Super-resolution of near-colliding point sources
SSUPER-RESOLUTION OF NEAR-COLLIDING POINT SOURCES
DMITRY BATENKOV, GIL GOLDMAN, AND YOSEF YOMDIN
Abstract.
We consider the problem of stable recovery of sparse signals of the form F p x q “ d ÿ j “ a j δ p x ´ x j q , x j P R , a j P C , from their spectral measurements, known in a bandwidth Ω with absolute error not exceeding (cid:15) ą
0. We consider the case when at most p ď d nodes t x j u of F form a cluster whose extent issmaller than the Rayleigh limit , while the rest of the nodes are well separated. Provided that (cid:15) Æ SRF ´ p ` , where SRF “ p Ω∆ q ´ and ∆ is the minimal separation between the nodes, we showthat the minimax error rate for reconstruction of the cluster nodes is of order SRF p ´ (cid:15) , whilefor recovering the corresponding amplitudes t a j u the rate is of the order SRF p ´ (cid:15) . Moreover, thecorresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are (cid:15) Ω and (cid:15) , respectively. These results suggest that stable super-resolution is possible in much more generalsituations than previously thought. Our numerical experiments show that the well-known MatrixPencil method achieves the above accuracy bounds. Introduction
Super-resolution of sparse signals.
The problem of mathematical super-resolution (SR)is to extract the fine details of a signal from band-limited and noisy measurements of its Fouriertransform [41]. It is an inverse problem of great theoretical and practical interest.The specifics of SR highly depend on the type of prior information assumed about the signalstructure. Many theoretical and practical studies assume signals of compact support , in which casethe SR problem is equivalent to analytic continuation (equivalently, extrapolation) of the Fouriertransform. However, it can be shown that the spectrum of a compactly supported function can beextrapolated from samples of accuracy (cid:15) by a factor which scales at most logarithmically with thesignal-to-noise ratio (cid:15) , see e.g. [34, 41, 9, 18] and references therein. On the other hand, in recentyears considerable progress has been made in studying SR for sparse signals , which are frequentlymodelled as idealized spike-trains (1.1) F p x q “ d ÿ j “ a j δ p x ´ x j q , x j P R , where δ is the ubiquitous Dirac’s δ -distribution. This particular type of signals is widely used inthe literature, as it is believed to capture the essential difficulty of SR with sparse priors, see e.g.[26, 21].Let F p F q denote the Fourier transform of F :(1.2) F p F qp s q “ ż F p x q e ´ πisx dx. Mathematics Subject Classification.
Primary 65H10, 94A12, 65J22.
Key words and phrases.
Signal reconstruction, spike-trains, Fourier transform, Prony systems, sparsity, super-resolution. a r X i v : . [ m a t h . NA ] J a n igure 1. The Rayleigh limit. For a signal F p x q “ ř j a j δ p x ´ x j q , its low resolution version is given by F Low p x q “ F ´ ` F p F q ¨ χ r´ Ω , Ω s ˘ — ÿ j a j sinc p Ω p x ´ x j qq .F Low p x q will have peaks of width « , and therefore it will be increasingly difficult to recover signals forwhich the minimal separation between the t x j u ’s is much smaller than . Further suppose that the spectral data is given as a function Φ satisfying, for some (cid:15) ą ą | Φ p s q ´ F p F qp s q| ď (cid:15), s P r´ Ω , Ω s . The sparse SR problem reads as follows: given Φ as above, estimate the unknown parameters of F , namely, the amplitudes t a j u and the nodes t x j u .If (cid:15) “
0, the problem can be solved exactly by a variety of parametric methods (Prony’s methodetc., see e.g. [51, 54] and Subsection 1.2 below). For (cid:15) ą
0, if f is any reconstruction algorithmreceiving Φ as an input, and producing an estimate F “ f p Φ q of the signal which satisfies (1.3),then, under an appropriate definition of the distance } F ´ F } , it is of great interest to have a goodestimate of the noise amplification factor (or the problem condition number ) K such that(1.4) } F ´ F } « K (cid:15). Rayleigh limit and minimal separation.
It has been well-established that the difficulty ofsparse SR is directly related to the minimal separation ∆ “ min ď i ă j ď d | x i ´ x j | , or, more precisely,to the relationship between ∆ and Ω.Without any a-priori information, the best attainable resolution from spectral data of bandwidthΩ is of the order , which is also known as the Rayleigh limit . Both classical methods of non-parametric spectral estimation [54], as well as modern convex optimization based methods solvethe problem under some sort of a separation condition of the form ∆ ě c Ω [21, 20, 28, 35, 27, 17, 5,19, 53, 55], and moreover these methods are generally considered to be stable.On the other hand, the case ∆ ! (and arbitrary signed/complex amplitudes t a j u ) is muchmore difficult (see Figure 1).The sparse SR problem has appeared already in the work by R. Prony [51], where he devised analgebraic scheme to recover the parameters t x j , a j u from 2 d equispaced measurements of F p F q , as-suming F is given by (1.1), and for arbitrary ∆ ą | a j | ą (cid:15) “
0, the question of their stability (the magni-tude of K in (1.4)) becomes of essential interest. For instance, if it so happens that an estimate F “ ř dj “ a j δ p x ´ x j q satisfies min ď j ď d | x j ´ x j | Ç ∆, then such F may be of little practical use inmany applications (because the inner structure of the sparse signal will be determined incorrectly).The first work which examined the stability of SR in the sub-Rayleigh regime was by D.Donoho[26]. The signal F was assumed to have an infinite number of spikes t x j u , constrained to a grid of tep size ∆, with less than one spike per unit interval on average, but whose local complexity wasconstrained to have no more than d spikes per any interval of length d (such d is called the Rayleighindex ). It was shown that the worst-case (cid:96) error of such F (i.e. the (cid:96) norm of the coefficientsequence of the difference) from continuous measurements with a band-limit Ω and perturbation ofsize (cid:15) (in L sense) scales like SRF α (cid:15) , where SRF “ ą super-resolution factor ,and α satisfies 2 d ´ ď α ď d `
1. In [24] the authors considered the case of d -sparse signalssupported on a grid, and showed that the correct exponent should be α “ d ´ d -sparse signals anddiscrete Fourier measurements.In the papers mentioned above, the error rate SRF d ´ (cid:15) is minimax , meaning that on one hand,it is attained by a certain algorithm for all signals of interest, and on the other hand, there existworst-case examples for which no algorithm can achieve an essentially smaller error. It turnsout that these worst-case signals all have the structure of a cluster, where all the d nodes t x j u appear consecutively, i.e. x j “ x ` p j ´ q ∆ , j “ , . . . , d . A natural question which arises is: if it is a-priori known that only a subset of the d spikes can become clustered, can we have betterreconstruction accuracy? In this paper we shall provide a positive answer to this question.1.3.
Main contributions.
In this paper we consider the case where the nodes t x j u can takearbitrary real values (the so-called off-grid setting), while the amplitudes t a j u can be arbitrarycomplex scalars. We further assume that exactly p nodes, x κ , . . . , x κ ` p ´ , form a small clusterof extent h ! and are approximately uniformly distributed inside the cluster, while the restof the nodes are well-separated from the cluster and from each other (see Definition 2.5 below).The approximate uniformity is expressed by the assumption that the minimal separation betweenany two cluster nodes is bounded from below by ∆ “ τ h for some fixed 0 ă τ ď
1. Under these p -clustered assumptions, we show in Theorem 2.10, that for small enough (cid:15) – and, in particular, for (cid:15) Æ p Ω∆ q p ´ , the worst case error rates of a minimax reconstruction algorithm (see Definition 2.2below), receiving Φ satisfying (1.3) as an input, and returning an estimate x j “ x j p Φ q , a j “ a j p Φ q ,satisfy (1) Non-cluster nodes: max j Rt κ,...,κ ` p ´ u | x j ´ x j | — (cid:15) Ω , max j Rt κ,...,κ ` p ´ u | a j ´ a j | — (cid:15). (2) Cluster nodes: max j Pt κ,...,κ ` p ´ u | x j ´ x j | — (cid:15) Ω p Ω∆ q ´ p ` , max j Pt κ,...,κ ` p ´ u | a j ´ a j | — (cid:15) p Ω∆ q ´ p ` . The constants appearing in our bounds depend on p, d , a-priori bounds on the magnitudes | a j | ,and additional geometric parameters, but neither on ∆ nor on Ω.Our results indicate, in particular, that the non-clustered nodes t x j u j Rr κ,...,κ ` p ´ s can be recoveredwith much better accuracy than the cluster nodes. Let the super-resolution factor be defined, asbefore, by SRF “ p Ω∆ q ´ , then the condition number of the cluster nodes scales like SRF p ´ inthe super-resolution regime SRF "
1, while the condition number of the non-cluster nodes doesnot depend on the SRF at all. We use the symbol — to denote order equivalence, up to constants: A p t q — B p t q , if and only if there exist positiveconstants c , c (depending on the specified parameters) such that c B p t q ď A p t q ď c B p t q for all specified values of t . ur approach is to reduce the continuous measurements problem to a certain “Prony-type”system of 2 d nonlinear equations, given by equispaced measurements of Φ p s q with a carefullychosen spacing λ « Ω, and analyze the sensitivity of this system to perturbations. The proofsinvolve techniques from quantitative singularity theory and numerical analysis. Some of the tools,in particular the “decimation-and-blowup” technique, were previously developed in [2, 6, 11, 7, 1,13, 12, 8]. The single-cluster case p “ d has been first analyzed in [7], while the lower bound (ina slightly less general formulation) has been essentially shown in [1]. One of the main technicalresults, Lemma 5.8, has been first proven in [8].Our numerical experiments in Section 3 show that the above bounds are attained by MatrixPencil (MP), a well-known high-resolution algorithm [37, 36].1.4. Related work and discussion.
Our main results generalize several previously availablebounds for both on-grid and off-grid SR [24, 39, 7], replacing the overall sparsity d with the “local”sparsity p . Compared with previous works, we also have an explicit control of the perturbation (cid:15) for which the stability bounds hold: (cid:15) ď C ¨ p Ω∆ q p ´ . So, given F satisfying the clusteringassumptions and Ω, we can choose (cid:15) “ c p Ω∆ q p ´ such that F can be accurately resolved, and c does not depend on Ω , ∆. But this also means that given (cid:15) ą
0, we can choose ∆ and Ω such that p Ω ∆ q p ´ ě (cid:15)c , and for any F satisfying the clustering assumptions with ∆ “ ∆ and Ω “ Ω , the SR problem can be accurately solved. Therefore, fixing (cid:15) , our results show thataccurate recovery is possible for all SRF values up to ` (cid:15) ˘ p ´ (but possibly also for higher valuesof SRF). On the other hand, a similar argument using the lower bounds for the minimax errorshows that with perturbation of magnitude (cid:15) , no algorithm can resolve signals having a cluster ofsize p and separation ∆ (cid:15) Æ (cid:15) p ´ , giving an upper bound for the attainable SRF values exactlymatching the lower bound above. To summarize, we obtain the best possible scaling of the attainableresolution with clustered sparsity p and absolute perturbation (cid:15) : (1.5) SRF — p ´ c (cid:15) . This H¨older-type scaling is much more favorable compared to SR by analytic continuation underthe prior of compact signal support, where the bandwidth extrapolation factor scales only as afractional power of log (cid:15) , see e.g. [9] and references therein. Also note that the sparse SR problemenjoys linear stability in (cid:15) (1.4), whereas analytic continuation exhibits stability of the form Error « (cid:15) γ , where γ ă for this quantity under the partial clusteringsetting (compare with [3, 45, 15, 29, 38]), and using these results, we have shown in the same paperthat the asymptotic scaling of the condition number for on-grid SR in this regime is SRF p ´ ,matching the off-grid setting of the present paper.The question of providing rigorous performance guarantees for high-resolution algorithms suchas MP, MUSIC, ESPRIT and others, in the super-resolution regime SRF ą
1, is of current interest.In two very recent works, [40, 39], the authors derive stability estimates for MUSIC and ESPRITalgorithms under similar clustering assumptions, finite sampling and white Gaussian perturbationmodel. Their results suggest that the corresponding noise amplification factors K for the nodes areof the order SRF p ´ with high probability. During the review of the present paper, the authors Our clustering model is distinct from Donoho’s model of sparse clumps on a grid [26], and so the two resultscannot be compared directly. Estimates for the smallest singular value were independently obtained in [39] giving same asymptotic order butbetter absolute constants. In [10] we have obtained optimal scalings of all the singular values by different techniques. f [40] established near-optimality of ESPRIT in the bounded noise model. In particular, theyshowed that ESPRIT is optimal up to a factor of 1 { Ω, i.e. | x j ´ ˜ x j | Æ p Ω∆ q ´ p ´ ε with discreteFourier measurements, however, requiring ε Æ p Ω∆ q p ´ { Ω. We also mention [16, 33], where theconnection between perturbation of (square) matrix pencil eigenvalues and the a-priori distributionof these eigenvalues was established via potential theory. It will be interesting to investigate thepossibility to applying these methods to the analysis of MP in the clustered setting.Turning to other techniques, the special case of a single cluster can be solved with optimal ac-curacy by polynomial homotopy methods, as described in [6], however in order to generalize thisalgorithm to configurations with non-cluster nodes, we need to know the optimal decimation param-eter λ . Nonlinear least-squares and related methods (e.g., Variable Projections [32, 47]) apparentlyprovide an optimal recovery rate, however they generally require very accurate initialization. Wehope that our methods may help in analyzing these techniques as well, and plan to pursue thisline of research in the future. For the case of positive point sources, stability rate SRF p has beenestablished for convex optimization techniques in [46], see also a related preprint [25].1.5. Organization of the paper.
In Section 2 we provide the necessary definitions and formulatethe main results. In Section 3 we present several numerical experiments confirming the optimalityof the Matrix Pencil algorithm. The proof of Theorem 2.6 (upper bound) is presented in Section5. The proof of Theorem 2.8 (lower bound) is given in Section 6.1.6.
Acknowledgements.
The research of GG and YY is supported in part by the Minerva Foun-dation. DB is supported in part by AFOSR grant FA9550-17-1-0316, NSF grant DMS-1255203,and a grant from the MIT-Skolkovo initiative.2.
Minimax bounds for clustered super-resolution
Notation and preliminaries.
We shall denote by P d the parameter space of signals F withcomplex amplitudes and real, pairwise distinct and ordered nodes, P d “ ! p a , x q : a “ p a , . . . , a d q P C d , x “ p x , . . . , x d q P R d , x ă x ă . . . ă x d ) , and identify signals F with their parameters p a , x q P P d . In particular, this induces a structure ofa linear space on P d . Throughout this text we will always use the maximum norm } ¨ } “ } ¨ } on C d , R d and P d , where for F “ p a , x q P P d } F } “ max ` } a } , } x } ˘ . We shall denote the orthogonal coordinate projections of a signal F to the j -th node and j -thamplitude, respectively, by P x ,j : P d Ñ R and P a ,j : P d Ñ C . We shall also denote the j -thcomponent of a vector v by v j .Let L r´ Ω , Ω s denote the space of bounded complex-valued functions defined on r´ Ω , Ω s withthe norm } e } “ max | s |ď Ω | e p s q| . Definition 2.1.
Given Ω ą and U Ď P d , we denote by F p Ω , U q the class of all admissiblereconstruction algorithms, i.e. F p Ω , U q “ " f : L r´ Ω , Ω s Ñ U * . efinition 2.2. Let U Ă P d . We consider the minimax error rate in estimating a signal F P U from Ω -bandlimited data as in (1.3) , with measurement error (cid:15) ą : E p (cid:15), U, Ω q “ inf f P F p Ω ,U q sup F P U sup } e }ď (cid:15) } F ´ f p F p F q ` e q } . Similarly the minimax errors of estimating the individual nodes, respectively, the amplitudes of F P U are defined by E x ,j p (cid:15), U, Ω q “ inf f P F p Ω ,U q sup F P U sup } e }ď (cid:15) | P x ,j p F q ´ P x ,j p f p F p F q ` e qq| , E a ,j p (cid:15), U, Ω q “ inf f P F p Ω ,U q sup F P U sup } e }ď (cid:15) | P a ,j p F q ´ P a ,j p f p F p F q ` e qq| . Let a signal F P P d be fixed. We define the (cid:15) -error set E (cid:15), Ω p F q as the following pre-image. Definition 2.3.
The error set E (cid:15), Ω p F q Ă P d is the set consisting of all the signals F P P d with ˇˇ F p F qp s q ´ F p F qp s q ˇˇ ď (cid:15), s P r´ Ω , Ω s . We will denote by E x ,j(cid:15) p F q “ E x ,j(cid:15), Ω p F q and E a ,j(cid:15) p F q “ E a ,j(cid:15), Ω p F q the projections of the error setonto the individual nodes and the amplitudes components, respectively: E x ,j(cid:15), Ω p F q “ (cid:32) x j P R : ` a , x ˘ P E (cid:15), Ω p F q ( ” P x ,j E (cid:15), Ω p F q ,E a ,j(cid:15), Ω p F q “ (cid:32) a j P C : ` a , x ˘ P E (cid:15), Ω p F q ( ” P a ,j E (cid:15), Ω p F q . (2.1)For any subset V of a normed vector space with norm } ¨ } , the diameter of V is diam p V q “ sup v , v P V } v ´ v } . The minimax errors are directly linked to the diameter of the corresponding projections of theerror set by the following easy computation, which is standard in the theory of optimal recovery[43, 42, 44] (see also [26, 24, 39]).
Proposition 2.4.
For U Ă P d , Ω ą , ď j ď d and (cid:15) ą we have
12 sup F : E (cid:15), Ω p F qĎ U diam ` E (cid:15), Ω p F q ˘ ď E p (cid:15), U, Ω q ď sup F P U diam ` E (cid:15), Ω p F qq (2.2) 12 sup F : E (cid:15), Ω p F qĎ U diam ` E x ,j (cid:15), Ω p F q ˘ ď E x ,j p (cid:15), U, Ω q ď sup F P U diam ` E x ,j (cid:15), Ω p F qq (2.3) 12 sup F : E (cid:15), Ω p F qĎ U diam ` E a ,j (cid:15), Ω p F q ˘ ď E a ,j p (cid:15), U, Ω q ď sup F P U diam ` E a ,j (cid:15), Ω p F qq (2.4) Proof.
We shall prove (2.2), the proof in the other cases is identical. We omit Ω from the followingto reduce clutter.
Upper bound:
Let (cid:15) ą
0. For any Φ P L r´ Ω , Ω s , let B p (cid:15), Φ q “ t F P U : } F p F q ´ Φ } ď (cid:15) u . Consider an oracle estimator f (cid:15) P F p Ω , U q defined as f (cid:15) p Φ q “ any element of B p (cid:15), Φ q if B p (cid:15), Φ q ‰ H ,F else , To ensure the minimax error rate is finite, depending on the noise level, we impose constraints on U Ă P d , namelylower and upper bounds on the magnitude of the amplitudes and the separation of the nodes. We will specify theseconstraints exactly in the statements of the accuracy bounds. here F is an arbitrary element of U . Now let F P U , and Φ “ F p F q ` e where } e } ď (cid:15) .Then by definition F P B p (cid:15), Φ q . Put F “ f (cid:15) p Φ q , thus } F p F q ´ Φ } ď (cid:15) , and therefore } F p F q ´ F p F q} ď } F p F q ´ Φ } ` } Φ ´ F p F q} “ (cid:15). We conclude that F P E (cid:15) p F q , and consequently E p (cid:15), U, Ω q ď } F ´ F } ď diam ` E (cid:15) p F q ˘ . Lower bound:
For the lower bound, let F P U such that E (cid:15) p F q Ď U . Let ξ ą F , F P E (cid:15) p F q with } F ´ F } “ diam ` E (cid:15) p F q ˘ ´ ξ . Let Φ “ F p F q ,and let F “ f p Φ q be the output of a certain estimator f corresponding to the input Φ. Wehave ›› Φ ´ F p F q ›› , ›› Φ ´ F p F q ›› ď (cid:15) . Consequently, there exist perturbation functions e , e satisfying } e } , } e } ď (cid:15) , while also F p F q “ Φ “ F p F q ` e “ F p F q ` e . By definition of the minimax error we therefore have E p (cid:15), U, Ω q “ inf f sup } e }ă (cid:15),F P U } F ´ f p F p F q ` e q}ě inf f max ` } F ´ F } , } F ´ F } ˘ ě inf f (cid:32) } F ´ F } ` } F ´ F } ( ě } F ´ F }“ diam ´ E (cid:15) p F q ¯ ´ ξ . The lower bound follows by letting ξ Ñ (cid:3) Uniform estimates of minimax error for clustered configurations.
The main goalof this paper is to estimate E p (cid:15), U, Ω q (in fact its component-wise analogues E x ,j p (cid:15), U, Ω q and E a ,j p (cid:15), U, Ω q ) where U Ă P d are certain compact subsets of P d containing signals with p ď d nodesforming a small, approximately uniform, cluster. In order to have explicit bounds, we describe suchsets U by additional parameters T, h, τ, η, m, M as follows.
Definition 2.5 (Uniform cluster configuration, Figure 2) . Given ă τ, η ď and ă h ď T , anode vector x “ p x , . . . , x d q P R d is said to form a p p, h, T, τ, η q -clustered configuration, if thereexists a subset of p nodes x c “ t x κ , . . . , x κ ` p ´ u Ă x , p ě , which satisfies the following conditions:(1) for each x j , x k P x c , j ‰ k , τ h ď | x j ´ x k | ď h ; (2) for x (cid:96) P x z x c and x j P x , (cid:96) ‰ j , ηT ď | x (cid:96) ´ x j | ď T. Our first main result provides an upper bound on diam p E (cid:15), Ω p F qq , and its coordinate projections,for any signal F forming a clustered configuration as above. Theorem 2.6. (Upper bound) Let F “ p a , x q P P d , such that x forms a p p, h, T, τ, η q -clusteredconfiguration and ă m ď } a } . Then there exist positive constants C , . . . , C , depending only on . . . . . . . . Figure 2.
A sketch of a uniform p p, h, T, τ, η q -clustered configuration x “ p x , . . . , x d q asin Definition 2.5. d, p, m , such that for each C ηT ď Ω ď C h and (cid:15) ď C p Ω τ h q p ´ , it holds that: diam p E x ,j(cid:15), Ω p F qq ď C Ω (cid:15) ˆ p Ω τ h q ´ p ` , x j P x c , , x j P x z x c ; diam p E a ,j(cid:15), Ω p F qq ď C (cid:15) ˆ p Ω τ h q ´ p ` , x j P x c , , x j P x z x c . Remark 2.7.
Our main focus is to investigate the error rates of the SR problem as the clustersize becomes small. Fixing the parameters p, d, m , the range of admissible Ω in Theorem 2.6, C ηT ď Ω ď C h , is non-empty for a sufficiently small cluster size h . Furthermore we comment herethat the constants C , C actually only depend on d . The above estimates are order optimal, as our next main theorem shows. For simplicity andwithout loss of generality, in the results below we assume that the index κ is fixed. Theorem 2.8. (Lower bound) Let m ď M, ď p ď d, τ ď p ´ , η ă d , T ą be fixed. Thereexist positive constants C . . . , C , depending only on d, p, m, M , such that for every Ω , h satisfying h ď C T and Ω h ď C there exists F “ p a , x q P P d , with x forming a p p, h, T, τ, η q -clustered config-uration, and with ă m ď } a } ď M ă 8 , such that for certain indices j , j P t κ, . . . , κ ` p ´ u and every (cid:15) ď C p Ω τ h q p ´ , it holds that: diam p E x ,j(cid:15), Ω p F qq ě C Ω (cid:15) ˆ p Ω τ h q ´ p ` , if j “ j , , @ j R t κ, . . . , κ ` p ´ u ; diam p E a ,j(cid:15), Ω p F qq ě C (cid:15) ˆ p Ω τ h q ´ p ` , if j “ j , , @ j R t κ, . . . , κ ` p ´ u . Remark 2.9.
The lower bounds for the quantities diam p E x ,j(cid:15), Ω p F qq were shown in [1] to hold for anysignal F with real amplitudes , however, at the expense of the implicit dependence of the constantson the separation parameter τ . While bounding diam p E (cid:15), Ω p F qq (and its projections) for all signals F is an interesting question in its own right, in this paper we use these to bound the minimaxerror rate, and therefore it is sufficient to show that there exist certain signals with large enough E (cid:15), Ω p F q . As it turns out, it is possible to obtain a more accurate geometric description of thesesets, which in turn can be used for reducing reconstruction error if additional a-priori informationis available. Work in this direction was started in [2] and we intend to provide further details ofthese developments in a future work. ombining Theorems 2.6 and 2.8 with Proposition 2.4, we obtain optimal rates for the minimaxerror E and its projections as follows. Theorem 2.10.
Let m ă M, ď p ď d, τ ă p p ´ q , η ă d , T ą be fixed. There exist constants c , c , c , depending only on d, p, m, M such that for all c ηT ď Ω ď c h and (cid:15) ď c p Ω τ h q p ´ , theminimax error rates for the set U : “ U p p, d, h, τ, η, T, m, M q“ tp a , x q P P d : 0 ă m ď } a } ď M ă 8 , x forms a p p, h, T, τ, η q -clustered configuration u , satisfy the following.(1) For the non-cluster nodes: @ j R t κ, . . . , κ ` p ´ u : E x ,j p (cid:15), U, Ω q — (cid:15) Ω , E a ,j p (cid:15), U, Ω q — (cid:15). (2) For the cluster nodes: max j “ κ,...,κ ` p ´ E x ,j p (cid:15), U, Ω q — (cid:15) Ω p Ω τ h q ´ p ` , max j “ κ,...,κ ` p ´ E a ,j p (cid:15), U, Ω q — (cid:15) p Ω τ h q ´ p ` . The proportionality constants in the above statements depend only on d, p, m, M .Proof.
Let C , C , C , C , C , C be the constants from Theorems 2.6 and 2.8. Put c “ C and c “ min p C , C , C C q . Let c ηT ď Ω ď c h , and (cid:15) ď c p Ω τ h q p ´ , where c ď min p C , C q will bedetermined below. It is immediately verified that Ω , h and (cid:15) as above satisfy the conditions of bothTheorems 2.6 and 2.8. Upper bound:
Directly follows from the upper bounds in Theorem 2.6 and Proposition 2.4.
Lower bound:
Denote U (cid:15) “ ! F P U : E (cid:15), Ω p F q Ď U ) . To prove the lower bounds on E , it issufficient to show that there exists an F P U (cid:15) ‰ H such that the conclusions of Theorem2.8 are satisfied for this F .It is not difficult to see that for any choice of the parameters as above, the set U has a non-empty interior, and furthermore that one can choose m , M satisfying m ă m ă M ă M ,and also T “ . T , τ “ τ and η “ η , such that U “ U p p, d, h, τ , η , T , m , M q Ă U, B U X B U “ H . By construction, there exist positive constants ˜ C , ˜ C , independent of Ω , h and τ, η , suchthat inf u PB U,u PB U ˇˇ P x ,j p u q ´ P x ,j p u q ˇˇ ě ˜ C ˆ τ h, x j P x c ,ηT, x j P x z x c ;inf u PB U,u PB U ˇˇ P a ,j p u q ´ P a ,j p u q ˇˇ ě ˜ C . (2.5) Now we use the fact that (cid:15) ă c p Ω τ h q p ´ . Applying Theorem 2.6 to an arbitrary signal F P U , and using the conditions ď ηTc and Ω τ h ď Ω h ď c , we obtain that diam ˆ E x ,j (cid:15) p F q ˙ ď C c τ h, x j P x c , C c p Ω τ h q p ´ ď C c c c p ´ ηT, x j P x z x c ; diam ˆ E a ,j (cid:15) p F q ˙ ď C c , x j P x c , C c c p ´ , x j P x z x c . (2.6) ow we set c “ min p C , C , C q where C “ min p , c q ˆ min p , c ´ p ` q ˆ min ˆ C C , C C ˙ . Combining (2.5) and (2.6) we obtain that F P U (cid:15) . Since F P U was arbitrary, we concludethat U Ď U (cid:15) . Since clearly U ‰ H , applying Proposition 2.4 and Theorem 2.8 finishes theproof. (cid:3) Numerical optimality of Matrix Pencil algorithm
The main theoretical result of this paper, Theorem 2.10, establishes the best possible scalings forthe SR problem with clustered nodes. In this section we provide some numerical evidence that acertain SR algorithm, the Matrix Pencil (MP) method [37, 36], attains these performance bounds.Our choice of MP is fairly arbitrary, as we believe that many high-resolution algorithms havesimilar behaviour in the regime SRF " N , so that the spectral data is sampled with unitspacing. Algorithm 3.1:
The Matrix Pencil algorithm
Input :
Model order d Input :
Sequence t ˜ m k u , k “ , , . . . , N ´ N ą d , of the form (3.1) Input : pencil parameter d ` ď L ď N ´ d Output:
Estimates for the nodes t x j u and amplitudes t a j u as in (3.1) Compute the matrices A “ r H Ò , B “ r H Ó ; Compute the truncated Singular Value Decomposition (SVD) of
A, B of order d : A “ U Σ V H , B “ U Σ V H , where U , U , V , V are L ˆ d and Σ , Σ are d ˆ d ; Generate the reduced pencil A “ U H U Σ V H V , B “ Σ where A , B are d ˆ d ; Compute the generalized eigenvalues ˜ z j of the reduced pencil p A , B q , and put t ˜ x j u “ π t = ˜ z j u , j “ , . . . , d ; Compute ˜ a j by solving the linear least squares problem ˜a “ arg min a P C d } ˜m ´ ˜ V a } , where ˜ V “ ˜ V p ˜ x q is the Vandermonde matrix ˜ V “ r exp p πı ˜ x j k qs j “ ,...,dk “ ,...,N ´ ; return the estimated ˜ x j and ˜ a j .3.1. The Matrix Pencil method.
Let F “ p a , x q P P d as in (1.1) with x j P “ ´ , ‰ . Given thenoisy Fourier measurements˜ m k “ F p F qp´ k q looooomooooon “ m k ` n k “ d ÿ j “ a j exp p πıx j k q ` n k , k “ , , . . . , N ´ , N ą d, (3.1) he Matrix Pencil method estimates ˜ F “ p ˜ a , ˜ x q as follows. Consider the Hankel matrix(3.2) H “ »———– m m . . . m N ´ L ´ m m . . . m N ´ L ... . . . . . . ...m L m L ` . . . . . . m N ´ fiffiffiffifl P C p L ` qˆp N ´ L q , and further let H Ò “ H r L ´ , : s and H Ó “ H r L, : s be the L ˆ p N ´ L q matrix obtainedfrom H by deleting the last (respectively, the first) row. Then it turns out that that the numbers z j “ exp p πıx j q are the d nonzero generalized eigenvalues (i.e. rank-reducing numbers) of thepencil H Ó ´ zH Ò . If we now construct the noisy matrices A “ r H Ò , B “ r H Ó from the available data t ˜ m k u k “ ,...,N ´ , we could apparently just solve the Generalized Eigenvalue Problem with A, B .However, if L ą d then the pencil B ´ zA is close to being singular, and so an additional stepof low-rank approximation is required. We summarize the MP method in Algorithm 3.1, and theinterested reader is referred to the widely available literature on the subject (e.g. [37, 36, 45, 54],and references therein) for further details. Note that there exist numerous variants of MP, but,again, we believe the particular details to be immaterial for our discussion.3.2. Experimental setup.
Clustered node configurations.
In our experiments presented below, we constructed p p, h, T, τ,η q -clustered configurations with τ “ p ´ , T “ π, η “ π ´ hπ p d ´ p ` q as follows:(1) The cluster nodes x c “ p x , . . . , x p q where x j “ p j ´ q ¨ ∆ and ∆ “ hp ´ for j “ , . . . , p .(2) The non-cluster nodes were chosen to be x p ` j “ p p ´ q ∆ ` j ¨ π ´ p p ´ q ∆ d ´ p ` , j “ , . . . , d ´ p. Choice of signal and perturbation.
Two different schemes were tested: S1 A generic signal with complex amplitude vector a p q “ ` ı , ı , ı , . . . ˘ P C d and a boundedrandom perturbation sequence t n k u , uniformly distributed in r´ (cid:15), (cid:15) s . S2 Worst-case scenario in accordance with the construction of Section 6 (and in particular ofTheorem 6.2): a real amplitude vector a p q “ p , ´ , , . . . , q P R d and the perturbed Fouriercoefficient sequence t ˜ m k u of the particular signal F (cid:15) “ p a , x q P P d constructed accordingto Algorithm 3.2:˜ m k “ F p F (cid:15) qp´ k q “ d ÿ j “ a j exp p πı x j k q , k “ , . . . , N ´ . Results.
Error amplification factors.
In the first set of experiments, we measured the actual erroramplification factors K x ,j , K a ,j as in Algorithm 3.3 (recall also (1.4)), choosing (cid:15), N, h randomlyfrom a pre-defined numerical range. The results are presented in Figures 3 and 4 for the testingschemes S1 and S2 , accordingly. The scalings of Theorem 2.10, in particular the dependence onSRF, are confirmed. lgorithm 3.2: The worst-case perturbation signal
Input :
Signal F “ p a , x q P P d with a “ a p q and cluster nodes x c “ p x , . . . , x p q Input :
Noise level (cid:15)
Output:
The perturbed signal F (cid:15) Compute the cluster center µ “ x ` x p and put ˜ x c “ x c ´ µ ; Construct the moment vector of the centered cluster: g “ ´ř pj “ a j ˜ x kj ¯ k “ , ,..., p ´ P R p ; Construct the vector g to be equal to g except the last entry: g k “ g k for k “ , , . . . , p ´ g p ´ “ g p ´ ` (cid:15) ; Solve the Prony problem of order p with the data g (for (cid:15) small enough, a unique solutionalways exists – see Proposition A.3 and [13]), obtaining a signal F “ p a , x q P P p ; Move the cluster nodes back and put F (cid:15) p x q “ d ÿ j “ p ` a j δ p x ´ x j q ` p ÿ j “ a j δ p x ´ p x j ` µ qq ; return the signal F (cid:15) . Algorithm 3.3:
A single experiment
Input : p, d, h, N, (cid:15)
Input :
Testing scheme (either S1 or S2 ) Construct the signal F and the sequence ˜ m k , k “ , . . . , N ´ Compute the actual perturbation magnitude (cid:15) “ max k “ ,...,N ´ | F p F qp´ k q ´ ˜ m k | ; Execute the MP method (Algorithm 3.1) with L “ P N T and obtain F MP “ p a MP , x MP q ; for each j do compute the error for node j : e j “ min (cid:96) | x MPj ´ x (cid:96) | ; The success for node j is defined as Succ j “ ˆ e j ă min (cid:96) ‰ j | x (cid:96) ´ x j | ˙ . if Succ j == true then let (cid:96) p j q “ arg min (cid:96) | x MPj ´ x (cid:96) | ; compute normalized node error amplification factor K x ,j “ | x j ´ x MP(cid:96) p j q | ¨ N(cid:15) ; compute normalized amplitude error amplification factor K a ,j “ | a j ´ a MP(cid:96) p j q | (cid:15) ; return (cid:15) , and p K x ,j , K a ,j , Succ j q for each node j “ , . . . , d . log(SRF) -20246 l o g () p=2,d=3, S MP x ,1 MP x ,2 MP x ,3 SRF p -0.5 0.0 0.5 1.0 1.5 2.0 2.5 log(SRF) l o g () p=2,d=3, S MP a ,1 MP a ,2 MP a ,3 SRF p Figure 3.
The error amplification factors. Algorithm 3.3 was executed 500 times with p “ , d “
3, scheme S1 and varying h, N, (cid:15) . For cluster nodes j “ ,
2, the node error amplification factors K x ,j (left panel) scalelike SRF p ´ , while the amplitude error amplification factors K a ,j (right panel) scale like SRF p ´ . For thenon-cluster node j “
3, both error amplification factors are bounded by a constant. -1 0 1 2 log(SRF) -10-505 l o g () p=2,d=3, S MP x ,1 MP x ,2 MP x ,3 SRF p -1 0 1 2 log(SRF) -15-10-505 l o g () p=2,d=3, S MP a ,1 MP a ,2 MP a ,3 SRF p Figure 4.
Same setup as in Figure 3, scheme S2 . Comparing with Figure 3, the variance of the factorscorresponding to the cluster nodes is much smaller than for the case of random perturbations, indicatingthat the construction is indeed worst-case. Noise threshold for successful recovery.
In the second set of experiments, we investigated thenoise threshold (cid:15) Æ SRF ´ p for successful recovery, as predicted by the theory. We have performed15000 random experiments with scheme S1 (the randomness was in the choice of h, N, (cid:15) and thenoise sequence t n k u ) according to Algorithm 3.3, recording the success/failure result of each suchexperiment. The results for d “ p “ , (cid:15) Æ SRF ´ p . .0 1.5 2.0 2.5 3.0 log SRF l o g / p=2,d=4, S SuccessFail
SRF p log SRF l o g / p=3,d=4, S SuccessFail
SRF p Figure 5.
Phase transition for successful recovery, random bounded perturbations (scheme S1 ) with d “ p “ ,
3. Each experiment is represented by either a blue triangle (if the recovery was successful, i.e.
Succ j ““ T rue, @ j “ , . . . , d as returned by Algorithm 3.3) or a red circle otherwise. The relationship (cid:15) crit « SRF ´ p for the critical value of (cid:15) is confirmed. log SRF -2024 l o g / p=2,d=8, j=6, S SuccessFail
Figure 6.
Phase transition for successful recovery of a non-cluster node. Comparing with Figure 5, thethreshold is approximately constant (cid:15) crit « const . Here p “ , d “
8, scheme S1 , plotted is the successfulrecovery of the node at index j “ Normalization
In the intermediate claims, instead of considering a general signal F “ p a , x q P P d , we shallusually assume that the node vector x “ p x , . . . , x d q is normalized to the interval “ ´ , ‰ , andcentered around the origin, i.e. x d “ ´ x . Let us briefly argue how to obtain the general resultfrom this special case.Let us define the scale and shift transformations on P d . Definition 4.1.
For F “ ř dj “ a j δ p x ´ x j q P P d and α P R , we define SH α : P d Ñ P d as follows: SH α p F qp x q “ d ÿ j “ a j δ p x ´ p x j ´ α qq . efinition 4.2. For F “ ř dj “ a j δ p x ´ x j q P P d and T ą , we define SC T : P d Ñ P d as follows: SC T p F qp x q “ d ÿ j “ a j δ ´ x ´ x j T ¯ . By the shift property of the Fourier transform, for any (cid:15), Ω ą
0, we have that(4.1) SH α p E (cid:15), Ω p F qq “ E (cid:15), Ω p SH α p F qq . By the scale property of the Fourier transform we have that for any (cid:15) ą SC T p E (cid:15), Ω p F qq “ E (cid:15), Ω T p SC T p F qq . Thus we have the following.
Proposition 4.3.
Let F “ p a , x q P P d , α P R and T ą . Then for any (cid:15) ą and ď j ď d wehave diam p E x ,j(cid:15), Ω p F qq “ T diam ´ E x ,j(cid:15), Ω T p SC T p SH α p F qqq ¯ (4.3) diam p E a ,j(cid:15), Ω p F qq “ diam ´ E a ,j(cid:15), Ω T p SC T p SH α p F qqq ¯ (4.4) 5. Upper bounds
Overview of the proof.
The proof of Theorem 2.6, presented in the next subsections andsome of the appendices, is somewhat technical. In order to help the reader, we provide an overviewof the essential ideas and steps.The main object of the study, the error set E (cid:15), Ω p F q Ă P d , is the pre-image of an (infinite-dimensional) (cid:15) -cube in the data space, under the Fourier transform mapping F (recall (1.2) andDefinition 2.3). However, it is not obvious how to obtain quantitative estimates on F ´ directly.Thus we replace F with certain finite-dimensional sampled versions of it, denoted F M λ : P d Ñ C d ,where the sampling parameter λ defines the rate at which 2 d equispaced samples of F p F q are taken.The pre-images of (cid:15) -cubes under F M λ define the corresponding λ -error sets E (cid:15), p λ q Ă P d , and infact the original E (cid:15), Ω p F q is contained in the intersection of all the E (cid:15), p λ q . Thus, it is sufficient tobound the diameter of a single such E (cid:15), p λ ˚ q (see remark in the next paragraph) with a carefullychosen λ ˚ so that the result will be as small as possible. Such quantitative estimates are obtainedby careful analysis of the row-wise norms of the Jacobian matrix of F M ´ λ ˚ and applying the so-called quantitative inverse function theorem (Theorem B.1). Using these estimates, the optimal λ ˚ is shown to be on the order of Ω, from which the upper bounds of Theorem 2.6 follow.An additional technical complication arises from the fact that F M ´ λ defines a multivaluedmapping, and the full pre-image E (cid:15), p λ q contains multiple copies of a certain “basic” set A “ A (cid:15),λ .However, when considering the intersection of all E (cid:15), p λ q ’s, the non-zero shifts for certain different λ ’s do not intersect, and therefore eventually only the diameter of the basic set A needs to beestimated.Below is a brief description of the different intermediate results, and the organization of theremainder of Section 5.(1) In Subsection 5.2 we formally define the λ -decimated maps F M λ , the corresponding errorsets E (cid:15), p λ q , and provide quantitative estimates on the Jacobian of F M ´ λ in Proposition 5.4(proved in Appendix C). These bounds essentially depend on the “effective separation” ofeach node in x from its neighbours, after a blowup by a factor of λ .(2) In Subsection 5.3 we show that for a signal F “ p a , x q , there exist a certain range ofadmissible λ ’s, denoted by Λ p x q , for which the effective separation (see previous item)between the nodes in x c is of the order of Ω h , while for the rest of the nodes, it is bounded rom below by a constant independent of Ω , h . These estimates are proved in Proposition5.9.(3) In Subsection 5.4 we study in detail the geometry of the error sets E (cid:15), p λ q for λ P Λ p x q .First, we consider (in Subsection 5.4.1) the local inverses F M ´ λ . For each λ P Λ p x q ,we show that the local inverse exists in a neighborhood V of radius R « p Ω h q p ´ around F M λ p F q , and provide estimates on the Lipschitz constants of F M ´ λ on V and the diameterof F M ´ λ p V q . The main bounds to that effect are proved in Proposition 5.15, using thepreviously established general estimates from Proposition 5.4 and the quantitative inversefunction theorem (Theorem B.1).(4) Next, denoting A “ A R,λ “ F M ´ λ p V q , we show in Proposition 5.17 that the set E (cid:15), p λ q is aunion of certain copies of A , where each such copy is obtained by shifting the nodes in A by an integer multiple of λ ´ , and/or by permuting them.(5) In Subsection 5.5 we complete the proof. At this point we consider the entire set Λ p x q . Themain technical step, Proposition 5.18 (proved in Appendix F), establishes that for a certain λ ˚ P Λ p x q and all possible permutations π and shifts (cid:96) P Z zt u , there exists a particular¯ λ “ ¯ λ p π, (cid:96) q P Λ p x q such that the intersection between π -permutation and (cid:96) -shift of A R,λ ˚ and the entire error set E R, p ¯ λ q is empty. From this fact it immediately follows that theoriginal error set E (cid:15), Ω p F q with (cid:15) “ R is contained in A R,λ ˚ (Proposition 5.19). The proofis finished by invoking the previously established estimates on the diameter of A R,λ ˚ andits projections. Remark 5.1.
We expect that the tools developed throughout the proof will also be useful to calculatethe minimal finite sampling rate required to achieve the minimax error rate stated in Theorem 2.6. λ -decimation maps. For the purpose of the following analysis, we extend the space of signals P d to include signals with complex nodes and denote the extended space by ¯ P d ,¯ P d “ ! p a , x q : a “ p a , . . . , a d q P C d , x “ p x , . . . , x d q P C d ) . We will be considering specific sets of exactly 2 d samples of the Fourier transform, made atconstant rate λ as follows. Definition 5.2.
For λ ą , we define the map F M λ : ¯ P d – C d Ñ C d by F M λ pp a , x qq “ µ “ p µ , . . . , µ d ´ q , µ k “ d ÿ j “ a j e πix j λk , k “ , . . . , d ´ . We call such map a λ -decimation map.For λ ą (cid:15) ą
0, we define the corresponding error set E (cid:15), p λ q as follows. Definition 5.3.
The error set E (cid:15), p λ q p F q Ă P d is the set consisting of all the signals F P P d with ›› F M λ p F q ´ F M λ p F q ›› ď (cid:15). Similarly we denote by E a ,j(cid:15), p λ q p F q , E x ,j(cid:15), p λ q p F q the projection of the error set E (cid:15), p λ q p F q onto the corre-sponding amplitudes and the nodes components (compare (2.1) ). Now consider the given spectrum F p F qp s q , s P r´ Ω , Ω s . Clearly for each λ ď Ω2 d ´ we have that E (cid:15), Ω p F q Ď E (cid:15), p λ q p F q giving(5.1) E (cid:15), Ω p F q Ď č λ Pp , Ω2 d ´ s E (cid:15), p λ q p F q . ence, to prove the upper bounds in Theorem 2.6, we shall show that there exists a certain subset S Ď ´ , Ω2 d ´ ı such that for each λ P S , diam ` E (cid:15), p λ q p F q ˘ can be effectively controlled.In the next proposition, we derive a uniform bound on the norms of the inverse Jacobian of F M λ near a signal with clustered nodes. The bounds explicitly depend on the distances between theso-called “mapped” nodes z j p λ q “ e πiλx j . Proposition 5.4 (Uniform Jacobian bounds) . Let F “ p a , x q P ¯ P d , a “ p a , . . . , a d q , x “p x , . . . , x d q and for λ ą let z “ e πiλx , . . . , z d “ e πiλx d . Suppose that for each j “ , . . . , d , wehave ă m ď | a j | and ď | z j | ď for some m ą .Further assume that for ˜ η, ˜ h with ě ˜ η ě ˜ h , and x c “ t x κ , . . . , x κ ` p ´ u Ă x , p ě , the nodes z , . . . , z d satisfy:(1) For each x j , x k P x c , j ‰ k , we have that | z j ´ z k | ě ˜ h .(2) For each x (cid:96) P x z x c and x j P x , (cid:96) ‰ j , we have that | z (cid:96) ´ z j | ě ˜ η .Then the Jacobian matrix of F M λ at F , denoted by J λ p F q , is non-degenerate. Furthermore, writethe inverse Jacobian matrix J ´ λ p F q in the following block form J ´ λ p F q “ „ A ˜ B , where A, ˜ B are d ˆ d . Then, the (cid:96) norms of the rows of the blocks A, ˜ B are bounded as follows: d ÿ k “ | A j,k | ď K p ˜ η, d, p q , x j P x z x c , (5.2) d ÿ k “ | ˜ B j,k | ď K p m, ˜ η, d, p q λ , x j P x z x c , (5.3) d ÿ k “ | A j,k | ď K p ˜ η, d, p q ˜ h ´ p ` , x j P x c , (5.4) d ÿ k “ | ˜ B j,k | ď K p m, ˜ η, d, p q λ ˜ h ´ p ` , x j P x c , (5.5) where K p¨ , . . . , ¨q , K p¨ , . . . , ¨q , K p¨ , .., , ¨q , K p¨ , . . . , ¨q are constants depending only on the parame-ters inside the brackets. The proof of Proposition 5.4 is given in Appendix C.5.3.
The existence of an admissible decimation.
In this section we shall prove the existenceof a certain blowup factors λ , such that the mapped nodes t e πiλx j u (see Proposition 5.4 above)attain “good” separation properties. This result will later be used to show that for any such λ ,the corresponding inverse λ -decimation map F M ´ λ will have the smallest possible coordinatewiseLipschitz constants with respect to Ω , h (up to constants) (see Proposition 5.4). Definition 5.5.
For each x P R and a ą consider the operation mod ` ´ a , a ‰ defined as x mod ´ ´ a , a ı “ x ´ ka, where k is the unique integer such that x ´ ka P ` ´ a , a ‰ . Using this notation the principal value ofthe complex argument function is defined as Arg p re iθ q “ θ mod p´ π, π s , for each θ P R and r ą . efinition 5.6. For α, β P C zt u , we define the angular distance between α, β as = p α, β q “ ˇˇˇˇ Arg ˆ αβ ˙ˇˇˇˇ “ ˇˇˇˇ p Arg p α q ´ Arg p β qq mod p´ π, π s ˇˇˇˇ , where for z P C zt u , Arg p z q P p´ π, π s is the principal value of the argument of z . Lemma 5.7.
For | x | “ | y | “ , we have (5.6) 2 π = p x, y q ď | x ´ y | ď = p x, y q . Proof.
First, | x ´ y | “ ˇˇˇˇ ´ xy ˇˇˇˇ “ ˇˇˇˇ
12 Arg xy ˇˇˇˇ “ ˇˇˇˇ = p x, y q ˇˇˇˇ . Then use the fact that for any | θ | ď π we have2 π | θ | ď sin | θ | ď | θ | (cid:3) . Let F “ p a , x q P P d such that the node vector x “ p x , . . . , x d q forms a p p, h, T, τ, η q -clusteredconfiguration, with x c “ t x κ , x κ ` , . . . , x κ ` p ´ u . According to Proposition 5.4, the the norms of therows of the inverse Jacobian J ´ λ p F q essentially depend on the the minimal distance between themapped nodes z j p λ q “ e πiλx j . After a blowup by a factor of λ ď h , the pairwise angular distances = p¨ , ¨q (and hence the euclidean distances) between the mapped cluster-nodes z κ , . . . , z κ ` p ´ arenow of order λh .On the other hand, the non-cluster nodes are at distance larger than ηT " h . Therefore, after theblowup by λ , the non-cluster nodes z , . . . , z κ ´ , z κ ` p , . . . , z d may in principle be located anywhereon the unit circle. For example, any of these mapped non-cluster nodes might coincide with, or bevery close to, a certain mapped cluster node, or yet another mapped non-cluster node.While this situation might occur for some values of λ , we will now show that there exist certainsets of λ ’s for which this does not happen. We shall require the following key estimate concerningthe pairwise angular distance between any two mapped nodes. Lemma 5.8 (A uniform blowup of two nodes) . Let x j , x k P R , x j ‰ x k , and let ∆ “ | x j ´ x k | .Consider the following blowups z j “ z j p λ q “ e πiλx j , z k “ z k p λ q “ e πiλx k . Then for ď α ď π andan interval I “ r a, b s Ă R , the set (5.7) Σ αj,k p I q “ (cid:32) λ P I : = ` z j p λ q , z k p λ q ˘ ď α ( is a union of N intervals I , . . . , I N with t | I | ∆ u ď N ď t | I | ∆ u ` , and | I j | ď απ , j “ , . . . , N. Proof.
For each λ P I we have(5.8) = p z j p λ q , z k p λ qq “ ˇˇˇˇ Arg ˆ z j p λ q z k p λ q ˙ˇˇˇˇ “ ˇˇˇ Arg p e πiλ ∆ q ˇˇˇ . y equation (5.8) we have (cid:32) λ P I : = ` z j p λ q , z k p λ q ˘ ď α ( “ ! λ P I : ˇˇˇ Arg p e πiλ ∆ q ˇˇˇ ď α ) “t λ P I : | πλ ∆ mod p´ π, π s| ď α u “t λ P I : ´ α ď p πλ ∆ mod p´ π, π sq ď α u “ " λ P I : ´ α π ď ˆ λ mod ˆ ´ , ˙ ď α π * . The last set above can be written as I X S α where(5.9) S α “ " λ P R : ´ α π ď ˆ λ mod ˆ ´ , ˙ ď α π * . Define the interval I α “ “ ´ α π , α π ‰ . Then the set S α is a union of intervals of length απ asfollows S α “ ď (cid:96) P Z ˆ I α ` (cid:96) ∆ ˙ “ ď (cid:96) P Z " λ ` (cid:96) ∆ : λ P I α * . The intersection of S α with any interval I is then a union of t | I | ∆ u ď N ď t | I | ∆ u ` απ . This concludes the proof of Lemma 5.8. (cid:3) Now we state and prove the main result of this subsection.
Proposition 5.9.
Let F “ p a , x q P P d , x “ p x , . . . , x d q Ă r´ , s , such that x forms a p p, h, , τ, η q -clustered configuration with x c “ t x κ , x κ ` , . . . , x κ ` p ´ u .Let Ω ď d ´ ¨ h . For each λ ą let z p λ q “ e πiλx , . . . , z d p λ q “ e πiλx d .Then each interval I Ă ”
12 Ω2 d ´ , Ω2 d ´ ı of length | I | “ η contains a sub-interval I Ă I of length | I | ě p d η q ´ such that for each λ P I :(1) For all x (cid:96) P x z x c and x j P x , x j ‰ x (cid:96) , = p z (cid:96) p λ q , z j p λ qq ě d . (5.10) (2) For all x j , x k P x c , x k ‰ x j , = p z j p λ q , z k p λ qq ě πλτ h ě πτ d ´ h. (5.11) Proof.
Let us first prove that assertion (5.11) holds for any
12 Ω2 d ´ ď λ ď Ω2 d ´ .Let x j , x k , j ą k , be two cluster nodes. The angular distance between the mapped cluster nodes z j “ z j p λ q “ e πiλx j , z k “ z k p λ q “ e πiλx k , is = p z j , z k q “ ˇˇˇ Arg p e πiλ p x j ´ x k q q ˇˇˇ . By assumption Ω h ď d ´ , then λ ď h and then 0 ď πλ p x j ´ x k q ď πλh ď π . With this wehave = p z j , z k q “ πλ p x j ´ x k q ě πλτ h. By assumption λ ě
12 Ω2 d ´ . Then, = p z j , z k q ě πτ d ´ Ω h . This concludes the proof of assertion (5.11).Using Lemma 5.8 we now prove that assertion (5.10) holds for any interval I “ r a, b s Ă R oflength | I | “ η . Let I be such an interval. For each 0 ă α ď π consider the setΣ α p I q “ " λ P I : D x (cid:96) P x z x c s.t. min ď j ď d,j ‰ (cid:96) = p z (cid:96) p λ q , z j p λ qq ď α * . e then have Σ α p I q “ ď x (cid:96) P x z x c ď x j ‰ x (cid:96) Σ α(cid:96),j p I q , where Σ α(cid:96),j are given by (5.7). By Lemma 5.8 each Σ α(cid:96),j p I q above is a union of at most t | I | η u ` “ απ η . Therefore Σ α p I q is a union of at most K “ ` d ˘ “ d p d ´ q intervals. Moreover, let ν denote the Lebesgue measure on R , then(5.12) ν p Σ α p I qq ď K απ η ď d p d ´ q απ η ď d α η . Put α “ d then by (5.12)(5.13) ν p Σ α p I qq ď η . Now consider the complement set of Σ α p I q with respect to I , p Σ α p I qq c “ " λ P I : @ x (cid:96) P x z x c , min ď j ď d,j ‰ (cid:96) = p z (cid:96) p λ q , z j p λ qq ą d * . By (5.13)(5.14) ν ` p Σ α p I qq c ˘ ě | I | ´ η “ η ´ η “ η . In addition, since Σ α p I q is a union of at most K “ d p d ´ q intervals, then ` Σ α p I q ˘ c is a union ofat most(5.15) L “ K ` “ d p d ´ q ` ď d intervals. Using (5.14) and (5.15), the average size of these intervals is bounded as follows: ν ` p Σ α p I qq c ˘ L ě d η . We therefore conclude that ` Σ α p I q ˘ c contains an interval of length greater or equal to d η . Thisproves assertion (5.10) of Proposition 5.9. (cid:3) Error sets of admissible decimation maps.
Throughout this section we fix a signal F “p a , x q P P d , a “ p a , . . . , a d q , x “ p x , . . . , x d q Ă “ ´ , ‰ , such that x forms a p p, h, , τ, η q -clusteredconfiguration, with x c “ t x κ , x κ ` , . . . , x κ ` p ´ u and } a } ě m ą
0. We also fix Ω ą h ď d .Proposition 5.9 demonstrated the existence of certain λ -decimation maps which achieve goodseparation of the non-cluster nodes. We define the set Λ p x q to consist of all such admissible λ ’s, asfollows. Definition 5.10 (Admissible blowup factors) . For each F “ p a , x q P P d , x “ p x , . . . , x d q , suchthat x forms a p p, h, , τ, η q -clustered configuration and z j “ z j p λ q “ e πiλx j , j “ , . . . , d and Ω ą ,we define the set of admissible blowup factors Λ p x q “ Λ Ω ,d p x q as the set of all λ P ”
12 Ω2 d ´ , Ω2 d ´ ı satisfying:(1) For all (cid:96) ‰ j such that x (cid:96) P x z x c and x j P x , = p z (cid:96) p λ q , z j p λ qq ě d . (5.16)
2) For all j ‰ k such that x j , x k P x c = p z j p λ q , z k p λ qq ě πλτ h ě π d ´ τ h. (5.17)5.4.1. The local geometry of admissible decimation maps.
The next result gives an explicit descrip-tion of a neighborhood around F where the map F M λ is injective (and, therefore, we can speakabout a local inverse). Definition 5.11.
For each α, β ą we denote by H α,β p F q the closed polydisc H α,β p F q “ (cid:32) p a , x q P ¯ P d : } a ´ a } ď α, } x ´ x } ď β ( , and by H o α,β p F q the interior of H α,β p F q . The following is proved in Appendix D.
Proposition 5.12 (One-to-one) . For each λ P Λ p x q the map F M λ is injective in the open polydisc U “ H o m, τh π p F q Ă ¯ P d . Next we can estimate the Lipschitz constants of the inverse map
F M ´ λ , using the previouslyestablished general bounds in Proposition 5.4. Proposition 5.13.
Let H “ H m , τh π p F q Ă U “ H o m, τh π p F q . Then, for each F P H :(1) The Jacobian matrix of F M λ at F , denoted by J λ p F q , is non-degenerate.(2) Put J ´ λ p F q “ „ A ˜ B , where A, ˜ B are d ˆ d . Then, the (cid:96) norms of the rows of the blocks A, ˜ B are bounded as follows: d ÿ k “ | A j,k | ď ˜ C, x j P x z x c , (5.18) d ÿ k “ | ˜ B j,k | ď ˜ C , x j P x z x c , (5.19) d ÿ k “ | A j,k | ď ˜ C p Ω τ h q ´ p ` , x j P x c , (5.20) d ÿ k “ | ˜ B j,k | ď ˜ C p Ω τ h q ´ p ` , x j P x c , (5.21) where ˜ C “ ˜ C p m, d, p q is a constant depending only on d, m, p .Proof. Let F “ p a , x q P H, a “ p a , . . . , a d q , x “ p x , . . . , x d q . Let z j “ z j p λ q “ e πiλx j and let z j “ z j p λ q “ e πiλx j , j “ , . . . , d .By the integral mean value theorem, for each j “ , . . . , d , | z j ´ z j | “ ˇˇˇ e πiλx j ´ e πiλx j ˇˇˇ ď λτ h. Let (cid:96) ‰ j such that x (cid:96) P x z x c and x j P x . Since λ P Λ p x q , = p z (cid:96) , z j q ě d . Then by (5.6) | z (cid:96) ´ z j | ě πd . e get that | z (cid:96) ´ z j | ě | z (cid:96) ´ z j | ´ | z (cid:96) ´ z (cid:96) | ´ | z j ´ z j | ě | z (cid:96) ´ z j | ´ λτ h ě πd ´ λτ h. With Ω h ď d and λ ď Ω2 d ´ by assumption, we have that 2 λτ h ď πd then | z (cid:96) ´ z j | ě πd ´ λτ h ě πd ´ πd ě d . We conclude that for each (cid:96) ‰ j such that x (cid:96) P x z x c and x j P x (5.22) | z (cid:96) ´ z j | ě d . Let j ‰ k such that x j , x k P x c . λ P Λ p x q then = p z j , z k q ě πλτ h. Then by (5.6) | z j ´ z k | ě λτ h. With a similar argument as above, we get that | z j ´ z k | ě | z j ´ z k | ´ λτ h ě λτ h. Using λ P Λ p x q ñ λ ě Ω2 p d ´ q , we conclude that for each j ‰ k such that x j , x k P x c (5.23) | z j ´ z k | ě λτ h ě d ´ τ h. Now using (5.22) and (5.23) we invoke Proposition 5.4 with ˜ h “ d ´ Ω τ h and ˜ η “ d and as aresult prove Proposition 5.13 with˜ C “ p d ´ q p ´ max „ K ˆ d , d, p ˙ , K ˆ m, d , d, p ˙ ,K ˆ d , d, p ˙ , K ˆ m, d , d, p ˙ . (cid:3) Definition 5.14.
For v P C d and r ą , we denote by Q r p v q the closed cube of radius r centeredat v : Q r p v q “ Q r,d p v q “ ! u P C d : } u ´ v } ď r ) . Proposition 5.15.
Let U “ H o m, τh π p F q and H “ H m , τh π p F q Ă U . Let λ P Λ p x q and let µ λ “ F M λ p F q , then there exists a constant ˜ C “ ˜ C p m, d, p q such that for R “ ˜ C p Ω τ h q p ´ , F M λ p H q Ě Q R p µ λ q . Furthermore for V λ “ F M λ p U q let F M ´ λ : V λ Ñ U be the local inverse of F M λ , i.e. for all F P U we have F M ´ λ p F M λ p F qq “ F . For each ď j ď d , let P a ,j , P x ,j : ¯ P d Ñ C be the projections onto the j th amplitude and the j th node oordinates respectively. Then F M ´ λ is Lipschitz on Q R p µ λ q with the following bounds: ˇˇ P x ,j F M ´ λ p µ q ´ P x ,j F M ´ λ p µ q ˇˇ ď ˜ C } µ ´ µ } ˆ x j P x z x c p Ω τ h q ´ p ` x j P x c , ˇˇ P a ,j F M ´ λ p µ q ´ P a ,j F M ´ λ p µ q ˇˇ ď ˜ C } µ ´ µ } ˆ x j P x z x c p Ω τ h q ´ p ` x j P x c , for each µ , µ P Q R p µ λ q , where ˜ C “ ˜ C p m, d, p q , ˜ C “ ˜ C p m, d, p q are constants depending onlyon d, m, p and ˜ C ˜ C ď .Proof. By Proposition 5.12
F M λ is injective in the open neighborhood U of the polydisc H “ H m , τh π p F q . In addition, for each F P H the inverse Jacobian norm bounds derived in Proposition5.13 apply. Finally one can verify (using a similar argument as in the proof of Proposition 5.13 )that J λ p F q is non-degenerate for each F P U . We can therefore invoke Theorem B.1 with U, H and f “ F M λ and the bounds (5.18), (5.19), (5.20), (5.21), and conclude that Proposition 5.15holds with ˜ C “ ˜ C “ ˜ C and ˜ C “ min ´ m C , π ˜ C ¯ . (cid:3) The global geometry of admissible decimation maps.
In this subsection we give a globaldescription of the geometry of the error set E (cid:15), p λ q p F q for any λ P Λ p x q and for (cid:15) ď R where R “ ˜ C p Ω τ h q p ´ , and ˜ C is as specified in Proposition 5.15.For each λ P Λ p x q let µ λ “ F M λ p F q , and put(5.24) A (cid:15),λ p F q “ F M ´ λ p Q (cid:15) p µ λ qq č P d , where F M ´ λ : V λ Ñ U is the local inverse of F M λ on U .Observe that A (cid:15),λ p F q Ă E (cid:15), p λ q p F q . The analysis of this subsection will reveal that globally E (cid:15), p λ q p F q is made from certain periodic repetitions of the set A (cid:15),λ p F q and its permutations.Consider the following example. Example 5.16.
Let F p x q “ δ p x ´ q ` δ p x ´ q and let λ “ . Applying F M λ on F we get that F M λ p F q “ p , e π i ` e ´ π i , e ´ π i ` e π i , q “ p , ´ , ´ , q . If we set F “ p a , x q with a “ p a , a q “ p , q and x “ p x , x q “ p , q then clearly the signal F “ p a , x q , x “ p x , x q “ p , q , that is attained by permuting the nodes of the signal F ,satisfies that F M λ p F q “ F M λ p F q . Observe that F R P since its nodes are not in ascending order(a condition that was posed on P d to avoid redundant solutions). However, the signal F “ p a, x q with x “ x ´ λ p , q “ x ´ p , q “ p´ , q , is in P and it holds that F M λ p F q “ F M λ p F q .One can verify that the set of signals G P P , which satisfies F M λ p G q “ F M λ p F q is given by " G “ p a , y q P P : y “ x ` λ p n , n q , n , n P Z * ď" G “ p a , y q P P : y “ x ` λ p n , n q , n , n P Z * . In order to formalize the statement regarding the global structure of E (cid:15), p λ q p F q , which is essentiallya generalization of the example above, we require some notation regarding permutation and shiftoperations.We denote the set of permutations of d elements byΠ “ Π d Ă t π : t , . . . , d u Ñ t , . . . , d uu . or a vector x “ p x , . . . , x d q P C d and a permutation π , we denote by x π the vector attained bypermuting the coordinates of x according to π x π “ p x π p q , . . . , x π p d q q . For a set A Ď P d and a permutation π P Π d , we denote by A π the set attained from A by permutingthe nodes and amplitudes of each signal in A according to πA π “ tp a π , x π q : p a , x q P A u . The following proposition gives a description of the global geometry of E (cid:15), p λ q p F q . Its proof ispresented in Appendix E. Proposition 5.17.
For each λ P Λ p x q and (cid:15) ď RE (cid:15), p λ q p F q “ ˜ ď π P Π d ď (cid:96) P Z d A π(cid:15),λ p F q ` λ (cid:96) ¸ č P d . Proof of the upper bound.
Fix F “ p a , x q P P d , a “ p a , . . . , a d q , x “ p x , . . . , x d q Ă “ ´ , ‰ , such that x forms a p p, h, , τ, η q -clustered configuration with x c “ t x κ , x κ ` , . . . , x κ ` p ´ u ,and } a } ě m ą p x q (see Definition 5.10). By the analysis ofSection 5.4, under the assumption that Ω h ď d , the following assertions hold:(1) By Proposition 5.12 there exists a neighborhood U of F such that for each λ P Λ p x q , F M λ is one-to-one on U .(2) By Proposition 5.15 there exists a constant ˜ C “ ˜ C p m, d, p q such for each λ P Λ p x q , V λ “ F M λ p U q contains a cube Q R p µ λ q , where µ λ “ F M λ p F q and R “ ˜ C p Ω τ h q p ´ .For each λ P Λ p x q consider the local inverse F M ´ λ : V λ Ñ U and let (as above) A R,λ p F q “ F M ´ λ p Q R p µ λ qq č P d . The following intermediate claim is proved in Appendix F.
Proposition 5.18.
There exist positive constants K and K ď d depending only on d , suchthat for K η ď Ω ď K h the following holds. There exists λ P Λ p x q such that for each pair p π, (cid:96) q P Π d ˆ ` Z d zt u ˘ , there exists λ π, (cid:96) P Λ p x q for which (5.25) ˆ A πR,λ p F q ` λ (cid:96) ˙ č E R, p λ π, (cid:96) q p F q “ H . With a bit of additional work, we obtain the main geometric result regarding the error set E (cid:15), Ω p F q . Proposition 5.19.
Let Ω as in Proposition 5.18, then there exists λ P Λ p x q such that (5.26) E R, Ω p F q Ă A R,λ p F q . Proof.
Using Proposition 5.18 fix λ ˚ P Λ p x q which satisfies (5.25). We will prove that λ ˚ satisfies(5.26).For each λ P Λ p x q , we have the following result due to Proposition 5.17:(5.27) E R, p λ q p F q Ă ď π P Π d ď (cid:96) P Z d ˆ A πR,λ p F q ` λ (cid:96) ˙ . Putting (cid:15) “ R in (5.1) we obtain(5.28) E R, Ω p F q Ď č λ Pp , Ω2 d ´ s E R, p λ q p F q . e then obtain (5.26) from (5.25), (5.27) and (5.28) by algebra of sets calculation as follows:First by (5.28)(5.29) E R, Ω p F q Ď č λ Pp , Ω2 d ´ s E R, p λ q p F q “ E R, p λ ˚ q p F q X ¨˚˝ č λ Pp , Ω2 d ´ s E R, p λ q p F q ˛‹‚ . By (5.27)(5.30) E R, p λ ˚ q p F q Ă ď π P Π d , (cid:96) P Z d ˆ A πR,λ ˚ p F q ` λ ˚ (cid:96) ˙ . Then by (5.29) and (5.30)(5.31) E R, Ω p F q Ď ¨˝ ď π P Π d , (cid:96) P Z d A πR,λ ˚ p F q ` λ ˚ (cid:96) ˛‚ X ¨˚˝ č λ Pp , Ω2 d ´ s E R, p λ q p F q ˛‹‚ . For each pair p π, (cid:96) q P Π d ˆ ` Z d zt u ˘ , let λ π, (cid:96) P Λ p x q be the value asserted by Proposition 5.18,i.e. satisfying (5.25) for λ “ λ ˚ . By this and by (5.31) we have E R, Ω p F q Ă ˜ ď π P Π d A πR,λ ˚ p F q ¸ ď ¨˝ ď p π, (cid:96) qP Π d ˆp Z d zt uq "ˆ A πR,λ ˚ p F q ` λ ˚ (cid:96) ˙ č E R, p λ π, (cid:96) q p F q *˛‚ “ ď π P Π d A πR,λ ˚ p F q . (5.32)By definition E R, Ω p F q Ă P d where we assume a canonical ascending order of the nodes. Then, weconclude from (5.32) that E R, Ω p F q Ă A R,λ ˚ p F q which proves (5.26) for λ “ λ ˚ . (cid:3) We have everything in place to estimate the diameter of the set E (cid:15), Ω p F q and its projections. Proposition 5.20.
Let F “ p a , x q P P d , x Ă “ ´ , ‰ , such that x forms a p p, h, , τ, η q -clusteredconfiguration and } a } ě m ą . Then there exist positive constants C , . . . , C , depending only on d, p, m, such that for each C η ď Ω ď C h and (cid:15) ď C p Ω τ h q p ´ , it holds that: diam p E x ,j(cid:15), Ω p F qq ď C p Ω τ h q ´ p ` (cid:15), x j P x c ,C (cid:15), x j P x z x c ,diam p E a ,j(cid:15), Ω p F qq ď C p Ω τ h q ´ p ` (cid:15), x j P x c ,C (cid:15), x j P x z x c . Proof.
Let Ω be such that K η ď Ω ď K h , where K “ K p d q , K “ K p d q are the constantsspecified in Proposition 5.18. Let (cid:15) ď ˜ C p Ω τ h q p ´ “ R , where ˜ C “ ˜ C p m, d, p q is as specified inProposition 5.15. Let F P E (cid:15), Ω p F q with F “ p a , x q . Using Proposition 5.19 fix λ ˚ P Λ p x q whichsatisfies (5.26), and put µ ˚ “ F M λ ˚ p F q . Consequently F P A R,λ ˚ p F q “ F M ´ λ ˚ p Q R p µ ˚ qq X P d . ut µ “ F M λ ˚ p F q . By Proposition 5.15 there exist constants ˜ C “ ˜ C p m, d, p q , ˜ C “ ˜ C p m, d, p q such that | x j ´ x j | “ } P x ,j F M ´ λ ˚ p µ ˚ q ´ P x ,j F M ´ λ ˚ p µ q} ď ˜ C p Ω h q ´ p ` (cid:15), x j P x c , ˜ C (cid:15), x j P x z x c . | a j ´ a j | “ } P a ,j F M ´ λ ˚ p µ ˚ q ´ P a ,j F M ´ λ ˚ p µ q} ď ˜ C p Ω h q ´ p ` (cid:15), x j P x c ˜ C (cid:15), x j P x z x c . Since F was an arbitrary signal in E (cid:15), Ω p F q , we repeat the above argument with F P E (cid:15), Ω p F q and consequently prove Proposition 5.20 with C “ C , C “ C , C “ ˜ C , C “ K and C “ K . (cid:3) We are now in a position to prove Theorem 2.6, essentially by combining Proposition 5.20 withProposition 4.3.
Proof of Theorem 2.6.
Let F “ p a , x q P P d such that x forms a p p, h, T, τ, η q -clustered configurationand } a } ě m ą
0. Let C ηT ď Ω ď C h where C “ C p d, p, m q , C “ C p d, p, m q are the constantsspecified in Proposition 5.20.Put α “ p x ` x d q{
2. The signal SC T p SH α p F qq “ p a , ˜ x q , ˜ x “ p ˜ x , . . . , ˜ x d q , ˜ x “ x ´ αT , . . . , ˜ x d “ x d ´ αT , is normalized such that ˜ x , . . . , ˜ x d P r´ , s . The node vector ˜ x forms a p p, hT , , τ, η q -clustered configuration. Applying Proposition 5.20 for ˜ F “ SC T p SH α p F qq , ˜ h “ hT , ˜Ω “ Ω T ě C η and ˜Ω˜ h “ Ω h ď C , we conclude that there exist constants C , C , C , depending only on d, p, m ,such that for any (cid:15) ď C p Ω τ h q p ´ diam ` E x ,j(cid:15), Ω T p SC T p SH α p F qqq ˘ ď C T p Ω τ h q ´ p ` (cid:15), x j P x c ,C T (cid:15), x j P x z x c ,diam ` E a ,j(cid:15), Ω T p SC T p SH α p F qqq ˘ ď C p Ω τ h q ´ p ` (cid:15), x j P x c ,C (cid:15), x j P x z x c . Applying Proposition 4.3 we conclude the proof Theorem 2.6. (cid:3) Lower bounds
In this section all the constants c , . . . , k , . . . , K , . . . are unrelated to those of the previoussection.The main technical result we need is the following. Proposition 6.1.
Let F “ p a , x q P P d , such that x forms a p p, h, , τ, η q -clustered configuration,with cluster nodes x c “ p x , . . . , x p q (according to Definition 2.5), and with a P R d satisfying m ď } a } ď M .Then there exist constants c , k , k , depending only on p d, τ, m, M q , such that for all (cid:15) ă c p Ω h q p ´ and Ω h ď , there exists a signal F (cid:15) P P d satisfying, for some j , j P t , . . . , p u , | P x ,j p F (cid:15) q ´ P x ,j p F q| ě k Ω p Ω h q ´ p ` (cid:15), (6.1) | P a ,j p F (cid:15) q ´ P a ,j p F q| ě k p Ω h q ´ p ` (cid:15), (6.2) | F p F (cid:15) q p s q ´ F p F qp s q| ď (cid:15), | s | ď Ω . (6.3)Assuming validity of Proposition 6.1, let us prove Theorem 2.8. roof of Theorem 2.8. Let a P R d be any real amplitude vector satisfying m ď } a } ď M . Let Ω , h satisfy Ω h ď
2, and choose x to be the configuration with cluster nodes x c “ p x “ , x “ τ h, . . . , x p “ p p ´ q τ h q , with the rest of the nodes equally spaced in pp p ´ q τ h, q . Now denote h “ p p ´ q τ h and τ “ p ´ . Clearly, x is a p p, h , , τ , η q -clustered configuration for all sufficiently small h (forinstance, h ă d ă ´ η p d ´ p ` q ). Now we apply Proposition 6.1 with the signal F “ p a , x q .Since τ does not depend on τ , and therefore the constants c , k , k depend only on d, p, m, M , weconclude that for (cid:15) ă c p p ´ q p ´ p Ω τ h q p ´ and Ω h ă p p ´ q τ , there exist j , j P t , . . . , p u suchthat diam p E x ,j (cid:15), Ω p F qq ě k Ω p p ´ q ´ p ` (cid:15) p Ω τ h q ´ p ` ,diam p E a ,j (cid:15), Ω p F qq ě k (cid:15) p p ´ q ´ p ` p Ω τ h q ´ p ` . Now we consider the case of a non-cluster node, x j P x z x c . Let F “ p a , x q be the signal above.Decompose F as follows: F p x q “ a j δ p x ´ x j q ` ÿ (cid:96) ‰ j a (cid:96) δ p x ´ x (cid:96) q loooooooomoooooooon F o . Now let (cid:15) be fixed. Define a j “ a j ` (cid:15) and x j “ x j ` (cid:15) π Ω M . Put F j p x q “ a j δ p x ´ x j q ` F o p x q . For | s | ď Ω, the difference between the Fourier transforms of F and F j satisfies ˇˇ F p F qp s q ´ F p F j qp s q ˇˇ “ ˇˇˇ a j e πix j s ´ a j e πix j s ˇˇˇ ď ˇˇˇ a j e πix j s ´ ´ e πi (cid:15) π Ω M s ¯ˇˇˇ ` ˇˇ a j ´ a j ˇˇ ď (cid:15) ` (cid:15) “ (cid:15). Since the constants do not depend on τ at all, and the above construction of F j can be repeatedfor each j R t κ, . . . , κ ` p ´ u , the proof of the non-cluster node case is finished.Again, the case of general T follows by rescaling and applying Proposition 4.3 (as was done inthe proof of Theorem 2.6).This finishes the proof of Theorem 2.8 with C “ max ´ k p p ´ q p ´ , πM ¯ , C “ max ´ , k p p ´ q p ´ ¯ , C “ c p p ´ q p ´ , C “ d and C “ (cid:3) In the rest of this section we prove Proposition 6.1.We start by stating the following result which has been shown in [2, Theorems 4.1 and 4.2].
Theorem 6.2.
Given the parameters ă h ď , ă τ ď , ă m ď M ă 8 , let the signal F “ p a , x q P P d with a P R d form a single uniform cluster as follows: ‚ (centered) x d “ ´ x ; ‚ (uniform) for ď j ă k ď d we have τ h ď | x j ´ x k | ď h ; ‚ m ď } a j } ď M .Then there exist constants K , . . . , K depending only on p d, τ, m, M q such that for every (cid:15) ă K h d ´ , there exists a signal F (cid:15) “ p b , y q P P d such that(1) m k p F q “ m k p F (cid:15) q for k “ , , . . . , d ´ , where m k are given by (A.1) ;(2) m d ´ p F (cid:15) q “ m d ´ p F q ` (cid:15) ;(3) K h ´ d ` (cid:15) ď } x ´ y } ď K h ´ d ` (cid:15) ; K h ´ d ` (cid:15) ď } b ´ a } ď K h ´ d ` (cid:15) .Proof of Proposition 6.1. Define F c and F nc to be the cluster and the non-cluster part of F corre-spondingly, i.e. F c “ ÿ x j P x c a j δ p x ´ x j q ,F nc “ ÿ x j P x z x c a j δ p x ´ x j q . Without loss of generality, suppose that F c is centered, i.e. x ` x p “
0. Next, define a blowup of F c by Ω as follows:(6.4) F c p Ω q “ SC p F c q “ ÿ x j P x c a j δ p x ´ Ω x j q . Put ˜ d “ p, ˜ h “ Ω h , and let c “ K ´ ˜ d, τ, m, M ¯ as in Theorem 6.2. Let (cid:15) ď c p Ω h q p ´ . Nowwe apply Theorem 6.2 with parameters ˜ d, ˜ h, τ, m, M, ˜ (cid:15) “ c (cid:15) and the signal F c p Ω q , where c ď G c p Ω q ,(cid:15) such that the following hold for the difference H “ G c p Ω q ,(cid:15) ´ F c p Ω q : m k p H q “ , k “ , , . . . , p ´ , (6.5) m p ´ p H q “ ˜ (cid:15) ;(6.6)while also, for some j , j P t , . . . , p u ˇˇˇ P x ,j ´ G c p Ω q ,(cid:15) ¯ ´ P x ,j ´ F c p Ω q ¯ˇˇˇ ě K p Ω h q ´ p ` ˜ (cid:15), (6.7) ˇˇˇ P x ,j ´ G c p Ω q ,(cid:15) ¯ ´ P x ,j ´ F c p Ω q ¯ˇˇˇ ď K p Ω h q ´ p ` ˜ (cid:15), j “ , . . . , p, (6.8) ˇˇˇ P a ,j ´ G c p Ω q ,(cid:15) ¯ ´ P a ,j ´ F c p Ω q ¯ˇˇˇ ě K p Ω h q ´ p ` ˜ (cid:15). (6.9)Now put F c p Ω q ,(cid:15) “ SC Ω ´ G c p Ω q ,(cid:15) ¯ . Applying the inverse blowup to the above inequalities, we obtain in fact that ˇˇˇ P x ,j ´ F c p Ω q ,(cid:15) ¯ ´ P x ,j p F c q ˇˇˇ ě K Ω p Ω h q ´ p ` ˜ (cid:15), (6.10) ˇˇˇ P a ,j ´ F c p Ω q ,(cid:15) ¯ ´ P a ,j p F c q ˇˇˇ ě K p Ω h q ´ p ` ˜ (cid:15). (6.11)From the above definitions we have H Ω “ SC Ω p H q “ F c p Ω q ,(cid:15) ´ F c . Let us now show that there isa choice of c such that(6.12) | F p H Ω q p s q| ď (cid:15), | s | ď Ω . Put ω “ s { Ω, then F p H Ω q p s q “ F p H q p ω q . Now we employ the fact that the Fourier transform of a spike train has Taylor series coefficientsprecisely equal to its algebraic moments (see [1, Proposition 3.1]):(6.13) F p H qp ω q “ ÿ k “ k ! m k p H q p´ πıω q k . ext we apply the following easy corollary of the Tur´an’s First Theorem [56, Theorem 6.1], ap-pearing in [14, Theorem 3.1], using the recurrence relation satisfied by the moments of H accordingto Proposition A.2. Theorem 6.3.
Let H “ ř pj “ β j δ p x ´ t j q , and put R “ min j “ ,..., p | t j | ´ ą . Then, for all k ě p we have the so-called “Taylor domination” property (6.14) | m k p H q| R k ď ˆ ek p ˙ p max (cid:96) “ , ,..., p ´ | m (cid:96) p H q| R (cid:96) . Proposition 6.4.
The constant R in Theorem 6.3 satisfies R ě C , where C does not depend on Ω , h .Proof. Recall that H “ G c p Ω q ,(cid:15) ´ F c p Ω q . The nodes of F c p Ω q are, by construction, inside the interval “ ´ Ω h , Ω h ‰ . The nodes of G c p Ω q ,(cid:15) , by (6.8), satisfy ˇˇˇ P x ,j ´ G c p Ω q ,(cid:15) ¯ˇˇˇ ď Ω h ` K p Ω h q ´ p ` ˜ (cid:15) ď Ω h ` K p Ω h q ´ p ` c p Ω h q p ´ “ p Ω h q ˆ c K ` ˙ . Since Ω h ď C “ p c K ` q . (cid:3) Therefore, by (6.14), (6.5) and (6.6) we have for k ě p | m k p H q| ď ˆ ep ˙ p k p R p ´ ´ k ˜ (cid:15) ď C C p ´ ´ k k p ˜ (cid:15). Now plugging this into (6.13) we obtain | F p H q p ω q| ď ˜ (cid:15) | πω | p ´ p p ´ q ! ` C C p ´ ˜ (cid:15) ÿ k ě p ˆ π | ω | C ˙ k k p k ! . Put ζ “ π | ω | C , then, since | ω | ď | F p H q p ω q| ď C ˜ (cid:15) ÿ k ě p ´ ζ k k p k ! ď C ˜ (cid:15). We can therefore choose c “ min ´ , C ¯ to ensure that | F p H q p ω q| ď (cid:15), | ω | ď , which shows (6.12).Finally, construct the signal F (cid:15) “ F nc ` F c p Ω q ,(cid:15) . Combining (6.12), together with (6.10) and (6.11)finishes the proof of Proposition 6.1 with k “ K and k “ K . (cid:3) eferences [1] Andrey Akinshin, Dmitry Batenkov, and Yosef Yomdin. Accuracy of spike-train Fourier reconstruction forcolliding nodes. In , pages 617–621. IEEE, 2015.[2] Andrey Akinshin, Gil Goldman, and Yosef Yomdin. Geometry of error amplification in solving Prony systemwith near-colliding nodes. arXiv:1701.04058 [math] , January 2017.[3] C´eline Aubel and Helmut B¨olcskei. Vandermonde matrices with nodes in the unit disk and the large sieve. Appliedand Computational Harmonic Analysis , August 2017.[4] Jon R Auton and Michael L Van Blaricum. Investigation of procedures for automatic resonance extraction fromnoisy transient electromagnetics data.
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IEEE trans-actions on Signal Processing , 50(6):1417–1428, 2002. ppendix A. Algebraic Prony system
The so-called Prony system of equations relates the parameters of the signal F as in (1.1) andits algebraic moments (A.1) m k p F q “ ż F p x q x k dx “ d ÿ j “ a j x kj , k “ , , . . . , . Extending the above to arbitrary complex nodes and amplitudes, we define the
Prony map
P M : C d Ñ C d as follows:(A.2) P M k p a , . . . , a d , w , . . . , w d q “ d ÿ j “ a j w kj , k “ , , . . . , d ´ . Now consider the system of equations defined by
P M , i.e. with unknowns t a j , z j u dj “ P C d anda given right hand side µ “ p µ , . . . , µ d ´ q P C d , P M k p a , . . . , a d , z , . . . , z d q “ µ k , k “ , , . . . , d ´ . (A.3)The following fact can be found in the literature about Prony systems and Pad´e approximation(see e.g. [13] Propositions 3.2 and 3.3). Proposition A.1.
If a solution p a , . . . , a d , z , . . . , z d q to System (A.3) exists with a j ‰ , j “ , . . . , d and for ď j ă k ď d , z j ‰ z k , it is unique up to a permutation of the nodes t z j u andcorresponding amplitudes t a j u . Clearly, the definition of
P M k is valid for arbitrary integer k P N . The next fact is very well-known, and it is the basis of Prony’s method of solving (A.3). Proposition A.2.
Let the sequence ν “ t ν k u k P N be given by ν k “ P M k p a , . . . , a d , z , . . . , z d q . Then each consecutive d ` elements of ν satisfy the following linear recurrence relation: (A.4) d ÿ (cid:96) “ ν k ` (cid:96) c (cid:96) “ , where the constants t c (cid:96) u d(cid:96) “ are the coefficients of the (monic) polynomial with roots t z , . . . , z d u (the “Prony polynomial”), i.e. (A.5) Q p z q “ d ź j “ p z ´ z j q ” d ÿ (cid:96) “ c (cid:96) z (cid:96) . Proof.
Let k P N , then d ÿ (cid:96) “ ν k ` (cid:96) c (cid:96) “ d ÿ (cid:96) “ c (cid:96) d ÿ j “ a j z k ` (cid:96)j “ d ÿ j “ a j z kj Q p z j q “ . (cid:3) Proposition A.3 (Prony’s method) . Let there be given the algebraic moments t m k p F qu d ´ k “ of thesignal F “ p a , x q where the nodes of x are pairwise distinct and } a } ą . Then the parameters p a , x q can be recovered exactly by the following procedure:
1) Construct the d ˆ p d ` q Hankel matrix H “ r m i ` j s ď j ď d ď i ď d ´ ;(2) Find a nonzero vector c in the null-space of H ;(3) Find x j to be the roots of the Prony polynomial (A.5) , whose coefficient vector is c ;(4) Find the amplitudes a by solving the linear system V a “ m , where V is the Vandermondematrix V “ ” x kj ı j “ ,...,dk “ ,...,d ´ .Proof. See e.g. [13]. (cid:3)
Appendix B. Quantitative Inverse Function Theorem
Here we prove a certain quantitative version of the inverse function theorem, which applies toholomorphic mappings C d Ñ C d (here d is a generic parameter).For a P C d and r , . . . , r d ą
0, let H r ,...,r d p a q Ă C d be the closed polydisc centered at a , H r ,...,r d p a q “ t x P C d : | x j ´ a j | ď r j , for all j “ , . . . , d u . For j “ , . . . , d, we denote by P j : C d Ñ C the orthogonal projection onto the j th coordinate.With some abuse of notation we will also treat P j as the d ˆ d matrix representing this projection.Finally recall Definition 5.14 of the hypercube Q r . Theorem B.1.
Let U Ď C d be open. Let f : U Ñ C d be a holomorphic injection with an invertibleJacobian J p x q , for all x P U . For a P U and r , . . . , r d ą , let H p a q “ H r ,...,r d p a q Ă U be suchthat for all x P H p a q , d ÿ k “ | J ´ j,k p x q| ď α j , j “ , . . . , d. Put b “ f p a q and f p U q “ V . Then:(1) For R “ min p r α , . . . , r d α d q , Q R p b q Ď f p H p a qq and f ´ : V Ñ U is holomorphic in an openneighborhood of Q R p b q .(2) For each j “ , . . . , d , f ´ j “ P j f ´ : Q R p b q Ñ C d is Lipschitz on Q R p b q with | f ´ j p y q ´ f ´ j p y q| ď α j } y ´ y } , for each y , y P Q R p b q .Proof. First we show that f p U q “ V is open and f ´ is holomorphic and provides a homeomorphismbetween U and V .By assumption f : U Ñ V is an injection, then f ´ : V Ñ U is well defined. By assumption f iscontinuously differentiable with non-degenerate Jacobians J p x q for all x P U . Then by the InverseFunction Theorem V is open and f ´ is continuously differentiable on V . We conclude that f is abiholomorphism between U and V . We now show that for R “ min p r α , . . . , r d α d q , Q R p b q Ď f p H p a qq . f is a homeomorphism between U and V , hence S “ f p H p a qq is a compact subset of V . We take Q R p b q Ď S as the maximal cubecentered at b that is contained in S .Then, there exists a point p such that p P B S X B Q R p b q . Put h “ p ´ b . f ´ is continuouslydifferentiable on V Ą Q R p b q , we can therefore apply the Mean Value Theorem in integral form It is an interesting fact that the condition that f has non-degenerate Jacobians on U can be dropped. Contraryto a real version of Theorem B.1 where this condition is necessary, it is true that if f is holomorphic and an injectionon the open set U then f is biholomorphism between U and f p U q (see e.g. [52], discussion at page 23). nd obtain (here the integral is applied to each component of the inverse Jacobian matrix) f ´ p b ` h q ´ f ´ p b q “ ˆż J ´ p b ` t h q dt ˙ h . Then for each coordinate j “ , . . . , d, (B.1) f ´ j p b ` h q ´ f ´ j p b q “ ˆż P j J ´ p b ` t h q dt ˙ h .f is a homeomorphism between U and V hence f ´ maps the boundary of S into boundary of f ´ p S q “ Q r p a q . Therefore there exists a coordinate ˆ j P t , . . . , d u such that ˇˇˇ f ´ j p b ` h q ´ f ´ j p b q ˇˇˇ “ r ˆ j . Then by equation (B.1) r ˆ j “ ˇˇˇ f ´ j p b ` h q ´ f ´ j p b q ˇˇˇ “ ˇˇˇˇˆż P ˆ j J ´ p b ` t h q dt ˙ h ˇˇˇˇ ď α ˆ j } h } “ α ˆ j R . Hence R ě r ˆ j α ˆ j ě min p r α , . . . , r d α d q “ R . We get that Q R p b q Ď Q R p b q Ď S “ f p H p a qq . Since we already argued that V Ą f p H p a qq Ě Q R p b q is open then clearly f ´ is holomorphic inan open neighborhood of Q R p b q . This proves item (1) of Theorem B.1.The second item of the Theorem is proved with a similar argument: let y , y P Q R p b q and put h “ y ´ y . Applying again the Mean Value Theorem ˇˇˇ f ´ j p y ` h q ´ f ´ j p y q ˇˇˇ “ ˇˇˇˇˆż P j J ´ p y ` t h q dt ˙ h ˇˇˇˇ ď α j } h } . This proves item (2) of the Theorem. (cid:3)
Appendix C. Norm bounds on the inverse Jacobian matrix
Let F “ p a , x q P ¯ P d , a “ p a , . . . , x d q , x “ p x , . . . , x d q . Put z j “ z j p λ q “ e πiλx j , j “ , . . . , d .By direct computation, the Jacobian matrix J “ J λ p F q “ J λ p a , x q , of F M λ at F is given by(C.1) J λ p a , x q “ »————– .. .. z .. z d .. z .. z d z .. z d ... . . . ... ... . . . ...z d ´ .. z d ´ d p d ´ q z d ´ .. p d ´ q z d ´ d fiffiffiffiffifl „ I d D , where D is a d ˆ d diagonal matrix, D j,j “ a j πiλz j , j “ , . . . , d , and I d is the d ˆ d identitymatrix.Denote the left hand matrix in the factorization (C.1) by U d “ U d p z , . . . , z d q . The matrix U d isan instance of a confluent Vandermonde matrix, whose inverses have been extensively studied in [30,31, 7]. In particular, the elements of U ´ d can be constructed using the coefficients of polynomialsfrom an appropriate Hermite interpolation scheme. Consequently, we have the following result dueto [31]. Theorem C.1 (Gautschi, [31], eqs. (3.10), (3.12)) . For z , . . . , z d P C pairwise distinct, put U ´ d p z , . . . , z d q “ „ AB , here A, B are d ˆ d . Then we have the following upper bounds on the 1-norm of the rows of theblocks A, B d ÿ k “ | A j,k | ď p ` p ` | z j |q| ∆ j |q Γ j , j “ , .., d, (C.2) d ÿ k “ | B j,k | ď p ` | z j |q Γ j , j “ , .., d, (C.3) where ∆ j “ d ÿ (cid:96) “ ,(cid:96) ‰ j | z j ´ z (cid:96) | , Γ j “ ¨˝ d ź (cid:96) “ ,(cid:96) ‰ j ` | z (cid:96) || z j ´ z (cid:96) | ˛‚ . Proof of Proposition 5.4.
By the factorization (C.1) J λ p F q “ U d p z , . . . , z d q „ I d D , where z “ e πiλx , . . . , z d “ e πiλx d and D “ D p z , . . . , z d q is the d ˆ d diagonal matrix, D j,j “ a j πiλz j , j “ , . . . , d .By assumption, the mapped nodes t z j u are pairwise distinct, and so it immediately follows that J λ p F q is non-degenerate.Put U ´ d “ U ´ d p z , . . . , z d q “ „ AB , where A, B are d ˆ d . Put ˜ B “ D ´ B . Then(C.4) J ´ λ p F q “ „ A ˜ B . By Theorem C.1 d ÿ k “ | A j,k | ď p ` p ` | z j |q| ∆ j |q Γ j , j “ , .., d, (C.5) d ÿ k “ | B j,k | ď p ` | z j |q Γ j , j “ , .., d, (C.6)where ∆ j “ d ÿ (cid:96) “ ,(cid:96) ‰ j | z j ´ z (cid:96) | , Γ j “ ¨˝ d ź (cid:96) “ ,(cid:96) ‰ j ` | z (cid:96) || z j ´ z (cid:96) | ˛‚ . ‚ Non-cluster node Let (cid:96) be such that x (cid:96) P x z x c .By assumptions we have | z (cid:96) ´ z j | ě ˜ η, @ x (cid:96) P x z x c , x j P x , (cid:96) ‰ j. Then we obtain(C.7) ∆ (cid:96) “ d ÿ j “ ,j ‰ (cid:96) | z (cid:96) ´ z j | ď d ´ η “ K p ˜ η, d q , hile Γ (cid:96) “ ¨˝ d ź j “ ,j ‰ (cid:96) ` | z j || z (cid:96) ´ z j | ˛‚ ď ¨˝ d ´ d ź j “ ,j ‰ (cid:96) | z (cid:96) ´ z j | ˛‚ ď ¨˚˝ d ´ ˜ η ´ d ` ´ t d ´ p u ! ¯ ˛‹‚ “ ¨˚˝ˆ η ˙ d ´ ´ t d ´ p u ! ¯ ˛‹‚ “ K p ˜ η, d, p q . (C.8) Inserting equations (C.7) and (C.8) into (C.5) and (C.6), we get d ÿ k “ | A (cid:96),k | ď p ` p ` | z (cid:96) |q| ∆ (cid:96) |q Γ (cid:96) ď p ` K q K “ K p ˜ η, d, p q , (C.9) and d ÿ k “ | B (cid:96),k | ď p ` | z (cid:96) |q Γ (cid:96) ď K “ K p ˜ η, d, p q , (C.10) for each (cid:96) such that x (cid:96) P x z x c .Now we are ready to bound the norms of rows of the blocks A, ˜ B for each non-clusternode index.For the block A , such bound is given in equation (C.9).For the block ˜ B , we have, using equation C.10, d ÿ k “ | ˜ B (cid:96),k | “ d ÿ k “ |p a (cid:96) πiλz l q ´ || B (cid:96),k | ď K πm λ “ K p m, ˜ η, d, p q λ , (C.11) for each (cid:96) such that x (cid:96) P x z x c .This completes the proof of equations (5.2) and (5.3) of Proposition 5.4. ‚ Cluster nodeWe now bound the norm of each row of J ´ λ p F q at an index corresponding to a clusternode.By assumptions | z j ´ z k | ě ˜ h, @ x j , x k , P x c , j ‰ k, | z j ´ z (cid:96) | ě ˜ η, @ x j P x c , x (cid:96) P x z x c . Then for each j such that x j P x c (C.12) ∆ j “ d ÿ (cid:96) “ ,(cid:96) ‰ j | z j ´ z (cid:96) | ď d ´ h , hile Γ j “ ¨˝ d ź (cid:96) “ ,(cid:96) ‰ j ` | z (cid:96) || z j ´ z (cid:96) | ˛‚ ď ¨˝ d ´ d ź (cid:96) “ ,(cid:96) ‰ j | z j ´ z (cid:96) | ˛‚ ď ¨˚˝ d ´ ˜ η ´ d ` p ˜ h ´ p ` ´ t d ´ p u ! ¯ ˛‹‚ “ K p ˜ η, d, p q ˜ h ´ p ` , (C.13) where K p ˜ η, d, p q “ ˆ d ´ η ´ d ` p p t d ´ p u ! q ˙ .Inserting equations (C.12) and (C.13) into (C.5) and (C.6), we get d ÿ k “ | A j,k | ď p ` p ` | z j |q| ∆ j |q Γ j ď p d ´ q K ˜ h ´ p ` “ K p ˜ η, d, p q ˜ h ´ p ` , (C.14) d ÿ k “ | B j,k | ď p ` | z j |q Γ j ď K ˜ h ´ p ` “ K p ˜ η, d, p q ˜ h ´ p ` , (C.15) for each j such that x j P x c .We now bound the norms of rows of the blocks A, ˜ B for each cluster node index.For the block A , the bound was given in equation (C.14).For the block ˜ B , we have, using equation C.15, d ÿ k “ | ˜ B j,k | “ d ÿ k “ |p a j πiλz j q ´ || B j,k | ď K πm λ ˜ h ´ p ` “ K p ˜ η, d, p, m q λ ˜ h ´ p ` , (C.16) for each j such that x j P x c .This completes the proof of equations (5.4) and (5.5) of Proposition 5.4. (cid:3) Appendix D. Proof of Proposition 5.12
Proof.
Let the map g “ g λ : ¯ P d » C d Ñ C d be defined as g k p a , . . . , a d , x , . . . , x d q “ a k , k “ , . . . , d, (D.1) g d ` k p a , . . . , a d , x , . . . , x d q “ e πiλx k , k “ , . . . , d. Consider the definition of the Prony map
P M from (A.2). We thus have(D.2)
F M λ “ P M ˝ g λ . Put W “ g λ p H o m, τh π p F qq “ g λ p U q . We will show that g λ is injective on U and that P M is injective on W . irst we show that P M is injective on W .Proposition A.1 gives sufficient conditions for P M to be one to one on a subset of C d , the nextProposition asserts that these conditions hold for W . Proposition D.1.
Let λ P Λ p x q . Then for each v , v P W “ g λ p H o m, τh π p F qq “ g λ p U q , with v “ p a , z q , a “ p a , . . . , a d q , z “ p z , . . . , z d q , v “ p a , z q , a “ p a , . . . , a d q , z “ p z , . . . , z d q ,and v ‰ v , it holds that:(1) a j ‰ for j “ , . . . , d. (2) z j ‰ z k for each ď j ă k ď d .(3) z j ‰ z k for all ď j ă k ď d .Proof. Let λ P Λ p x q and let v , v P g λ p H o m, τh π p F qq as specified in Proposition D.1.The first assertion is apparent from the fact that } a ´ a } ă m and the assumption that | a j | ě m for j “ , . . . , d .We now prove assertions 2 and 3.Let z “ p z , . . . , z d q , with z “ e πiλx , . . . , z d “ e πiλx d .As a first step we argue that for each pair of mapped nodes z j , z k , ď j ă k ď d , | z j ´ z k | ě λτ h, ď j ă k ď d. (D.3)Indeed with the assumption that Ω h ď d we have that(D.4) π ą d ą πλτ h. By (D.4) and since λ P Λ p x q (D.5) = p z j , z k q ě πλτ h. Then by (D.4), (D.5) and (5.6) | z j ´ z k | ě λτ h. Next we claim that(D.6) W Ă H o m, λτh p a , z q “ ! p a , z q P C d : } a ´ a } ă m, } z ´ z } ă λτ h ) . Let p a , x q P H o m, τh π p F q . To show (D.6), we need to verify that g λ p a , x q P H o m, λτh p a , z q . Forthis purpose put g λ p a , x q “ p a , z q , z “ p e πiλx , . . . , e πiλx d q . Then using the integral meanvalue bound, for any j “ , . . . , d , ˇˇˇ e πiλx j ´ e πiλx j ˇˇˇ ď max c Pt x j ` t p x j ´ x j q : t Pr , su ˇˇˇˇ ddx e πiλx ˇˇˇ c ˇˇˇˇ τ h π ď λτ he λh ă λτ h, where in the last step we used the assumption Ω h ď d and the fact that λ ď Ω2 d ´ , which thenimplies that e λh ă
2. This in turn proves (D.6).We now prove assertion 2.Let 1 ď j ă k ď d and assume by contradiction that z j “ z k . By (D.6), p a , z q P H o m, λτh p a , z q then | z j ´ z j | ă λτ h and | z k ´ z j | “ | z k ´ z k | ă λτ h . Then | z j ´ z k | ď | z j ´ z j | ` | z k ´ z j | ă λτ h, which is a contradiction to (D.3). inally we prove assertion 3.Assume by contradiction that for 1 ď j ă k ď d , z j “ z k . By (D.6) | z j ´ z j | ă λτ h . Byassumption | z k ´ z j | “ | z k ´ z k | then by (D.6) | z k ´ z j | ă λτ h . Using these | z j ´ z k | ď | z j ´ z j | ` | z k ´ z j | ă λτ h, which is a contradiction to (D.3).This completes the proof of Proposition D.1. (cid:3) Now by Propositions D.1 and A.1 we have that
P M is injective on W .We now show that g λ is injective on U . Proposition D.2.
For each λ ą , the map g λ is injective in the polydisc H o m, λ p F q .Proof. Let p a , x q , p a , x q P H o m, λ p F q such that g p a , x q “ g p a , x q . We will show that p a , x q “p a , x q .For the amplitudes coordinates k “ , . . . , d , g k p a , . . . , a d , x , . . . , x d q “ a k therefore a “ a .For coordinates d ` , . . . , d , g d ` j p a , . . . , a d , x , . . . , x d q “ g d ` j p x j q “ e πiλx j , j “ , . . . , d. Fix a certain 1 ď j ď d and set x j “ α j ` β j i , α j , β j P R . The set of complex numbers w “ α ` βi such that g d ` j p w q “ g d ` j p x j q “ e πiλx j is equal to S j “ " α ` βi : β “ β j , α “ α j ` (cid:96)λ , @ (cid:96) P Z * . Since p a , x q , p a , x q P H o m, λ p F q implies that | x j ´ x j | ă λ then x j “ x j and because j was chosenarbitrarily we have x “ x . (cid:3) By assumption λ ď Ω2 d ´ and Ω h ď d then λ ą h . Using the former, U “ H o m, τh π p F q Ă H o m, λ p F q then by Proposition D.2 g λ is injective on U .We have shown that g λ is injective on U and that P M is injective on W “ g λ p U q then by (D.2) F M λ is injective on U .This completes the proof of Proposition 5.12. (cid:3) Appendix E. Proof of Proposition 5.17
Proof.
First observe that if F P P d is of the form F “ p a π , x π q ` λ (cid:96) , with π P Π d and (cid:96) P Z d ,and p a , x q P A (cid:15),λ p F q then F M λ p F q “ F M λ ˆˆ a π , x π ` λ (cid:96) ˙˙ “ d ÿ j “ a π p j q e πiλ p x π p j q ` (cid:96) jλ q “ d ÿ j “ a π p j q e πiλ x π p j q “ d ÿ j “ a j e πiλ x j “ F M λ ` p a , x q ˘ . ince by definition of A (cid:15),λ p F q (see equation (5.24) ), p a , x q P A (cid:15),λ p F q implies that p a , x q P E (cid:15), p λ q p F q , then the above shows that E (cid:15), p λ q p F q Ě ˜ ď π P Π d ď (cid:96) P Z d A π(cid:15),λ p F q ` λ (cid:96) ¸ č P d . For the other direction, let F “ p a , y q P E (cid:15), p λ q p F q with a “ p a , . . . , a d q and y “ p y , . . . , y d q .Put µ “ F M λ p F q , then µ P Q (cid:15) p µ λ q (with µ λ “ F M λ p F q as above).By definition of the set A (cid:15),λ p F q , there exists a signal F P A (cid:15),λ p F q such that F M λ p F q “ µ ,and put F “ p a , x q with a “ p a , . . . , a d q and x “ p x , . . . , x d q .Recall that by (D.2) (see (A.2) and (D.1)) F M λ “ P M ˝ g λ . Put g λ p F q “ p a , z q with z “ p z , . . . , z d q , z j “ e πiλx j for j “ , . . . , d . By Proposition D.1each point in W “ g λ p U q has non-vanishing amplitudes and pairwise distinct nodes. We have that F P A (cid:15),λ p F q Ď U and hence p a , z q satisfies the above properties. Then by Proposition A.1 theset of all solutions to the equation P M pp a , z qq “ µ is given by(E.1) (cid:32) p a π , z π q : π P Π d ( . By (E.1) there exists π P Π d such that g λ p F q “ g λ ` p a , y q ˘ “ p a π , z π q . Finally since x , . . . , x d are real, the set of all solutions to the equation g λ pp a , x qq “ p a π , z π q isgiven by " p a π , x π ` λ (cid:96) q : (cid:96) P Z d * . By the above, F is of the form ` a π , x π ` λ (cid:96) ˘ for some π P Π d and (cid:96) P Z d .This concludes the proof of Proposition 5.17. (cid:3) Appendix F. Proof of Proposition 5.18
Within the course of the proof we will make appropriate assumptions of the form C η ď Ω ď C h ,with C , C being constants depending only on d , for which some arguments of the proof hold. Itis to be understood that K is the maximum of the constants C and K is the minimum of theconstants C .Assume that Ω ě p d ´ q η . Then the length of the interval ”
12 Ω2 d ´ , Ω2 d ´ ı is larger than η and byProposition 5.9 there exists an interval I Ď ”
12 Ω2 d ´ ,
12 Ω2 d ´ ` η ı such that(F.1) I Ă Λ p x q , | I | “ p d η q ´ . Fix I “ r λ , λ ` p d η q ´ s Ď Λ p x q X „
12 Ω2 d ´ ,
12 Ω2 d ´ ` η to be the sub-interval of Λ p x q X ”
12 Ω2 d ´ ,
12 Ω2 d ´ ` η ı with the minimal starting point λ whichsatisfies (F.1). We will show that there exists λ P I that satisfies (5.25).We require the following intermediate results.As in Section 5.3 we denote by ν the Lebesgue measure on R . emma F.1. Let ď a ă and I “ r a, s . Then for each (cid:15), α, c P R such that ă α ď , ă (cid:15) ď α and | c | ě (cid:15)α | I | , it holds that ν ` t x P I : D k P Z such that | kx ´ c | ď (cid:15) u ˘ ă α | I | . Lemma F.2.
Consider the interval r a, b s Ă p , and let S Ď r a, b s be a union of N disjointsub-intervals S “ Ť Ni “ r a i , b i s . Set I ´ “ r b , a s and S ´ “ Ť Ni “ r b i , a i s . Then ν p S q ν p I q ď ba ν p S ´ q ν p I ´ q . Proposition F.3.
There exists constants K , K depending only on d such that for K η ď Ω ď K h the following holds. For each h ă | c | ď η , there exists an interval I Ă Λ p x q of length | I | “ p d η q ´ such that for all λ P I and for all k P Z (F.2) ˇˇˇˇ c ´ kλ ˇˇˇˇ ą h. We now complete the proof of Proposition 5.18 using the claims above, and provide their proofsthereafter.
Step 1:
First it is shown, using Lemma F.1 and Lemma F.2, that there exists λ ˚ P I such that for allpair of distinct nodes i, j with not both x i , x j in x c , it holds that(F.3) ˇˇˇ x i ´ x j ` nλ ˚ ˇˇˇ ą p d q ´ λ , for all n P Z . Put I ´ “ „ λ ` p d η q ´ , λ , ˜ I ´ “ λ I ´ “ „ λ λ ` p d η q ´ , . Fix any distinct indices i, j such that not both x i , x j are in x c . Put c i,j “ x i ´ x j and observethat under the cluster assumption(F.4) | c i,j | ě η. Put I “ ˜ I ´ , c “ c i,j λ , (cid:15) “ p d q ´ and α “ d . We now validate that under appropriateassumptions on the size of Ω we have that I, c, (cid:15), α satisfy the conditions of Lemma F.1. Put a asthe left end point of the interval I then with Ω ě ηd we have that a ě . With d ě (cid:15) “ d ă d . With Ω ě ηd we have that(F.5) | I | “ | ˜ I ´ | ě p d ηλ q ´ . Now with (F.4) and (F.5) we have that | c | ě (cid:15)α | I | . Having validated the conditions of Lemma F.1hold for I, c, (cid:15), α we now invoke it and get that ν ` ! t P ˜ I ´ : D k P Z such that | kt ´ c i,j λ | ď p d q ´ ) ˘ ă d | ˜ I ´ | . Then ν ` " t P I ´ : D k P Z such that | kt ´ c i,j | ď p d q ´ λ * ˘ ă d | I ´ | . Now we apply Lemma F.2 and conclude from the above that(F.6) ν ` " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c i,j ˇˇˇˇ ď p d q ´ λ * ˘ ă d | I | . efine the set E “ ď ď i ă j ď d (cid:32)p x i P x c ^ x j P x c q " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c i,j ˇˇˇˇ ď p d q ´ λ * . Then using (F.6) and the union bound(F.7) ν ` E ˘ ă ˆ d ˙ d | I | ă | I | . We conclude from (F.7) that there exists λ ˚ P I which satisfies (F.3). Step 2:
Now we show that in fact λ ˚ satisfies (5.25), i.e. it satisfies the condition of Proposition 5.18.Let p ˜ π, ˜ (cid:96) q P Π d ˆ p Z d zt uq . We will show that there exists λ ˜ π, ˜ (cid:96) P Λ p x q such that for all π P Π d and for all (cid:96) P Z d (F.8) ˆ A ˜ πR,λ ˚ p F q ` λ ˚ ˜ (cid:96) ˙ X ˜ A πR,λ ˜ π, ˜ (cid:96) p F q ` λ ˜ π, ˜ (cid:96) (cid:96) ¸ “ H . Proposition 5.18 will then follow by Proposition 5.17.We can assume without loss of generality that ˜ π “ id . Accordingly we put A ˜ πR,λ ˚ p F q “ A R,λ ˚ p F q and we will prove that there exists λ ˜ (cid:96) P Λ p x q such that for all π P Π d and for all (cid:96) P Z d (F.9) ˆ A R,λ ˚ p F q ` λ ˚ ˜ (cid:96) ˙ X ˆ A πR,λ ˜ (cid:96) p F q ` λ ˜ (cid:96) (cid:96) ˙ “ H . Fix i such that ˜ (cid:96) i ‰ n “ ˜ (cid:96) i . Assume that x i P x c , and one can verify that the casewhere x i P x z x c is proved using a similar argument to the one that is given below.In the cases considered below we will use the following fact about the “radius” of the set A R,λ p F q for each λ P Λ p x q , established in Proposition 5.15. For each F “ p a , x q P A R,λ p F q with x “p x , . . . , x d q , ˇˇ x j ´ x j ˇˇ ď ˜ C p Ω τ h q ´ p ` R ď h, j “ , . . . , d. (F.10)We consider the following mutually exclusive and collectively exhaustive cases: Case 1: nλ ˚ ď η .Put c “ nλ ˚ . Then under the assumption of this case and with Ω ě d h we have that 3 h ă | c | ď η .We can therefore apply Proposition F.3 for c and (under appropriate further assumptions on Ω)get that there exists an interval I Ă Λ p x q of length | I | “ p d η q ´ , such that for all λ P I andfor all k P Z it holds that(F.11) ˇˇˇˇ c ´ kλ ˇˇˇˇ “ ˇˇˇˇ nλ ˚ ´ kλ ˇˇˇˇ ą h. Put I “ r λ , λ ` p d η q ´ s , I ´ “ „ λ ` p d η q ´ , λ , ˜ I ´ “ λ I ´ . Let 1 ď j ď d be any index such that x j P x z x c . Put c j “ p x i ` nλ ˚ ´ x j q . Then(F.12) | c j | “ | x i ` nλ ˚ ´ x j | ě | x i ´ x j | ´ nλ ˚ ě η ´ nλ ˚ ě η ´ η ě η, here in the second inequality we used the fact that x j is a non-cluster node and in the thirdinequality we used the assumption of case 1.Put I “ I ´ , c “ c j λ , (cid:15) “ hλ and α “ d . By (F.12) we have that | c | ě ηλ . Using theformer, one can validate that there exists positive constants C p d q , C p d q such that if C p d q η ď Ω ď C p d q h , then I, c, (cid:15), α meet the conditions of Lemma F.1. We then invoke Lemma F.1 and get that ν ` ! t P ˜ I ´ : D k P Z such that | kt ´ c j λ | ď hλ ) ˘ ă d | ˜ I ´ | . Then ν ` (cid:32) t P I ´ : D k P Z such that | kt ´ c j | ď h ( ˘ ă d | I ´ | . By the above and using Lemma (F.2)(F.13) ν ` " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c j ˇˇˇˇ ď h * ˘ ă d | I | . Define the set E “ ď ď j ď d,x j R x c " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c j ˇˇˇˇ ď h * . Using the union bound and (F.13)(F.14) ν p E q ă | I | . We conclude from the above that there exists λ P I such that for any non-cluster node x j and forany k P Z ˇˇˇˇ x i ` nλ ˚ ´ x j ´ kλ ˇˇˇˇ ą h. On the other hand we have that for all k P Z (see (F.11)) ˇˇˇˇ nλ ˚ ´ kλ ˇˇˇˇ ą h. Fix λ ˜ (cid:96) “ λ . Then using the above, for any π P Π d and any k P Z , if x π p i q is a cluster node then(F.15) ˇˇˇˇ x i ` nλ ˚ ´ x π p i q ´ kλ ˜ (cid:96) ˇˇˇˇ ě ˇˇˇˇ nλ ˚ ´ kλ ˜ (cid:96) ˇˇˇˇ ´ ˇˇ x i ´ x π p i q ˇˇ ą h ´ h “ h, and if x π p i q is a non-cluster node then(F.16) ˇˇˇˇ x i ` nλ ˚ ´ x π p i q ´ kλ ˜ (cid:96) ˇˇˇˇ ą h. Now by combing (F.10), (F.15) and (F.16), we get that λ ˜ (cid:96) satisfies (F.9) . This completes theproof of case 1. Case 2: nλ ˚ ą η and @ y P x z x c : | x i ` nλ ˚ ´ y | ą η . We show that in this case there exists λ P I such that λ ˜ (cid:96) “ λ satisfies (F.9).Put (as above) I ´ “ „ λ ` p d η q ´ , λ , ˜ I ´ “ λ I ´ “ „ λ λ ` p d η q ´ , . Put I “ ˜ I ´ , c “ nλ ˚ λ , (cid:15) “ hλ and α “ . By the assumptions of this case we have nλ ˚ ą η , then c “ nλ ˚ λ ą η λ . Using the former, one can validate that there exist positive constants C p d q , C p d q uch that if C p d q η ď Ω ď C p d q h , then I, c, (cid:15), α meet the conditions of Lemma F.1. We then invokeLemma F.1 and get that ν ` ! t P ˜ I ´ : D k P Z such that ˇˇˇ kt ´ nλ ˚ λ ˇˇˇ ď hλ ) ˘ ă | ˜ I ´ | . Then ν ` ! t P I ´ : D k P Z such that ˇˇˇ kt ´ nλ ˚ ˇˇˇ ď h ) ˘ ă | I ´ | . By the above and using Lemma (F.2)(F.17) ν ` " λ P I : D k P Z such that ˇˇˇˇ kλ ´ nλ ˚ ˇˇˇˇ ď h * ˘ ă | I | . Now for any index j such that x j is a non-cluster node put c j “ x i ` nλ ˚ ´ x j . Put I “ ˜ I ´ , c “ c j λ , (cid:15) “ hλ and α “ d . Then by the assumptions of this case | c | ą η λ and with this onecan validate that there exist positive constants C p d q , C p d q such that if C p d q η ď Ω ď C p d q h , then I, c, (cid:15), α meet the conditions of Lemma F.1. Invoking it and using Lemma (F.2) we have that(F.18) ν ` " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c j ˇˇˇˇ ď h * ˘ ă d | I | . Define the set E “ ď ď j ď d,x j R x c " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c j ˇˇˇˇ ď h * . Using the union bound and (F.18)(F.19) ν p E q ă | I | . Now combing (F.17) and (F.19) we get that there exists λ P I such that for all k P Z ˇˇˇˇ kλ ´ nλ ˚ ˇˇˇˇ ą h, ˇˇˇˇ x i ` nλ ˚ ´ x j ´ kλ ˇˇˇˇ ą h, @ x j P x z x c . Finally setting λ ˜ (cid:96) “ λ we get from the above and (F.10) that λ ˜ (cid:96) satisfies (F.9). Case 3: nλ ˚ ą η and D y P x z x c : | x i ` nλ ˚ ´ y | ď η . First we note that since the non-cluster nodes are each separated from any other node by atleast η , there can be at most one node y P x z x c such that | x i ` nλ ˚ ´ y | ď η . Therefore let j be theindex of the non-cluster node for which we have | x i ` nλ ˚ ´ x j | ď η . By the choice of λ ˚ we alsohave that | x i ` nλ ˚ ´ x j | ą p d q ´ λ (see (F.3)). We conclude that p d q ´ λ ď | x i ` nλ ˚ ´ x j | ď η , and for Ω ď d h we then have that3 h ă | x i ` nλ ˚ ´ x j | ď η . e now invoke Proposition F.3 and get that there exists an interval I P Λ p x q of length | I | “p d η q ´ such that for all λ P I and for all k P Z (F.20) ˇˇˇˇ x i ` nλ ˚ ´ x j ´ kλ ˇˇˇˇ ą h. Put I “ r λ , λ ` p d η q ´ s , I ´ “ „ λ ` p d η q ´ , λ , ˜ I ´ “ λ I ´ . For each index 1 ď (cid:96) ď d, (cid:96) ‰ j put c (cid:96) “ x i ` nλ ˚ ´ x (cid:96) and note that | c (cid:96) | “ | x i ` nλ ˚ ´ x j ` x j ´ x (cid:96) | ě | x j ´ x (cid:96) | ´ | x i ` nλ ˚ ´ x j | ě η. Put I “ ˜ I ´ , c “ c (cid:96) λ , (cid:15) “ hλ and α “ d . Then with the above | c | ě ηλ and then followingsimilar computations as in the previous cases (see cases 1,2), one can validate that I, c, (cid:15), α meetthe conditions of Lemma F.1 for C η ď Ω ď C h where C , C are constants depending only on d .Invoking Lemma F.1 with I, c, (cid:15), α we get that ν ` ! t P ˜ I ´ : D k P Z such that | kt ´ c (cid:96) λ | ď hλ ) ˘ ă d | ˜ I ´ | . Then ν ` (cid:32) t P I ´ : D k P Z such that | kt ´ c (cid:96) | ď h ( ˘ ă d | I ´ | . By the above and using Lemma (F.2)(F.21) ν ` " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c (cid:96) ˇˇˇˇ ď h * ˘ ă d | I | . Define the set E “ ď ď (cid:96) ď d, (cid:96) ‰ j " λ P I : D k P Z such that ˇˇˇˇ kλ ´ c (cid:96) ˇˇˇˇ ď h * . Using the union bound and (F.21) ν p E q ă | I | . We conclude from the above that there exists λ P I such that for all k P Z and for any index1 ď (cid:96) ď d, (cid:96) ‰ j ,(F.22) ˇˇˇˇ x i ` nλ ˚ ´ x (cid:96) ´ kλ ˇˇˇˇ ą h. Put λ ˜ (cid:96) “ λ . Recall that I satisfies (F.20). Then with (F.20) and (F.22) λ ˜ (cid:96) satisfies that for all k P Z and for any index 1 ď (cid:96) ď d ˇˇˇˇ x i ` nλ ˚ ´ x (cid:96) ´ kλ ˜ (cid:96) ˇˇˇˇ ą h. Using the above and (F.10) we get that that λ ˜ (cid:96) satisfies (F.9). (cid:3) We now prove the intermediate claims: Lemma F.1, Lemma F.2 and Proposition F.3.
Proof of Lemma F.1.
Let a, (cid:15), α, c and I “ r a, s as specified in Lemma F.1. Without loss ofgenerality we assume that c ą
0, consequently it is sufficient to prove that ν ` t x P I : D k P N such that | kx ´ c | ď (cid:15) u ˘ ă α | I | . If 0 ă c ă ν ` t x P I : D k P N such that | kx ´ c | ď (cid:15) u ˘ ď (cid:15). hen under this condition and with the assumption that c ě (cid:15)α | I | , we have that 2 (cid:15) ă α | I | , therefore ν ` t x P I : D k P N such that | kx ´ c | ď (cid:15) u ˘ ď (cid:15) ă α | I | . We now prove the case c ě N P N be the unique integer such that(F.23) c t c u ` N ď a ă c t c u ` N ´ . Then ν ` t x P I : D k P Z such that | kx ´ c | ď (cid:15) u ˘ ď N ÿ k “ (cid:15) t c u ` k “ (cid:15) N ÿ k “ t c u ` k . (F.24)If N ď c ě (cid:15)α | I | (cid:15) N ÿ k “ t c u ` k ď (cid:15) ÿ k “ t c u ` k ă (cid:15)c ď α | I | . Combining (F.24) with the above proves the claim for this case.We are left to prove the case N ě , c ě H n the n th partial sum of the Harmonic series we have thatlog p n q ` γ ă H n ă log p n ` q ` γ, where log is the base 2 logarithm. Then2 (cid:15) N ÿ k “ t c u ` k ď (cid:15) p log p t c u ` N ` q ´ log p t c u ´ qq“ (cid:15) log ˆ t c u ` N ` t c u ´ ˙ “ (cid:15) log ˆ ` N ` t c u ´ ˙ . (F.25)Using (F.23) and since by assumption a ě we have that(F.26) N ď t c u ` . Then by (F.23) and (F.26) (and assuming N ě c ě | I | “ ´ a ě N ´ t c u ` N ´ ě N ´ t c u ` ě p N ` q t c u ` ě p N ` q t c u ´ . Inserting (F.27) into (F.25) and using the assumption that 100 (cid:15) ď α (cid:15) log ˆ ` N ` t c u ´ ˙ ď (cid:15) log p ` | I |q“ (cid:15) log p e q ln p ` | I |qă (cid:15) | I |ď α | I | , (F.28)which then proves the claim using (F.24) and (F.25).This completes the proof of Lemma F.1. (cid:3) roof of Lemma F.2. For any sub-interval r c, d s Ď I we have that(F.29) ν pr c, d sq ν p I q “ d ´ cb ´ a “ cdab c ´ d a ´ b ď ba ν p “ d , c ‰ q ν p I ´ q . Using the above ν p S q ν p I q “ ÿ i ν pr a i , b i sq ν p I q ď ba ÿ i ν pr b i , a i sq ν p I ´ q “ ba ν p S ´ q . This completes the proof of Lemma F.2. (cid:3)
Proof of Proposition F.3.
Without loss of generality assume that c ą T “ cλ .We will use the following inequality repeatably below. For each k ě ď α ď λ we have(F.30) kα λ ď k ˆ λ ´ λ ` α ˙ ď kαλ . Put β “ T ´ t T u and consider the following cases: Case 1 : ď β ď .We show that in this case I “ I Ă Λ p x q satisfies (F.2) provided that Ω h ă d and Ω ě dη . Tosee this recall that I “ r λ , λ ` p d η q ´ s . Put λ p α q “ λ ` α , 0 ď α ď p d η q ´ . We have thatfor each integer k ď t T u ˇˇˇˇ c ´ kλ p α q ˇˇˇˇ “ Tλ ´ kλ p α q ě βλ ě λ . On the other hand, for each integer k ě r T s ˇˇˇˇ c ´ kλ p α q ˇˇˇˇ ě k ´ Tλ ´ k ˆ λ ´ λ p α q ˙ ě k ´ Tλ ´ kαλ “ p k ´ T q ˆ λ ´ αλ ˙ ´ T αλ ě p ´ β q ˆ λ ´ αλ ˙ ´ T αλ ě ˆ λ ´ αλ ˙ ´ T αλ , (F.31)where in the second inequality we used (F.30). Using Ω ě dη ñ αλ ď , Tλ ď η and Ω h ă d wehave that 18 ˆ λ ´ αλ ˙ ´ T αλ ě λ ´ λ “ λ ą h. We conclude from the above that for ď β ď (and under the assumptions on Ω and Ω h ) I “ I Ă Λ p x q satisfies (F.2). Case 2 : β ď .First if t T u “ I “ I Ă Λ p x q satisfies (F.2) for Ω h ď d . For k “ ˇˇˇˇ c ´ kλ ˇˇˇˇ “ c ą h. For k ą λ P I ˇˇˇˇ c ´ kλ ˇˇˇˇ “ ˇˇˇˇ βλ ´ kλ ˇˇˇˇ ě λ ´ βλ ě λ ´ λ “ λ ą h, where in the last inequality we used the assumption that Ω h ď d . ow assume that t T u ą T ˆ λ ´ λ p α q ˙ ą h, (F.32) t T u ˆ λ ´ λ p α q ˙ ă λ . (F.33)We show that if for 0 ď α ď λ , λ p α q satisfies both (F.32) and (F.33) then λ p α q satisfies (F.2),provided that Ω h ď d .For any integer k ď t T u we have using (F.32) that Tλ ´ kλ p α q ě T ˆ λ ´ λ p α q ˙ ą h. For any integer k ą t T u kλ p α q ´ Tλ ě t T u λ p α q ´ Tλ ` λ p α qě ´ t T u ˆ λ ´ λ p α q ˙ ´ βλ ` λ p α qą ´ λ ´ βλ ` λ p α qě ´ λ ` λ ě λ ě h, where in the 3 rd inequality we used (F.33), in the 4 th inequality we used both β ď and 0 ď α ď λ ,and in last inequality we used Ω h ď d .We then conclude that when Ω h is small enough, each λ p α q with 0 ď α ď λ which satisfies both(F.32) and (F.33) satisfies (F.2). We now solve (F.32) and (F.33) for α . By (F.30) T α λ ą h ñ T ´ λ ´ λ p α q ¯ ą h , then each 0 ď α ď λ such that α ą λ hT satisfies (F.32). By (F.30) t T u αλ ă λ ñ t T u ´ λ ´ λ p α q ¯ ă λ , then each 0 ď α ď λ such that α ă λ t T u , satisfies (F.33).We conclude from the above that for α P ˆ λ hT , λ t T u ˙ “ I ,λ p α q satisfies (F.2).Now we recall that by Proposition 5.9, every interval I Ă ”
12 Ω2 d ´ , Ω2 d ´ ı of size η contains asub-interval I of size p d η q ´ such that I Ă Λ p x q . Put I “ λ ` I and I “ I X ”
12 Ω2 d ´ , Ω2 d ´ ı . e will now validate that | I | ą η for Ω h ă d . To prove that we show that λ ` λ hT ` η ă min ˆ λ ` λ t T u , Ω2 d ´ ˙ . First we show that λ ` λ hT ` η ă λ ` λ t T u : λ t T u ´ λ hT ě λ T ˆ ´ λ h ˙ ě η ˆ ´ λ h ˙ ą η , where in the penultimate inequality we used the proposition assumption that η ě c “ Tλ and inthe last inequality we used Ω h ă d . Next we show that λ ` λ hT ` η ă Ω2 d ´ for Ω ą p d ´ q η andΩ h ă d : λ ` λ hT ` η ď λ p ` λ h q ` η ď λ ` η ď ˆ Ω2 p d ´ q ` η ˙ ` η ă Ω2 d ´ . We conclude that | I | ą η and I Ă ”
12 Ω2 d ´ , Ω2 d ´ ı then by Proposition 5.9 I contains a sub-interval I of size p d η q ´ such that I Ă Λ p x q . Since by construction I satisfies (F.2) this completes theproof of the case β ď of Proposition (F.3).We are left to prove the case ď β . This case is proved similarly to the case β ď . We thereforeomit the proof of this case. (cid:3) Department of Applied Mathematics, School of Mathematical Sciences, Tel-Aviv University, P.O.Box 39040, tel-aviv 6997801, Israel
E-mail address : [email protected] Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
E-mail address : [email protected] Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
E-mail address : [email protected]@weizmann.ac.il