.g. a space of band-limited functions or a Fock space). The signal itself can not be fully measured but only some discrete sample of it (e.g. regularly spaced samples in Shannon sampling) and we seek conditions that guarantee that this sample determines the signal fully (in continuous time) and, ideally, be reconstructed from the sample. This subject is also related to super-resolution (see e.g. [CFG]) where one wants to reconstruct a finer sample. An other field of application is that of unique continuation properties of PDE s. Here we will look at uniqueness properties of solutions of a PDE P (D)u(t, x) = 0 satisfying Let us now describe a bit more precisely the tasks CHARGE member will undergo next year. TASK 1: GEOMETRY AND STABILITY OF GABOR SYSTEMS In this task we study the Gabor systems i.e. systems of time frequency shifts of a single window g G(g, Λ) = {π λ g; λ Λ}, and their generalizations to multiple windows (Gabor molecules). In this case, the frame constants are denoted by A(g, Λ), B(g, Λ) and the stability is the conditional number, i.e. the ratio B(g, Λ)/A(g, Λ). This quantity is crucial in practical implementation as it controls the stability of the expansion (1). In the case Λ is a lattice, it is known by the Balian-Low theorem that there is a critical density of Λ such that 0 < A(g, λ) B(g, λ) < + implies bad localization properties of g. For instance when Λ a = {a(m + in); m.n Z}, the critical density is a = 1 and, for g(t) = e πt2, [BLG] showed

3 AURORA PROJECT CHARGE 3 that c(1 a) 1 B(Λ) A(Λ) (1 a) 1, as a 1 ( ) for some constants 0 < c < C < when a 1. This inequality together with the values of constants c, C (still unknown, except some numerical experiments) defines the stability of the frame G(g, Λ a ). The natural question is then if by using such bases of different Gabor molecules one can improve the frame constants (conditional number) of the frame generated by a lattice. Further questions we adress in CHARGE are: obtain a similar results for other regular sequences Λ, specifically for hexagonal lattice; study the stability of frames which correspond to irregular sequences Λ, thus clarify the connection between frame stability and perturbation of the lattice; address the stability questions to frames generated by other windows in particular hyperbolic secant and g(t) = exp( t ) as such frames are related to frames generated by the Gaussian window. This should in particular allow to understand if this relation allows to extended stability properties; study frame constants for system of Gabor molecules, for example one defined by rectangle tiling of the phase space that appears naturally in a number of application, generalized Gabor frames adjusted to such tiling has been recently described, the idea is to use eigenfunctions of the corresponding phase space localization operators. Study of these questions will demand developing of specific techniques on entire functions see [L1, BLG], and specific techniques adapted to Zak transform [G1, L2], it will also require new applications of the localization operators, see [GM]. A further area of research in CHARGE is that of multidimensional Gabor frames. This is a very interesting topic related to applications of Gabor analysis techniques in image processing. Many non-trivial phenomena appear already for the classical case of the Gaussian window leading to delicate problems for sampling/interpolation of analytic functions in several variables. For this particular problem analytic techniques can be related to those developed in frame theory which gives hope to study this problem (intractable in the general setting) Preliminary results in this direction are already obtained in [G2]. Most of the known results on localization properties of more general systems of Gabor molecules in higher-dimension have one-dimensional nature (Balian-Low theorem, density inequalities for Gabor and generalized Gabor frames). At the same time recent research demonstrates deep geometrical properties of the multidimensional phase space and its connections to symplectic capacities. These results should give a better understanding of phase space localization of generating systems in higher dimensions. It would be also interesting to study the problems in a more general context of compact abelian groups. Local concentration of generating systems. One of the eldest and simplest generating systems in time-frequency analysis is the family of Hermite functions. It is the best locally concentrated basis in many aspects. The Hermite functions are also used as building blocks for Gabor and Gabor-like systems. On the other hand, as it has been shown by Shapiro and Byrnes, there exists another simple and useful in application basis with good local concentration properties, that is in some sense even better than one consisting of Hermite functions. The construction of Shapiro and Byrnes is based on Rudin polynomials. Here we will study extremal properties of the Shapiro-Byrnes basis and its multidimensional generalizations; compare the properties of this basis to localization properties of arbitrary generating system and investigate the Gabor systems corresponding to this basis.

4 4 AURORA PROJECT CHARGE TASK 2: SAMPLING, INTERPOLATION AND RIESZ BASES The aim of this task is to investigate sampling and interpolation properties in weighted spaces of entire functions such as the Fock spaces or the Paley-Wiener spaces. In such spaces, interpolation and sampling can be expressed in terms of geometric properties of reproducing kernels: interpolation means that the sequence of the associated reproducing kernels is unconditional, sampling means completeness. It turns out that in the Fock space (spaces with rapidly decaying weights), no sequence is both interpolating and sampling so that there is no Riesz basis of reproducing kernels (see Seip [Se]). When the weight has very moderate growth, the existence of Riesz bases was established by Borichev and Lyubarskii [BL]. In a recent work, Baranov, Dumont, Hartmann and Kellay [BDHK] were able to give a characterization of Riesz bases in the spirit of the Kadets- Ingham 1/4-theorem and its generalizations, following earlier work in [CFT, F] in model and de Branges Rovnyak spaces. With these encouraging results in mind, we intend to characterize in a unified way Riesz bases in more general spaces that include Paley-Wiener type spaces and model spaces, as well as Fock spaces with even more slowly decreasing weights. This would also constitute a generalization of some results by Belov, Mengestie, Seip [BMS] who studied the discrete Hilbert transform on certain sequence spaces. A central piece in our investigations will be the behavior of the reproducing kernel in the space under consideration, which constitutes a research direction of independent interest. As a matter of fact interpolation and sampling problems are strongly related with zero and uniqueness sequences. So, another aim in this connection of our project is to improve the understanding of the delicate behavior of these kinds of sequences. A further direction in CHARGE is that of multiple interpolation, that is interpolating a function and its derivative (in the spirit of Bézier curves). Thanks to the work of several members of CHARGE, the situation is now rather well understood when the maximal number of derivatives is bounded, but only partial results are known for the problem when the multiplicities are unbounded. Finally, let us mention the connection between quasi-crystals and interpolation as discovered by CHARGE member B. Matei (see [MM]) and that will be further explored in this project. Simple quasicrystals (defined by a cut and project scheme ) are sets of points that provide universal sets of stable sampling and universal sets of stable intrepolation. These sets are of particular interest when we need to adapt the samping and the intrepolation to the geometry of the spectrum. In signal and image processing applications we only have a partial information on the spectrum (its Lebesgue dimension does not exceed some fixed constant). In such situations by using simple quasicrystals we obtain optimal sets of stable sampling. So far this theory concerns only the simple quasicrystals, it will further analyze other types of quasicrystals, which are not defined by a cut a project scheme. In particular the quasicrytals defined by mean of local rules. This study would be a bridge between the analysis of nonlinear subdivision scheme used in geometric design and super resolution problems. TASK 3: ANNIHILATING AND HEISENBERG UNIQUENESS PAIRS One of the mathematical formulations of the uncertainty principle is to say that a function and its Fourier transform cannot be both supported on sets (S, Σ) that are too small. Such a pair is then called an annihilating pair. The concept has many applications, for instance in compressed sensing (where a discrete version is used) and it is crucial in Lebeau-Robianno s [LR] approach to control theory (where a quantitative version of this notion is used). In this last case, the sets under consideration are open sets while results of members of CHARGE suggest that one may consider sets of positive measure.

5 AURORA PROJECT CHARGE 5 A further extension of the notion of annihilating pairs has been introduced in [HMR]. There one considers curves Γ and finite measures µ supported on the curve that are absolutely continuous with respect to arc length and one considers their Fourier transform µ. Such objects occur very naturally when one determines solutions of PDEs via the Fourier method (e.g. for the Shrödinger or Klein-Gordon equations). The question of uniqueness of solutions when restricted on a set Λ (the set of sensors) of the corresponding PDE can then be reduced to asking knowing whether µ(ξ) = 0, ξ Λ implies µ = 0. In this case, we say, following Hedenmalm and Montes-Rodríguez, that (Γ, Λ) is a Heisenberg Uniqueness pair. Current research in CHARGE shows that, when Λ is a set of lines, this property reduces to a geometric property of the curve Γ in the direction orthogonal to the lines. However, the proper exploitation of this property is not yet satisfactory excepted in a few cases (convex curves). Our aim is to extend this to more general curves as well as to obtain more quantitative bounds in order to obtain control of PDEs over thin sets. A further issue here is to put the sensors not on a full line but only on a discrete sampling set i.e. to assume µ(ξ) = 0, ξ Λ only for Λ in a discrete subset of a family of lines. Finally, the results so far are restricted to two dimensional curves and it is natural to ask for extensions to higher dimensional manifolds. REFERENCES [BDHK] A. BARANOV, A. DUMONT, A. HARTMANN & K. KELLAY Sampling, interpolation and Riesz bases in small Fock spaces. preprint. [BMS] YU. BELOV, T.Y. MENGESTIE & K. SEIP Discrete Hilbert transforms on sparse sequences. Proc. Lond. Math. Soc. (3) 103 (2011), no. 1, [BLG] A. BORICHEV, K. GRÖCHENIG & YU. LYUBARSKII Frame constants of Gabor frames near the critical density. J. Math. Pures Appl. 94 (2010) [BL] A. BORICHEV, YU. LYUBARSKII, Riesz bases of reproducing kernels in Fock type spaces Journal of the Institute of Mathematics of Jussieu 9 (2010) [CFG] E. J. CANDÈS & C. FERNANDEZ-GRANDA Towards a mathematical theory of super-resolution. To appear in Communications on Pure and Applied Mathematics. [CRT] E. J. CANDÈS, J. ROMBERG & T. TAO Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59 (2005) [CFT] N. CHEVROT, E. FRICAIN & D. TIMOTIN On certain Riesz families in vector-valued de Branges Rovnyak spaces. J. Math. Anal. and Appl., 355 (2009), [F] E. FRICAIN Bases of reproducing kernels in de Branges spaces. J. Funct. Anal. 226 (2005), [EKPV] L. ESCAURIAZA, C. E. KENIG, G. PONCE, G. & L. VEGA Hardy s uncertainty principle, convexity and Schrdinger evolutions. J. Eur. Math. Soc. 10 (2008), [FS] G. B. FOLLAND & A. SITARAM The uncertainty principle a mathematical survey. J. Fourier Anal. Appl. 3 (1997), [G1] K. GRÖCHENIG Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhuser Boston, Inc., Boston, MA, [G2] K. GRÖCHENIG Multivariate Gabor frames and sampling of entire functions of several variables. Appl. Comput. Harmon. Anal. 31 (2011), [GM] K. GRÖCHENIG & E. MALINNIKOVA Phase space localization of Riesz bases for L 2 (R d ). Rev. Mat. Iberoam. 29 (2013), [HMR] H. HEDENMALM & A. MONTES-RODRÍGUEZ Heisenberg uniqueness pairs and the Klein-Gordon equation. Annals Math. 173 (2011) [LR] G. LEBEAU & L. ROBBIANO Contrôle exact de l équation de la chaleur. Comm. Differential Equations 20 (1995), [L1] YU. I. LYUBARSKII Frames in the Bargmann space of entire functions. Entire and subharmonic functions Adv. Soviet Math., 11, Amer. Math. Soc., Providence, RI, [L2] YU. I. LYUBARSKII & N.P. NES Gabor frames with rational density. Appl. Comput. Harmon. Anal. 34 (2013), [MM] B. MATEI & Y. MEYER A variant of compressed sensing. Rev. Mat. Iberoam. 25 (2009), [Se] K. SEIP Interpolation and sampling in spaces of analytic functions. University Lecture Series, 33 American Mathematical Society, Providence, RI, 2004.