https://doi.org/10.1007/s12220-021-00617-0
Sampling the Flow of a Bandlimited Function
Akram Aldroubi1·Karlheinz Gröchenig2·Longxiu Huang3· Philippe Jaming4,5·Ilya Krishtal6·José Luis Romero2,7
Accepted: 27 April 2020 / Published online: 8 February 2021
© The Author(s) 2021
Abstract
We analyze the problem of reconstruction of a bandlimited function f from the space–
time samples of its states ft = φt ∗ f resulting from the convolution with a kernel φt. 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, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function 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-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.
Keywords Dynamical sampling·Mobile sampling·Bandlimited function·Heat flow·Remez-Turan inequality
Mathematics Subject Classification 42C05·42C30·42C40·65M99·65T99
1 Introduction
In this paper, we consider the sampling and reconstruction problem of signalsu = u(t,x)that arise as an evolution of an initial signal f = f(x)under the action of convolution operators. The initial signal f is assumed to be in the Paley-Wiener space
Extended author information available on the last page of the article
P Wc,c>0 (fixed throughout this paper) given by P Wc:=
f ∈L2(R):supp(f)⊆ [−c,c]
with the Fourier transform normalized as f(ξ)=
R f(t)e−i tξdt.
The functionsu 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{φt :t >0}such thatu(t,x)=φt∗f(x),φt+s =φt∗φsfor allt,s∈(0,∞), and f = lim
t→0+φt ∗ f, f ∈ L2(R).
As we are primarily interested in physical systems, we typically consider the fol- lowing set of kernels:
c= {φ∈L1(R):there existsκφ>0 such thatκφ≤φ(ξ)≤1 for|ξ| ≤c,φ(0)=1}.
(1.1) Observe thatφ∈ L1implies thatφis continuous and, therefore, the existence of κφ>0 such thatφ≥κφon[−c,c]is equivalent toφ >0 on[−c,c]. We remark that some of our results hold for a less restrictive class of kernels.
Example 1.1 A prototypical example is the diffusion process withφt(ξ) =e−tσ2ξ2, t >0 It corresponds to the initial value problem (IVP) for the heat equation (with a diffusion parameterσ =0)
∂tu(x,t)=σ2∂x2u(x,t) forx∈Randt>0
u(x,0)= f(x) , (1.2)
for which the solution is given byu(x,t)=(φt∗ f)(x).
Other examples include the IVP for the fractional diffusion equation ∂tu(x,t)=(∂2x)α/2u(x,t) forx∈Randt >0
u(x,0)= f(x), 0< α≤1,
for which the solution is given byu(x,t)=(φt∗ f)(x)withφt(ξ)=e−t|ξ|α, and the IVP for the Laplace equation in the upper half plane
u(x,y)=0 forx ∈Randy>0
u(x,0)= f(x) ,
for which the solution is given byu(x,y)=(φy∗ f)(x)withφy(ξ):=e−y|ξ|. The following problem serves as a motivation for this paper.
Problem 1 Letφ ∈ ,L >0, and ⊂ Rbe a discrete subset ofR. What are the conditions that allow one to recover a function f ∈P Wcin a stable way from the data set
{(f ∗φt)(λ):λ∈,0≤t ≤L}? (1.3) The set of measurements (1.3) is the image of an operatorT :P Wc→L2
×[0,L]
. Thus, the stable recovery of f from (1.3) amounts to finding conditions on, φand L such that T has a bounded inverse fromT(P Wc) to P Wc or, equivalently, the existence ofA,B>0 such that
Af22≤ L
0
λ∈
|(f ∗φt)(λ)|2dt≤Bf22, for all f ∈ P Wc. (1.4)
If for a givenφandLthe frame condition (1.4) is satisfied, we say that=φ,L is a stable sampling set.
Remark 1.2 It was shown in [5, Theorem 5.5] thatφ,L is a stable sampling set for someL >0, if and only ifφ,1is a stable sampling set.
Thus, for qualitative results, we will only consider the case ofL =1. For quanti- tative results, however, we may keepLin 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 Problem1in 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 Wc:
ft := f ∗φt, 0≤t ≤1, whereφt is the heat kernel at timet:
φt(ξ)=e−tσ2ξ2, (1.5)
with a parameterσ =0. According to Shannon’s sampling theorem, f can be stably reconstructed from equispaced samples{f(k/T):k∈Z}if and only if the sampling rateT is bigger than or equal to the critical valueT = c
π, known as theNyquist 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 ⊆R, then thelower Beurling density
D−():=lim inf
r→∞ inf
x∈R
#(
[x−r,x+r]) 2r
satisfiesD−()≥ c
π. Recall that theupper Beurling densityis defined by D+():=lim sup
r→∞ sup
x∈R
#(
[x−r,x+r])
2r .
We are interested to know if the spatial sampling rate can be reduced by using the information provided by the following spatio-temporal samples:
{ft(k/T):k∈Z,0≤t≤1}. (1.6) Observe that the amount of collected data in (1.6) is not smaller than that in the case of sampling at the Nyquist rateT = c
π. IfT < 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 allT < 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 ⊆Rthat do lead to a meaningfulspatio-temporal trade-off: there exist sets⊆R that have sub-Nyquist density and, yet, lead to the frame inequality:
Af22≤ 1
0
λ∈
|(ft)(λ)|2dt ≤ Bf22, for all f ∈ P Wc, (1.7)
with A,B >0; see Example4.1for a concrete construction. The emerging field of dynamical sampling investigates such phenomena in great generality (see, e.g., [1–5]).
As follows from Example4.1, 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 themaximal gapof a stable sampling set to the bounds from (1.7).
Theorem 1.4 Let ⊆ R be such that (1.7) holds. Then there exists an absolute constant K > 0 such that, for R ≥ Kmax
B A,1
c
and every a ∈ R, we have [a −R,a +R] ∩ = ∅. In particular, we have D−() ≥ K−1minA
B,c and D+()≤ K B.
Theorem4.4, which is a more general version of the above result, provides a more explicit dependence ofK on the parameters of the problem.
Besides the constraints implied by Theorem1.4, 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) withT = c mπ, m ∈ N, and restrict the sampling/reconstruction problem to a subset V ⊆ P W ,
aiming for an inequality of the form:
Af22≤ 1
0
k∈Z
ft
mπ
c k2 dt ≤ Bf22, f ∈V. (1.8) Specifically, we consider the following signal models.
Away from blind spots. We will identify a set Ewith measure arbitrarily close to 1 such that (1.8) holds withV =VE = {f ∈ P Wc :supp f ⊆ E}. In effect,E is the set[−c,c] \OwhereOis a small open neighborhood of a finite set, i.e.,Eavoids a certain number of “blind spots.”
Theorem 1.5 Letφ∈and m ≥2be an integer. Then for any r >0there exists a certain compact set E ⊆ [−c,c]of measure at least2c−r such that any f ∈VEcan be recovered from the samples
M= ft
mπ c k
:k∈Z,0≤t ≤1
in a stable way.
The setE in the above theorem depends only onφand the choice ofr. The stable recovery in this case means that (1.8) holds with B =1 and some A >0 which is estimated in a more explicit version of the above result, Theorem2.8.
Prolate spheroidal wave functions. The Prolate Spheroidal Wave Functions (PSWFs) are eigenfunctions of an integral operator known as the time-band liminting operator or sinc-kernel operator
Qcf(x)= 1
−1
sinπc(y−x)
π(y−x) f(y)dy.
Using the min-max theorem, we get thatψn,cis the norm-one solution of the following extremal problem
max
fL2(−1,1)
fL2(R) : f ∈ P Wc, f ∈span{ψk,c: k<n}⊥
where the condition f ∈span{ψk,c: k<n}⊥is void forn =0. The family(ψn,c)n≥0
forms an orthogonal basis for P Wc and has the property to form an orthonormal sequence inL2(−1,1).
We consider theN-dimensional space
VN =span{ψ0c, . . . , ψNc} ⊂ P Wc. (1.9)
The Landau-Pollak-Slepian theory shows that this subspace provides an optimal approximation of a bandlimited function that is concentrated on[−1,1]. More pre- cisely,V =VN minimizes the approximation error
sup
f∈P Wc
f2=1 ginf∈V
1
−1
|f(x)−g(x)|2dx,
among allN-dimensional subspaces ofP Wc.
Sparse sinc translates with free nodes. In this model, we let
VN = N
n=1
cnsincc(x−λn) :c1, . . . ,cN ∈C, λ1, . . . , λN∈R
(1.10)
be the class of linear combinations ofNarbitrary translates of the sinc kernel sinc(x)=
sinx
x Note thatVNis not a linear space. However,VN −VN ⊆V2N. Therefore, (1.8) withV =V2Nimplies
Af −g22≤ 1
0
k∈Z
ft
mπ c k
−gt
mπ
c k2 dt ≤ Bf −g22, f,g∈VN, which ensures the numerical stability of the sampling problem f → {ft(mπk/c): k ∈ Z: 0 ≤ t ≤ 1}restricted non-linearly to the classVN. In other words, if (1.8) holds withV =V2N then any f ∈VN 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 mostNrestricted to the unit interval. Explicitly,
VN = N
n=0
cnDnsincc· :c0, . . . ,cN ∈C
, (1.11)
whereD: P Wc→ P Wcis the differential operatorD f = f. Observe that the union of suchVN,N ∈N, is dense inP Wc.
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≥2be an integer,be given by(ξ)=e−σ2ξ2. Let V =VNbe given by(1.9),(1.10), or(1.11). Then(1.8)holds with
A= cκ0(c) (σc)2+mexp
−κ1(c)N−m2
−κ2(c)lnσ +κ3(c)σ2+lnm
, B =1, where theκ ’s are positive constants that depend on c only.
We provide a more precise expression for the lower frame constant in Theorem3.5.
Note that the lower bound deteriorates whenσ2→0 (no diffusion) andσ2→ +∞
(very rapid diffusion). This agrees with the intuition and numerical experiments for (non-bandlimited) sparse initial conditions presented in [20]: if σ2 is very small, because of spatial undersampling, some components of f may be hidden from the sensors, while for largeσ2the diffusion completely blurs out the signal and no infor- mation can be extracted.
Remark 1.7 To simplify the discussion we take c = 1/2 in this remark. There are instances when Theorem1.6applies for a signal f ∈VNwhich cannot be recovered simply from its samples on, say, 2Z. As an example, we offer V1 given by (1.10) withλ1 = 1. The samples at timet = 0 are not sufficient to identify each signal since sinc(· −1) ∈ VN vanishes on mZ,m ≥ 2. Similarly, for Theorem 1.5: the function sin(ω·)sinc(a·), with an appropriately chosena andω, belongs toVE and vanishes onmZform≥2. In finite dimensional subspacesVN, e.g., given by (1.9) and (1.11), sampling at timet=0 with anym∈Nmay 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 att =0. Theorems1.5and1.6can be used together. For example, a function f can be reconstructed away from the blind spots using Theorem1.5and approximated around the blind spots using Theorem1.6.
1.2 Technical Overview
Lu and Vetterli explain the impossibility of subsampling the heat-flow of a bandlimited function on a grid (1.6) as follows [16]. The function with Fourier transform
f :=δ−T −δT
is formally bandlimited to I = [−c,c]if T < c, and vanishes on the lattice πTZ.
Moreover, f is an eigenfunction of the diffusion operator since ft =e−tσ2(−T)2δ−T −e−tσ2T2δT =e−tσ2T2f,
see (1.2) and (1.5). Hence, all the diffusion samples (1.6) vanish, although f ≡ 0.
While no Paley-Wiener function is infinitely concentrated at{−T,T}, a more formal argument can be given by regularization. Ifη:R→Ris continuous and supported on [−1,1]andηε(x)=ε−1η(x/ε), then f ·ηε ∈ P Wcand provides a counterexample to (1.4), provided thatεis sufficiently small.
As we show below in Sect. 2.1, a similar phenomenon holds for more general diffusion kernelsφ 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 Wc
do not lead to a stable recovery of f. However, these samples do allow for the stable recoveryaway from certain blind spotsdetermined byφ; that is, one can effectively recover f ·1E, for a certain subset E ⊆ I of positive measure (1E denotes the characteristic function of the setE). If we, furthermore, restrict the sampling problem to one of the finite dimensional spacesV =VN given by (1.9), (1.10), or (1.11), we
may then infer all other values of f. The main tools, in this case, areRemez-Turán-like inequalities of the form:
f1I ≤CEf1E, f ∈V.
For Fourier polynomials (1.11) the classical Remez-Turán inequality provides an explicit constantCE, 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 Sect.2, we show that uniform dynamical samples at sub-Nyquist rate allow one to stably reconstruct the function f away from certain, explicitly described blind spots determined by the kernelφ. 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 Sect.3, we restrict the problem to the setsV =VN 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 Sect.4, we discuss the case of irregular spacial sampling. We recall that a stable reconstruction may be possible with setsthat 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 BA of the frame bounds).
2 Recovering a Bandlimited Function Away from the Blind-Spot 2.1 Dynamical Sampling inPWc
In this section, we recall some of the results on dynamical sampling from [4,5] and adapt them for problems studied in this paper.
Forφ∈L1, consider the function φp(x)=
k∈Z
φ(x−2ck)1[−c,c)(x−2ck),
that is, the 2c-periodization of the piece ofφ supported in[−c,c). Recall that we consider kernels from the setgiven by (1.1). Hence,
κφ≤φp(ξ)≤1, ξ ∈R.
We also write
ft(ξ):= f(ξ)φt(ξ), f ∈P Wc.
Next, we introduce thesampled diffusion matrix, which is them×mmatrix-valued function given by
Bm(ξ)=
⎛
⎝
1
0
(φ)tp 2c
m(ξ+ j)
(φ)tp 2c
m(ξ+k)
dt
⎞
⎠
0≤j,k≤m−1
= 1
0
A∗m(ξ,t)Am(ξ,t)dt, (2.12)
where
Am(ξ,t)=
(φ)tp
2c m(ξ+k)
k=0,...,m−1
=
(φ)tp
2c mξ
· · ·(φ)tp
2c
m(ξ+m−1)
∈M1,m(C).
Remark 2.1 Observe that the matrix functionBm ism-periodic. Its eigenvalues, how- ever, are 1-periodic because the matricesBm(ξ)andBm(ξ+k),k∈Z, are similar via a circular shift matrix.
The following lemma explains the role of the sampled diffusion matrix. In the lemma, we let
f(ξ)=
(f)p
2c m(ξ+j)
j=0,...,m−1=
⎛
⎜⎜
⎜⎜
⎜⎝
(f)p
2c mξ
... (f)p
2c
m(ξ+m−1)
⎞
⎟⎟
⎟⎟
⎟⎠
∈Mm,1(C). (2.13)
Note that if we recoverf(ξ)forξ ∈ [0,1]then we can recover fp. Observe also that
1
0 f(ξ)2dξ=
m−1
j=0 1 0
(f)p
2c
m(ξ+ j)
2dξ= m 2c
m−1
j=0
2c(j+1)/m
2cj/m |(f)p(u)|2du
= m 2c
2c 0
|(f)p(s)|2ds= m 2c
c
−c|f(s)|2ds
(2.14) In other words, f →
2c
mf : P Wc → L2([0,1],Mm,1(C))is an isometric isomor- phism.
Lemma 2.2 For f ∈ P Wc,
1 0
k∈Z
ft
mπ
c k2dt= c mπ
2 1 0
f(ξ)∗Bm(ξ)f(ξ)dξ. (2.15)
Proof Observe that it suffices to prove the result inP Wc∩S(R)(the Schwarz class).
Consider the function
b(ξ,t)=
k∈Z
ft
mπ c k
e−2iπkξ.
Using the Poisson summation formula and the definition of ft, we get
b(ξ,t)= c mπ
j∈Z
ft 2c
m(ξ+j)
= c mπ
−m2−ξ≤j<m2−ξ
φt 2c m(ξ+j)
f
2c m(ξ+j)
= c mπ
m−1
j=0
(φ)tp
2c m(ξ+j)
(f)p
2c m(ξ+j)
,
Note that the functionsb(·,t)are 1-periodic, b(ξ,t)= c
mπAm(ξ,t)f(ξ), (2.16)
and thus
1 0
|b(ξ,t)|2dt= c mπ
2
f(ξ)∗Bm(ξ)f(ξ), ξ ∈R.
Combining the last equation with the Parseval’s relation
1
0 |b(ξ,t)|2dξ =
k∈Z
ft
mπ
c k2. (2.17)
yields the desired conclusion.
Remark 2.3 Lemma2.2shows that the stability of reconstruction from spatio-temporal samples is controlled by the condition number of the self-adjoint matricesBm(ξ)in (2.12). For symmetricφ∈andm≥2, however,
ξ∈[inf0,1]λmin
Bm(ξ)
=λmin
Bm(0)
=0,
which precludes the stable reconstruction of all f ∈ P Wc, see, e.g., [4]. This adds to our explanation of the phenomenon of blind spots in Sect.1.2. We can nonetheless hope to find a large set E⊆ [0,1]such thatλmin
Bm(ξ)
≥ κ for ξ ∈ E. Then, repeating the computation in (2.14), we get
1 0
k∈Z
ft
mπ
c k2dt= c mπ
2 1 0
f(ξ)∗Bm(ξ)f(ξ)dξ.≥κ c mπ
2
˜
Ef(ξ)2dξ
= cκ
2mπ2 Ef(ξ)2dξ
(2.18) whereE =
2c
m(E˜+Z)
∩ [−c,c].
In the following example, we offer some numerics. To simplify the computations, we representBm(ξ)in (2.12) as a Pick matrix (see, e.g., [7,10]). Forξ ∈ [−c,c), we writeφ(ξ)=e−ψ(ξ),so thatψ≥0 andψ(0)=0,and obtain forj,k=0, . . . ,m−1,
(Bm)j k(ξ)= 1
0
φt 2c
m(ξ+j) φt
2c
m(ξ+k)
dt where the indices j,kare in the set
Iξ =
n∈Z: ξ+n
m ∈ [−1/2,1/2)
, (2.19)
mdivides|j−j|and|k−k|, and j,k, andξ are not 0 simultaneously. Thus
(Bm)j k(ξ)= 1
0
e−t
ψ 2c m(ξ+j)
+ψ 2c m(ξ+k)
dt
=
ψ 2c
m(ξ+j)
+ψ 2c
m(ξ+k) −1
1−e−
ψ 2c m(ξ+j)
+ψ 2c m(ξ+k)
(2.20) Observe that(Bm)00(0)=1.
Example 2.4 Here, we chooseφto be the Gaussian function, i.e., φ(ξ)=φ1(ξ)=e−σ2ξ2
for various values ofσ =0. Hence,ψ(ξ)=σ2ξ2, and we get
(Bm)j k(ξ)= m2
4c2σ2· 1−e−
σ2 m2
(ξ+j)2+(ξ+k)2
(ξ+j)2+(ξ+k)2 with j,k, and(Bm)00(0)as above.
In Figure1, we show the condition numbers of the matricesBm(ξ)withξ =0.45, c=1/2,m∈ {2,3,5}, andσ varying from 1 to 200.
In Figure2, we also show the condition numbers of the matricesBm(ξ). This time, however, stillc = 1/2, the parameterσ is fixed to be 200, whereas the point ξ is allowed to vary from 0.35 to 0.49. We still havem∈ {2,3,5}.
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 eigenvalueλ(minm)(ξ)of the matricesBm(ξ)in (2.12). We also present an upper estimate forλ(minm)(ξ), which follows from the general theory of Pick matrices [7,10].
We begin with the following auxiliary result.
Lemma 2.5 Let v0, v1, . . . vm−1 be m distinct non-zero real numbers and let v = (v0, . . . , vm−1). For k∈N, define a functionk :R→Rbyk(t)= 11−−tt22/k if t =1 andψk(1)=k. For j =0, . . . ,m−1, define
σ2j =
m−1 k=0
v2kj =m(vmj)=
⎧⎨
⎩
m ifvj =1
1−v2mj
1−v2j
otherwise.
Letσ =m−1 j=0σ2j1/2
,γ−=minj|vj|>0,γ+=maxj|vj|and let
α=
m−1 σ2
m−1
2
0≤j<k≤m−1
|vj−vk|.
For N ∈ N, let WN be the (m N)×m Vandermonde matrix associated to vN = (v0N1, v1N1, . . . vmN1−1), i.e.,
WN =
! vji−1N
"
1≤i≤m N,0≤j≤m−1
.
Then for each x∈Cm, we have
α2N(γ−)x2≤ WNx2≤σ2N(γ+)x2. Proof letV be them×mVandermonde matrix associated tov:
V = [vij]0≤i≤m−1,0≤j≤m−1.
Note that the Frobenius norm ofV and its determinant are given by VF =σ and |detV| = |vj−vk|.
Fig.1ConditionnumbersofBm(ξ)form∈{2,3,5},c=1/2,ξ=0.45,andσ∈[1,200]
2ConditionnumbersofBm(ξ)form∈{2,3,5},c=1/2,σ=200,andξ∈[0.35,0.49]
Recall from [23] an estimate for the minimal singular value of anm×mmatrixA:
σmin(A)≥
# m−1 A2F
$(m−1)/2
|detA|. (2.21)
Specifying this toV we getσmin(V)≥ α. AsV ≤ VF, it follows that, for all x∈Cm,
α2x2≤ V x2≤σ2x2. (2.22) LetDNbe the diagonal matrix withvN on the main diagonal. Since
WNx2= WN∗WNx,x =
N−1
=0
(DN)∗V∗V DNx,x =
N−1
=0
V DNx2,
we deduce from (2.22) that
N−1
=0
α2DNx2≤ WNx2≤
N−1
=0
σ2DNx2.
Moreover, we haveγ−2Nx2≤ DNx2 ≤γ+2Nx2by definition of DN. The con- clusion now follows by summing the two geometric sequences.
Note that the functionNis increasing on(0,+∞)and that, fort =1,t>0
Nlim→∞
1
NN(t)= 1−t2
limN→∞N(1−e2 lnt/N)= 1−t2
2 lnt
. (2.23)
Corollary 2.6 With the notation of Lemma2.5, assume further that0 < ν ≤vj ≤ 1 and m≥2. Let
α=e−1/2m−m2−1
0≤j<k≤m−1
|vj−vk|. (2.24)
Then for each x∈Cm, we have
α2N(ν)x2≤ WNx2≤m2Nx2.
Proof Indeed,ν≤γ−≤γ+≤1 soN(ν)≤N(γ−)andN(γ+)≤N(1)=N. Further, since vj ≤ 1, σ2 ≤ m2. Moreover, the derivative of t−1
t
(t−1)/2
=
1−1t(t−1)/2
is 1 2
1−1
t
(t−1)/2 1 t +ln
1−1
t
≤0
fort ≥1. Thus, m−1
m
(m−1)/2
≥ lim
t→+∞exp
!t−1 2 ln
1−1
t
"
=e−1/2.
It follows thatαin the statement of Lemma2.5satisfies α≥
%
0≤j<k≤m−1|vj−vk|
√em(m−1)/2 ,
and the result is established.
Proposition 2.7 Letφ∈. Define
m(ξ)=
0≤j<k≤m−1
φp
2c m(ξ+j)
−φp
2c
m(ξ+k) .
Then, for each x ∈Cm, we have 1
2emm2 m(ξ)2·1−κφ2/m
|lnκφ| x2≤ Bm(ξ)x,x ≤mx2.
Proof We fixξ and apply Corollary2.6tovj =(φ)p
2c m(ξ+j)
1
m
. Withαgiven by (2.24),
α=e−1/2m−m−21
0≤j<k≤m−1
φp
2c m(ξ+j)
1/m
−φp
2c m(ξ+k)
1/m ,
we get
α2N(κφ1/m)x2≤ WNx2≤m2Nx2.
On the other hand, m N1 WN∗WNequals the left-endm N-term Riemann sum for the integral definingBm(ξ). It follows that
Bm(ξ)x,x = lim
N→∞
1
m NWN∗WNx,x = lim
N→∞
1
m NWNx2. Using (2.23), we get
α21−κφ2/m
| κ |x2≤ Bm(ξ)x,x ≤mx2.
Finally, note that if 0<a,b≤1, using the mean value theorem, there is anη∈(a,b) such that
|a1/m−b1/m| = 1
m|a−b|η−1+1/m ≥ 1
m|a−b|.
Therefore
α=e−1/2m−m−21
0≤j<k≤m−1
φp
2c m(ξ+j)
1/m
−φp
2c m(ξ+k)
1/m
≥e−1/2m−m−12 −m(m2−1) (ξ)=e−1/2m−m
2−1
2 (ξ)
establishing the postulated estimates.
For an upper estimate of the minimal eigenvalue λ(minm)(ξ) we use the estimates of the singular values of Pick matrices by Beckerman-Townsend [7]. For pj ∈ C,
j=1, . . . ,m, and 0<a ≤x1<x2<· · ·<xm≤blet (Pm)j k = pj +pk
xj +xk, j,k=1, . . . ,m, (2.25) be the corresponding Pick matrix. Then the smallest singular value smin of Pm is bounded above by
smin≤min
⎧⎨
⎩1,4
&
exp
# π2 2 ln4b
a
$'−2m/2⎫
⎬
⎭smax, (2.26) wheresmaxis the largest singular value.
If(Pm)j k = 1x−j+cjxckk, then Pm is related to a Pick matrix of the form (2.25) via the diagonal matrixD=diag(1+cj):
1
2D−1PmD−1=Pm
withpj =11−+ccjj,cj = −1.
In our case, see (2.20),xj =ψ2c
m(ξ+j)
andcj =e−xj ∈(0,1], so Id≤ D≤ 2Id and the singular values ofBm(ξ)and the corresponding Pick matrixPm differ at most by a factor 4. Therefore, (2.26) holds witha(ξ)=min+
ψ2c
m(ξ+k)
:k∈ Iξ, andb(ξ) = max+
ψ2c
m(ξ+k)
:k∈Iξ,
, Iξ defined in (2.19), and an additional factor 4 provided thata(ξ) =0.
For our main examples, we haveψ(ξ)= |ξ|α,α >0. This yields b(ξ)≤cα and a(ξ)=min
2c
m(ξ−k) α : 2c
m|ξ−k| ≤c, |ξ| ≤ 1 2
= 2c
m|ξ| α
So for the smallest singular value ofBm(ξ)we obtain the estimate
λ(minm)(ξ)≤42
⎡
⎢⎣exp
⎛
⎜⎝ π2 2 ln 4
m 2|ξ|
α
⎞
⎟⎠
⎤
⎥⎦
−2m/2
m
≤16m exp
#
− (m−1)π2 ln 16+2αln2m|ξ|
$ .
(2.27)
Observe that the Beckerman-Townsend estimate (2.26) holds forallPick matrices with the same values fora =minxj andb=maxxjand is completely independent of the particular distribution of thexj. Regardless, it shows that the condition number grows nearly exponentially withm, establishing limitations on how well the space–
time trade-off can work numerically. Of course, the condition number may be much worse if two valuesxjandxj+1are close together (ifxj =xj+1, thenPmis singular).
Thus, (2.27) is an optimistic upper estimate forλ(minm)(ξ). By comparison, our lower estimate in Proposition2.7depends crucially on the distribution of the parameters xj 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.
2.3 Partial Recoverability
Theorem 2.8 Letφ ∈ , m ≥ 2 an integer and E ⊆ I = [0,1]be a compact set.
Assume that there existsδ > 0such that, for every0 ≤ j <k ≤ m−1and every ξ ∈ E
φp
2c m(ξ+j)
−φp
2c
m(ξ+k) ≥δ.
Let E = 2c
m(E˜+Z)
∩ [−c,c]. Then for any f ∈ P Wc, the function f1Ecan be recovered from the samples
M= ft
mπ c k
:k∈Z,0≤t ≤1
(2.28)
in a stable way. Moreover, we have
Af1E2≤ 1ft
mπ
c k2 dt ≤ c
2π2f2, (2.29)
where
A= c 4eπ2
δm(m−1) m1+m2
κφ2/m−1 lnκφ . Proof Recall from (2.15) that we need to estimate
1 0
k∈Z
ft
mπ
c k2dt = c mπ
2 1 0
f(ξ)∗Bm(ξ)f(ξ)dξ.
The upper bound follows directly from Proposition2.7, and (2.14):
1 0
f(ξ)∗Bm(ξ)f(ξ)dξ ≤m
1 0
f(ξ)2dξ = m2 2cf2.
Let us now prove the lower bound using (2.18). First, m(ξ)≥δm(m2−1). It follows from Proposition2.7that, ifξ ∈Ethen
f(ξ)∗Bm(ξ)f(ξ)≥ κφ2/m−1
2emm2lnκφδm(m−1)f(ξ)2.
Takingκ = κφ2/m−1
2emm2lnκφδm(m−1)in (2.18) gives the result.
Remark 2.9 The condition number implied by the above theorem is not the best pos- sible one can obtain through this method. For instance, a better estimate for theσmin
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 asmincreases. This follows from the Beckerman-Townsend estimate (2.26) we discussed in the previous subsection.
Corollary 2.10 Assume that φ ∈ ,φ is even and strictly decreasing onR+, and m ≥ 2 is an integer. Given η ∈ (0,14), let E = [−12 +η,−η] ∪ [η,12 −η] and E =
2c
m(E˜+Z)
∩ [−c,c]. Then there exists A>0such that, for any f ∈ P Wc,
Af1E2≤ 1
0
k∈Z
ft
mπ
c k2 dt ≤ f2.
Proof We look into the main condition of Theorem2.8:there existsδ >0such that, for every0≤ j <k≤m−1and everyξ ∈E
φp
2c m(ξ+ j)
−φp
2c
m(ξ+k)
≥δ. (2.30)
