Sampling the flow of a bandlimited function

We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=\phi_t\ast f$ resulting from the convolution with a kernel $\phi_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space-time samples, % with irregular spatial sampling, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $\widehat f$ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Tur\'an type inequalities. En route, we obtain Remez-Tur\'an inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.


Introduction
In this paper, we consider the sampling and reconstruction problem of signals u " upt, xq that arise as an evolution of an intial signal f " f pxq under the action of convolution operators. The intial signal f is assumed to be in the Paley-Wiener space P W c , c ą 0 (fixed throughout this paper) given by P W c :" ! f P L 2 pRq : suppp p f q Ď r´c, cs ) with the Fourier transform normalized as p f pξq " ş R f ptqe´i tξ dt. The functions u are solutions of initial value problems stemming from a physical system. Thus, due to the semigroup properties of such solutions, there is a family of kernels tφ t : t ą 0u such that upt, xq " φ t˚f pxq, φ t`s " φ t˚φs for all t, s P p0, 8q, and f " lim tÑ0`φ t˚f , f P L 2 pRq. As we are primarily interested in physical systems, we typically consider the following set of kernels: Φ c " tφ P L 1 pRq : there exists κ φ ą 0 such that κ φ ď p φpξq ď 1 for |ξ| ď c, p φp0q " 1u. (1.1) Observe that φ P L 1 implies that p φ is continuous and, therefore, the existence of κ φ ą 0 such that p φ ě κ φ on r´c, cs is equivalent to p φ ą 0 on r´c, cs. We remark that some of our results hold for a less restrictive class of kernels. Problem 1. Let φ P Φ, L ą 0, and Λ Ă R be a discrete subset of R. What are the conditions that allow one to recover a function f P P W c in a stable way from the data set (1.3) tpf˚φ t qpλq : λ P Λ, 0 ď t ď Lu?
The set of measurements (1.3) is the image of an operator T : P W c Ñ L 2`Λˆr 0, Ls˘. Thus, the stable recovery of f from (1.3) amounts to finding conditions on Λ, φ and L such that T has a bounded inverse from T pP W c q to P W c or, equivalently, the existence of A, B ą 0 such that 2 2 , for all f P P W c .
If for a given φ and L the frame condition (1.4) is satisfied, we say that Λ " Λ φ,L is a stable sampling set.
Remark 1.2. It was shown in [5,Theorem 5.5] that Λ φ,L is a stable sampling set for some L ą 0, if and only if Λ φ,1 is a stable sampling set. Thus, for qualitative results, we will only consider the case of L " 1. For quantitative results, however, we may keep L in order to estimate the optimal time length of measurements.
Remark 1.3. Whenever (1.4) holds, standard frame methods can be used for the stable reconstruction of f [11].
Let us discuss Problem 1 in more detail in the case of our prototypical example.
1.1. Sampling the heat flow. Consider the problem of sampling the temperature in a heat diffusion process initiated by a bandlimited function f P P W c : where φ t is the heat kernel at time t: p φ t pξq " e´t σ 2 ξ 2 , (1.5) with a parameter σ " 0. According to Shannon's sampling theorem, f can be stably reconstructed from equispaced samples tf pk{T q : k P Zu if and only if the sampling rate T is bigger than or equal to the critical value T " c π , known as the Nyquist rate. The Nyquist bound is universal in the sense that it also applies to irregular sampling patterns: if a bandlimited function can be stably reconstructed from its samples at Λ Ď R, then the lower Beurling density We are interested to know if the spatial sampling rate can be reduced by using the information provided by the following spatio-temporal samples: (1.6) tf t pk{T q : k P Z, 0 ď t ď 1u.
Observe that the amount of collected data in (1.6) is not smaller than that in the case of sampling at the Nyquist rate T " c π . If T ă c π , however, the density of sensors is smaller, and thus such a sampling procedure may provide considerable cost savings.
Lu and Vetterli showed [16] that for all T ă c π there exist bandlimited signals with norm 1 that almost vanish on the samples (1.6), i.e. stable reconstruction is impossible from (1.6). As a remedy, they introduced periodic, nonuniform sampling patterns Λ Ď R that do lead to a meaningful spatio-temporal trade-off : there exist sets Λ Ď R that have sub-Nyquist density and, yet, lead to the frame inequality: with A, B ą 0; see Example 4.1 for a concrete construction. The emerging field of dynamical sampling investigates such phenomena in great generality (see, e.g., [1,2,3,4,5]). As follows from Example 4.1, the estimates (1.7) may hold with an arbitrary small sensor density. The meaningful trade-off between spatial and temporal resolution, however, is limited by the desired numerical accuracy. For example, in the following theorem we relate the maximal gap of a stable sampling set to the bounds from (1.7). Theorem 1.4. Let Λ Ď R be such that (1.7) holds. Then there exists an absolute constant K ą 0 such that, for R ě K maxˆB A , 1 c˙a nd every a P R, we have ra´R, a`Rs X Λ ‰ H.
In particular, we have D´pΛq ě K´1 min`A B , c˘and D`pΛq ď KB.
Theorem 4.4, which is a more general version of the above result, provides a more explicit dependence of K on the parameters of the problem.
Besides the constraints implied by Theorem 1.4, the special sampling configurations of Lu and Vetterli that lead to (1.7) lack the simplicity of regular sampling patterns. In this article, we explore a different solution to the diffusion sampling problem. We consider sub-Nyquist equispaced spatial sampling patterns (1.6) with T " c mπ , m P N, and restrict the sampling/reconstruction problem to a subset V Ď P W c , aiming for an inequality of the form: Specifically, we consider the following signal models.
Away from blind spots. We will identify a set E with measure arbitrarily close to 1 such that (1.8) holds with V " V E " tf P P W c : supp p f Ď Eu. In effect, E is the set r´c, cszO where O is a small open neighborhood of a finite set, i.e., E avoids a certain number of "blind spots." Theorem 1.5. Let φ P Φ and m ě 2 be an integer. Then for any r ą 0 there exists a certain compact set E Ď r´c, cs of measure at least 2c´r such that any f P V E can be recovered from the samples in a stable way.
The set E in the above theorem depends only on φ and the choice of r. The stable recovery in this case means that (1.8) holds with B " 1 and some A ą 0 which is estimated in a more explicit version of the above result, Theorem 2.8.
Prolate spheroidal wave functions. The Prolate Spheroidal Wave Functions (PSWFs) are eigenfunctions of an integral operator known as the time-band liminting operator or sinckernel operator sin πcpy´xq πpy´xq f pyq dy.
Using the min-max theorem, we get that ψ n,c is the norm-one solution of the following extremal problem where the condition f P spantψ k,c : k ă nu K is void for n " 0. The family pψ n,c q ně0 forms an orthogonal basis for P W c and has the property to form an orthonormal sequence in L 2 p´1, 1q.
We consider the N -dimensional space The Landau-Pollak-Slepian theory shows that this subspace provides an optimal approximation of a bandlimited function that is concentrated on r´1, 1s. More precisely, V " V N minimizes the approximation error sup f PP Wc Sparse sinc translates with free nodes. In this model, we let c n sinc cpx´λ n q : c 1 , . . . , c N P C, λ 1 , . . . , λ N P R + be the class of linear combinations of N arbitrary translates of the sinc kernel sincpxq " sin x which ensures the numerical stability of the sampling problem f Þ Ñ tf t pmπk{cq : k P Z : 0 ď t ď 1u restricted non-linearly to the class V N . In other words, if (1.8) holds with V " V 2N then any f P V N can be stably reconstructed from the samples (1.6).
Fourier polynomials. As our last model, we consider the Fourier image of the space of polynomials of degree at most N restricted to the unit interval. Explicitly, where D : P W c Ñ P W c is the differential operator Df " f 1 . Observe that the union of such In this article, we show that each of the above-mentioned signal models regularizes the diffusion sampling problem, albeit with possibly very large condition numbers. Theorem 1.6. Let m ě 2 be an integer, Φ be given by p Φpξq " e´σ 2 ξ 2 . Let V " V N be given by (1.9), (1.10), or (1.11). Then (1.8) holds with A " cκ 0 pcq pσcq 2`m exp´´κ 1 pcqN´m 2`´κ 2 pcq ln σ`κ 3 pcqσ 2`l n m˘¯, where the κ j 's are positive constants that depend on c only.
We provide a more precise expression for the lower frame constant in Theorem 3.5. Note that the lower bound deteriorates when σ 2 Ñ 0 (no diffusion) and σ 2 Ñ`8 (very rapid diffusion). This agrees with the intuition and numerical experiments for (non-bandlimited) sparse initial conditions presented in [20]: if σ 2 is very small, because of spatial undersampling, some components of f may be hidden from the sensors, while for large σ 2 the diffusion completely blurs out the signal and no information can be extracted.
Remark 1.7. To simplify the discussion we take c " 1{2 in this remark. There are instances when Theorem 1.6 applies for a signal f P V N which cannot be recovered simply from its samples on, say, 2Z. As an example, we offer V 1 given by (1.10) with λ 1 " 1. The samples at time t " 0 are not sufficient to identify each signal since sincp¨´1q P V N vanishes on mZ, m ě 2. Similarly, for Theorem 1.5: the function sinpω¨q sincp¨a q, with an appropriately chosen a and ω, belongs to V E and vanishes on mZ for m ě 2. In finite dimensional subspaces V N , e.g., given by (1.9) and (1.11), sampling at time t " 0 with any m P N may be sufficient for stable recovery. However, the expected error of reconstruction in the presence of noise will be reduced if temporal samples are used in addition to those at t " 0. Theorems 1.5 and 1.6 can be used together. For example, a function f can be reconstructed away from the blind spots using Theorem 1.5 and approximated around the blind spots using Theorem 1.6.
1.2. Technical overview. Lu and Vetterli explain the impossibility of subsampling the heatflow of a bandlimited function on a grid (1.6) as follows [16]. The function with Fourier transform p f :" δ´T´δ T is formally bandlimited to I " r´c, cs if T ă c, and vanishes on the lattice π T Z. Moreover, f is an eigenfunction of the diffusion operator since (1.5). Hence, all the diffusion samples (1.6) vanish, although f ı 0. While no Paley-Wiener function is infinitely concentrated at t´T, T u, a more formal argument can be given by regularization. If η : R Ñ R is continuous and supported on r´1, 1s and η ε pxq " ε´1ηpx{εq, then f¨p η ε P P W c and provides a counterexample to (1.4), provided that ε is sufficiently small. As we show below in Subsection 2.1, a similar phenomenon holds for more general diffusion kernels φ as in (1.1). Indeed, an analysis along the lines of the Papoulis sampling theorem [18] shows that the diffusion samples (1.6) of a function f P P W c do not lead to a stable recovery of p f . However, these samples do allow for the stable recovery away from certain blind spots determined by φ; that is, one can effectively recover p f¨1 E , for a certain subset E Ď I of positive measure (1 E denotes the characteristic function of the set E). If we, furthermore, restrict the sampling problem to one of the finite dimensional spaces V " V N given by (1.9), (1.10), or (1.11), we may then infer all other values of p f . The main tools, in this case, are Remez-Turán-like inequalities of the form: For Fourier polynomials (1.11) the classical Remez-Turán inequality provides an explicit constant C E , while the case of sparse sinc translates (1.10) is due to Nazarov [17]. The corresponding inequality for prolate spheroidal wave functions (1.9) is new and a contribution of this article (our technique relies on [15]).
1.3. Paper organization and contributions. In Section 2, we show that uniform dynamical samples at sub-Nyquist rate allow one to stably reconstruct the function p f away from certain, explicitly described blind spots determined by the kernel φ. We also provide an upper and lower estimate for the lower frame bound in (1.8). The upper estimate relies on the standard formulas for Pick matrices (see, e.g. [7,10]). The lower estimate is far more intricate and is based on the analysis of certain Vandermonde matrices. We also provide some numerics and explicit estimates in the case of the heat flow problem.
In Section 3, we restrict the problem to the sets V " V N given by (1.9), (1.10), or (1.11). We provide quantitative estimates for the frame bounds in (1.8). En route, we obtain an explicit Remez-Turán inequality for prolate spheroidal wave functions -a result which we find interesting in its own right.
In Section 4, we discuss the case of irregular spacial sampling. We recall that a stable reconstruction may be possible with sets Λ that have an arbitrarily small (but positive) lower density. Nevertheless, we show that the maximal gap between the spacial samples (and, hence, the lower Beurling density) is controlled by the condition number of the problem (i.e. the ratio B A of the frame bounds).

2.
Recovering a bandlimited function away from the blind-spot 2.1. Dynamical sampling in P W c . In this section, we recall some of the results on dynamical sampling from [4,5] and adapt them for problems studied in this paper. For φ P L 1 , consider the function that is, the 2c-periodization of the piece of p φ supported in r´c, cq. Recall that we consider kernels from the set Φ given by (1.1). Hence, We also write Next, we introduce the sampled diffusion matrix, which is the mˆm matrix-valued function given by (2.12) Remark 2.1. Observe that the matrix function B m is m-periodic. Its eigenvalues, however, are 1-periodic because the matrices B m pξq and B m pξ`kq, k P Z, are similar via a circular shift matrix.
The following lemma explains the role of the sampled diffusion matrix. In the lemma, we let . .
Note that if we recover f pξq for ξ P r0, 1s then we can recover f p . Observe also that In other words, f Þ Ñ b 2c m f : P W c Ñ L 2 pr0, 1s, M m,1 pCqq is an isometric isomorphism.
Proof. Observe that it suffices to prove the result in P W c XSpRq (the Schwarz class). Consider the function bpξ, tq " Using the Poisson summation formula and the definition of f t , we get bpξ, tq " c mπ Note that the functions bp¨, tq are 1-periodic, Combining the last equation with the Parseval's relation yields the desired conclusion.
Remark 2.3. Lemma 2.2 shows that the stability of reconstruction from spatio-temporal samples is controlled by the condition number of the self-adjoint matrices B m pξq in (2.12). For symmetric φ P Φ and m ě 2, however, inf ξPr0,1s λ min`Bm pξq˘" λ min`Bm p0q˘" 0, which precludes the stable reconstruction of all f P P W c , see, e.g., [4]. This adds to our explanation of the phenomenon of blind spots in Subsection 1.2. We can nonetheless hope to find a large set r E Ď r0, 1s such that λ min`Bm pξq˘ě κ for ξ P r E. Then, repeating the computation in (2.14), we get c m pẼ`Zq˙X r´c, cs.
In the following example, we offer some numerics. To simplify the computations, we represent B m pξq in (2.12) as a Pick matrix (see, e.g., [7,10]). For ξ P r´c, cq, we write p φpξq " e´ψ pξq , so that ψ ě 0 and ψp0q " 0, and obtain for j, k " 0, . . . , m´1, pB m q jk pξq " where the indices j 1 , k 1 are in the set * , m divides |j´j 1 | and |k´k 1 |, and j, k, and ξ are not 0 simultaneously. Thus pB m q jk pξq "
In Figure 1, we show the condition numbers of the matrices B m pξq with ξ " 0.45, c " 1{2, m P t2, 3, 5u, and σ varying from 1 to 200. In Figure 2, we also show the condition numbers of the matrices B m pξq. This time, however, still c " 1{2, the parameter σ is fixed to be 200, whereas the point ξ is allowed to vary from 0.35 to 0.49. We still have m P t2, 3, 5u.

2.2.
Estimating the minimal eigenvalue of the sampled diffusion matrix. In this subsection, we use Vandermonde matrices to obtain a lower estimate for the eigenvalue λ pmq min pξq of the matrices B m pξq in (2.12). We also present an upper estimate for λ pmq min pξq, which follows from the general theory of Pick matrices [7,10].
Then for each x P C m , we have Proof. let V be the mˆm Vandermonde matrix associated to v: V " rv i j s 0ďiďm´1,0ďjďm´1 . Note that the Frobenius norm of V and its determinant are given by Recall from [23] an estimate for the minimal singular value of an mˆm matrix A: Specifying this to V we get σ min pV q ě α. As V ď V F , it follows that, for all x P C m , Let D N be the diagonal matrix with v N on the main diagonal. Since Moreover, we have γ 2 Ń }x} 2 ď }D N x} 2 ď γ 2 Ǹ }x} 2 by definition of D N . The conclusion now follows by summing the two geometric sequences.
Note that the function Ψ N is increasing on p0,`8q and that, for t ‰ 1, t ą 0 Corollary 2.6. With the notation of Lemma 2.5, assume further that 0 ă ν ď v j ď 1 and m ě 2. Let Then for each x P C m , we have Proof. Indeed, ν ď γ´ď γ`ď 1 so Ψ N pνq ď Ψ N pγ´q and Ψ N pγ`q ď Ψ N p1q " N . Further, since v j ď 1, σ 2 ď m 2 . Moreover, the derivative of`t´1 t˘p t´1q{2 "`1´1 t˘p t´1q{2 is It follows that α in the statement of Lemma 2.5 satisfies and the result is established.
Then, for each x P C m , we have Proof. We fix ξ and apply Corollary 2.6 to v j " p p φq pˆ2 c m pξ`jq˙1 m . With r α given by (2.24), On the other hand, 1 mN WN W N equals the left-end mN -term Riemann sum for the integral defining B m pξq. It follows that Using (2.23), we get Finally, note that if 0 ă a, b ď 1, using the mean value theorem, there is an η P pa, bq such that For an upper estimate of the minimal eigenvalue λ pmq min pξq we use the estimates of the singular values of Pick matrices by Beckerman-Townsend [7]. For p j P C, j " 1, . . . , m, and 0 ă a ď be the corresponding Pick matrix. Then the smallest singular value s min of P m is bounded above by , . - where s max is the largest singular value. If p r P m q jk " 1´c j c k x j`xk , then r P m is related to a Pick matrix of the form (2.25) via the diagonal matrix D " diagp1`c j q: In our case, see (2.20), x j " ψ`2 c m pξ`jq˘and c j " e´x j P p0, 1s, so Id ď D ď 2Id and the singular values of B m pξq and the corresponding Pick matrix P m differ at most by a factor 4. Therefore, (2.26) holds with apξq " min ψ`2 c m pξ`kq˘: k P I ξ ( and bpξq " max ψ`2 c m pξ`kq˘: k P I ξ ( , I ξ defined in (2.19), and an additional factor 4 provided that apξq ‰ 0.

(2.27)
Observe that the Beckerman-Townsend estimate (2.26) holds for all Pick matrices with the same values for a " min x j and b " max x j and is completely independent of the particular distribution of the x j . Regardless, it shows that the condition number grows nearly exponentially with m, establishing limitations on how well the space-time trade-off can work numerically. Of course, the condition number may be much worse if two values x j and x j`1 are close together (if x j " x j`1 , then P m is singular). Thus, (2.27) is an optimistic upper estimate for λ pmq min pξq. By comparison, our lower estimate in Proposition 2.7 depends crucially on the distribution of the parameters x j and is much harder to obtain. It does, however, establish an upper bound on the condition number and, thus, shows that the space-time trade-off may be useful. A precise result is formulated in the following subsection.

Partial recoverability.
Theorem 2.8. Let φ P Φ, m ě 2 an integer and r E Ď I " r0, 1s be a compact set. Assume that there exists δ ą 0 such that, for every 0 ď j ă k ď m´1 and every ξ P r Ěˇˇˇp Let E "ˆ2 c m pẼ`Zq˙X r´c, cs. Then for any f P P W c , the function p f 1 E can be recovered from the samples in a stable way. Moreover, we have Proof. Recall from (2.15) that we need to estimate The upper bound follows directly from Proposition 2.7, and (2.14): Let us now prove the lower bound using (2.18 2em m 2 ln κ φ δ mpm´1q }f pξq} 2 . Taking κ " κ 2{m φ´1 2em m 2 ln κ φ δ mpm´1q in (2.18) gives the result.
Remark 2.9. The condition number implied by the above theorem is not the best possible one can obtain through this method. For instance, a better estimate for the σ min of a Vandermonde matrix may be used in place of (2.21). However, the method will always lead to a deteriorating estimate of the condition number as m increases. This follows from the Beckerman-Townsend estimate (2.26) we discussed in the previous subsection. Corollary 2.10. Assume that φ P Φ, p φ is even and strictly decreasing on R`, and m ě 2 is an integer. Given η P p0, 1 4 q, let r E " r´1 2`η ,´ηsYrη, 1 2´η s and E "ˆ2 c m pẼ`Zq˙Xr´c, cs.
Then there exists A ą 0 such that, for any f P P W c , Proof. We look into the main condition of Theorem 2.8: there exists δ ą 0 such that, for every 0 ď j ă k ď m´1 and every ξ P r E For a general φ P Φ, the function p φ is continuous and, therefore, p φ p is continuous, except possibly on c`2cZ where a jump discontinuity occurs if p φp´cq ‰ p φpcq. Under current assumptions, however, p φ is even and, therefore p φ p is continuous everywhere.
For 0 ď ď m´1 and ξ P I, we have´1 2m ď ξ` m ď 1´1 2m and Thus, the condition of Theorem 2.8 would be satisfied with r E " I if | p φ| were one-to-one on I, that is, either strictly decreasing or strictly increasing. However, p φ is even and strictly decreasing on R`, so that p φ p is continuous, strictly decreasing on r0, cs and strictly increasing on r´c, 0s. It follows that (2.30) may only fail in small intervals around the points ξ P I where p φ pˆ2 c m pξ`jq˙´p φ pˆ2 c m pξ`kq˙" 0 for some j, k P Z. Such points must satisfy Thus, we need ξ " 1 2 pm´j´ q, i.e. ξ P t0,˘1 2 u. In view of the continuity of p φ p , it follows that there exists η ą 0 such that (2.30) holds for ξ P r E " r´1 2`η ,´ηs Y rη, 1 2´η s. It remains to observe that with any given η P p0, 1 4 q inequality (2.30) will hold for δ sufficiently small.

Explicit quantitative estimates for the Gaussian.
To obtain explicit estimates, we need to establish a precise relation between η and δ in the proof of Corollary 2.10. In other words, we need to estimate min ξP r E ψpξq, where, as above, r E " r´1 2`η ,´ηs Y rη, 1 2´η s, η P p0, 1 4 q, and the function ψ is given by With this, the assertion of the lemma follows immediately from the mean value theorem.
The above observation leads to the following explicit estimate for the Gaussian kernel.
We remark that the estimate in the above proposition is quite pessimistic. Our numerical experiments showed that the true bound may be much better.

Remez-Turán Property and Fixing the Blind Spots
In Theorem 2.8, the main issue is that the lower bound is only in terms of p f 1 E and not p f so that stability is not obtained. In this section, we consider a certain class of subsets of P W c for which Theorem 2.8 does lead to stable reconstruction.

Remez-Turán Property.
Definition 3.1. Let V Ă P W c and write p V " t p f : f P V u Ă L 2 pr´c, csq. We will say that p V has the Remez-Turán property if, for every E Ă r´c, cs of positive Lebesgue measure, there exists C " CpE, V q such that, for every f P V , When V is a finite dimensional subspace of P W c such that p V consists of analytic functions (restricted to I), then p V has the Remez-Turán property since p f 1 E 2 is then a norm on V which, by finite dimensionality of V , is equivalent to p f 1 r´c,cs 2 . However, the previous argument does not provide any quantitative estimate on the constant CpE, V q. Let us start with two fundamental examples of spaces that have the Remez-Turán property, and for which quantitative estimates are known.

Fourier polynomials.
Let V N be given by (1.11), so that p V N " tP 1 r´c,cs , P P C N rxsu is the space of polynomials of degree at most N , restricted to I. The quantitative form of the Remez-Turán property for p V N is then known as the Remez Inequality [9]: for every polynomial of degree at most N , 3.3. Sparse sinc translates with free nodes. Let V N be given by (1.10), so that p V N " # P 1 r´c,cs : P pξq " N ÿ n"1 c n e 2iπλnξ + . Recall that p V N is not a linear subspace. The fact that p V N has the Remez-Turán property is a deep result of Nazarov [17]: for every exponential polynomial of order at most N , i.e. every P of the form P pξq " ř N n"1 c n e 2iπλnξ one has where γ is an absolute constant.
3.4. Prolate spheroidal wave functions (PSWF). The Prolate spheroidal wave functions (PSWFs) denoted by pψ n,c p¨qq ně0 , are defined as the bounded eigenfunctions of the Sturm-Liouville differential operator L c , defined on C 2 pr´1, 1sq, by They are also the eigenfunctions of the finite Fourier transform F c , as well as the ones of the operator Q c " c 2π Fc F c , which are defined on L 2 pr´1, 1sq by They are normalized so that }ψ n,c } L 2 pr´1,1sq " 1 and ψ n,c p1q ą 0. We call pχ n pcqq ně0 the corresponding eigenvalues of L c , µ n pcq the eigenvalues of F c (3.38) µ n pcqψ n,c pxq " ż 1 1 ψ n,c pyqe´i cxy dy, x P r´1, 1s. and λ n pcq the ones of Q c which are arranged in decreasing order. They are related by λ n pcq " c 2π |µ n pcq| 2 .
A well known property is then that ψ n,c L 2 pRq " 1 ?
λnpcq . Further, their Fourier transform is given by (3.39) y ψ n,c pξq " ż R ψ n,c pxqe´i xξ dx " p´1q k 2π c µ n |µ n pcq| 2 ψ n,cˆξ c˙1 |ξ|ďc The crucial commuting property of L c and Q c has been first observed by Slepian and coauthors [21], whose name is closely associated with all properties of PSWFs, the spectrum of the operators L c and Q c and almost time-and band-limited functions. Among the basic properties of PSWFs, we cite their analytic extension to the whole real line and their unique properties to form an orthonormal basis of L 2 pr´1, 1sq and an orthogonal basis of P W c . The prolate spheroidal wave functions admit a good representation in terms of the orthonormal basis of Legendre polynomials. In agreement with the standard practice, we will be denoting by P k the classical Legendre polynomials, defined by the three-term recursion with the initial conditions P 0 pxq " 1, P 1 pxq " x.
These polynomials are orthogonal in L 2 pr´c, csq and are normalized so that P k p1q " 1 and We will denote by P k,c the normalized Legendre polynomial r P k,c pxq " c 2k`1 2c P k´x c¯a nd the P k,c 's then form an orthonormal basis of L 2 pr´c, csq. We start from the following identity relating Bessel functions of the first kind to the finite Fourier transform of the Legendre polynomials, see [6]: for every x P R, where j k is the spherical Bessel function defined by j k pxq " p´xq kˆ1 x d dx˙k sin x x . Note that j k has the same parity as k and recall that, for x ě 0, j k pxq " a π 2x J k`1{2 pxq where J α is the Bessel function of the first kind. In particular, from the well-known bound |J α pxq| ď |x| α 2 α Γpα`1q , valid for all x P R, we deduce that Using the bound Γpxq ě ? 2πx x´1{2 e´x we get 2p2k`3q k`1 |x| k , k P N.
We have the following lemma.
We will also need the following estimate. .
Proof. Precise pointwise estimates of the λ n pcq's have been obtained in [15,Section 4 & Appendix C] and have been further improved in [8] to λ n pcq ďˆe c 2pn`1q˙2 n`1 for n ě max´n, ec 2¯.
while we always have λ n pcq ă 1.

It follows that
The result follows.
We can now prove our Remez lemma for Prolate spheroidal wave functions. Then, for every ψ P p V N and every E Ď r´c, cs of positive measure, Proof. Let ψ " ř N n"0 c ψ n,c so that, by orthogonality and the fact that ψ n,c " λ n pcq´1 {2 , On the other hand Let K be an integer that will be fixed later and write Note that F K is a polynomial of degree K so that ψ by Cauchy-Schwarz. We now apply Lemma 3.2 to get Using Lemmas 3.3 and 3.2 we can rewrite this in the form First, if N ď maxpec, 2q, then we want so that it is enough that e 2K`5 ď c 3 48 2 p3`ecq and e 2K`5 ď |E| 8c so we take On the other hand, if N ě maxpec, 2q, then we want It remains to fix the blind spots p f 1 E 2 with the help of a Remez type inequality. For V " V N given by (1.10), (1.11) and f P V N , we simply have p Adding the estimates for fixing the blind spot yields (3.47).
Remark 3.7. Theorem 3.5 immediately implies Theorem 1.6. We also note that if V " V N is given by (1.9) or (1.11), the reconstruction can be done from measurements at a finite number of spacial locations. Indeed, our results imply that in this case one can find the coefficients of f in its decomposition in a basis of V via simple least squares.

Sensor density, maximal spatial gaps and condition numbers
In this section, we discuss irregular spatio-temporal sampling. We establish that stable reconstruction from dynamical samples may occur when the set Λ has an arbitrarily small density. More importantly, however, we show that the density cannot be arbitrarily small for fixed frame bounds in (1.4). In fact, we provide an explicit estimate for the maximal spatial gap in terms of the condition number B A . Example 4.1. In this example, we take c " 1{2 to simplify discussion. Assume that φ P Φ is such that p Φ is real, even, and decreasing on r0, 1{2s. Let Λ 0 " mZ, with m P N odd, Λ k " mnZ`k, where n is any fixed odd number and k " 1, . . . m´1
The claim in the last example follows by stringing together several theorems on dynamical sampling. Firstly, [4, Theorems 2.4 and 2.5] yield that any f P 2 pZq can be recovered from the space-time samples tφ j˚f px k q : j " 0, . . . , m´1, x k P Λu and that the problem of sampling and reconstruction in P W c on subsets of Z is equivalent to the sampling and reconstruction problem of sequences in 2 pZq. Secondly, combining [5,Theorems 5.4 and 5.5] shows that for φ P Φ, f P P W c can be stably reconstructed from tφ j˚f px k q : j " 0, . . . , m´1, x k P Λu if and only if (1.7) is satisfied.
Example 4.1 thus shows that (1.7) can hold with sets having arbitrarily small densities. The goal of this section is to show that the maximal gap in such sets is controlled by the condition number B{A.
Theorem 4.4. Let φ P Φ and assume that p φ is C 1 -smooth on r´c, cs. Assume that Λ Ď R is a stable sampling set for Problem 1 with frame bounds A, B (i.e., (1.4) holds: Proof. Denoting I a " ra´π{4c, a`π{4cs, a P R, let us bound the covering number n Λ :" sup aPR # pΛ X I a q .
As a first consequence, this implies that D`pΛq ď 4 B c Φ,L . Now we assume that for some a 0 P R, and some R ě π c , Λ X ra 0´R , a 0`R s " H. As the Paley-Wiener space is invariant under translation, if (1.4) holds for Λ, it also holds for its translates, so that we may assume that a 0 " 0.
since we assumed that R ě π{c. It follows that R ď 8c π B A C Φ,L c Φ,L . Finally, note that this implies that D´pΛq ě 1 2R .
Remark 4.5. Computing the explicit estimate for C Φ,L c Φ,L , we observe that the maximal allowed gap in spacial measurements grows with L, which is to be expected. For the Gaussian, we may take the constant C Φ,L c Φ,L to be OpL 2 q (see (4.52)). The above results also shows that for C 1 -smooth functions φ, stable sampling sets must have positive lower density. The authors are also grateful for the hospitality of various conferences, where we were able to work together on this project. We thank the hosts and organizers of all those events, in particular: ICERM, University of Bordeaux, and Vanderbilt University.
Finally, it gives us great pleasure to dedicate this paper to Guido Weiss on the occasion of his 90 th birthday. To a dear friend who taught so much to so many: Merry Guidmas!