Sampling the flow of a bandlimited function

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Sampling the flow of a bandlimited function

Akram Aldroubi, Karlheinz Gröchenig, Longxiu Huang, Philippe Jaming, Ilya Krishtal, José-Luis Romero

To cite this version:

Akram Aldroubi, Karlheinz Gröchenig, Longxiu Huang, Philippe Jaming, Ilya Krishtal, et al.. Sam- pling the flow of a bandlimited function. 2020. �hal-02557625�

AKRAM ALDROUBI, KARLHEINZ GR ¨OCHENIG, LONGXIU HUANG, PHILIPPE JAMING, ILYA KRISHTAL, AND JOS ´E LUIS ROMERO

Abstract. We analyze the problem of reconstruction of a bandlimited functionf from the space-time samples of its statesft“φt˚f resulting from the convolution with a kernelφt. It is well-known that, in natural phenomena, uniform space-time samples off 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 functionfpaway 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.

1. 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 “ fpxq under the action of convolution op- erators. 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 Wc :“

!

f PL²pRq: supppfpq Ď r´c, cs )

with the Fourier transform normalized as fpξq “p ş

Rfptqe^´itξdt.

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 kernelstφ_t :t ą0u such thatupt, xq “ φ_t˚fpxq,φ_t`s“φ_t˚φ_s for allt, sP p0,8q, and f “ lim

tÑ0`φ_t˚f,f P L²pRq.

As we are primarily interested in physical systems, we typically consider the following set of kernels:

Φ_c“ tφ PL¹pRq: there exists κ_φą0 such thatκ_φ ďφpξq ďp 1 for |ξ| ďc,φp0q “p 1u.

(1.1)

Observe that φ P L¹ implies that φp is continuous and, therefore, the existence of κ_φ ą 0 such that φpě κ_φ on r´c, cs is equivalent to φpą0 on r´c, cs. We remark that some of our results hold for a less restrictive class of kernels.

1

Example 1.1. A prototypical example is the diffusion process with φp_tpξq “ e^´tσ2ξ2, t ą 0 It corresponds to the initial value problem (IVP) for the heat equation (with a diffusion parameter σ­“0)

(1.2)

#

Btupx, tq “σ²B_x²upx, tq for xPR and tą0

upx,0q “ fpxq ,

for which the solution is given by upx, tq “ pφ_t˚fqpxq.

Other examples include the IVP for the fractional diffusion equation

#

Btupx, tq “ pB_x²q^α{2upx, tq for xPR and tą0

upx,0q “ fpxq ,0ăαď1,

for which the solution is given by upx, tq “ pφ_t˚fqpxq with φp_tpξq “ e^´t|ξ|α, and the IVP for the Laplace equation in the upper half plane

#

∆upx, yq “0 for xP R and yą0 upx,0q “ fpxq

,

for which the solution is given by upx, yq “ pφ_y ˚fqpxqwith φp_ypξq:“e^´y|ξ|. The following problem serves as a motivation for this paper.

Problem 1. Let φPΦ, Lą0, and ΛĂRbe a discrete subset of R. What are the conditions that allow one to recover a functionf PP Wc in a stable way from the data set

(1.3) tpf˚φ_tqpλq:λP Λ,0ďtďLu?

The set of measurements (1.3) is the image of an operator T :P W_c ÑL²`

Λˆ r0, Ls˘

. Thus, the stable recovery of f from (1.3) amounts to finding conditions on Λ, φ and Lsuch that T has a bounded inverse from TpP W_cqto P W_c or, equivalently, the existence ofA, B ą0 such that

(1.4) Akfk²₂ ď

żL 0

ÿ

λPΛ

|pf ˚φtqpλq|²dt ďBkfk²₂, for all f PP Wc.

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 Lą0, 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 re- construction 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 PP W_c:

f_t:“f˚φ_t, 0ďtď1, where φ_t is the heat kernel at time t:

φp_tpξq “ e^´tσ2ξ2, (1.5)

with a parameter σ ­“0. According to Shannon’s sampling theorem, f can be stably recon- structed from equispaced samplestfpk{Tq:k PZuif and only if the sampling rateT 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

D^´pΛq:“lim inf

rÑ8 inf

xPR

#pΛŞ

rx´r, x`rsq 2r

satisfies D^´pΛq ě c

π. Recall that the upper Beurling density is defined by D^`pΛq:“lim sup

rÑ8

sup

xPR

#pΛŞ

rx´r, x`rsq

2r .

We are interested to know if the spatial sampling rate can be reduced by using the infor- mation provided by the following spatio-temporal samples:

(1.6) tftpk{Tq:k PZ,0ďtď1u.

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 ΛĎR that do lead to a meaningful spatio-temporal trade-off: there exist sets Λ ĎRthat have sub-Nyquist density and, yet, lead to the frame inequality:

(1.7) Akfk²₂ ď

ż1 0

ÿ

λPΛ

|pftqpλq|²dt ďBkfk²₂, for all f PP Wc,

withA, B ą0; 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 ĎR be such that (1.7) holds. Then there exists an absolute constant K ą0 such that, for RěKmax

ˆB A,1

c

˙

and every aP R, we have ra´R, a`Rs XΛ‰ H.

In particular, we have D^´pΛq ěK^´1min`_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 subsetV ĎP Wc, aiming for an inequality of the form:

A}f}²₂ ď ż1

0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dtďB}f}²₂, f PV.

(1.8)

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 : suppfpĎ Eu. In effect, E is the set r´c, cszO whereOis a small open neighborhood of a finite set, i.e., E avoids a certain number of “blind spots.”

Theorem 1.5. Let φPΦ andm ě2 be an integer. Then for any rą0there exists a certain compact set E Ď r´c, cs of measure at least 2c´r such that any f PV_E can be recovered from the samples

M“

! f_t

´mπ c k

¯

:k PZ,0ďtď1 )

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 withB “1 and some Aą0 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 sinc- kernel operator

Qcfpxq “ ż1

´1

sinπcpy´xq

πpy´xq fpyqdy.

Using the min-max theorem, we get that ψ_n,c is the norm-one solution of the following ex- tremal problem

max

"kfk_L²_p´1,1q

kfk_L²_pRq :f P P W_c, f Pspantψ_k,c : kănu^K

*

where the conditionf Pspantψ_k,c: k ănu^K is void forn “0. The familypψ_n,cqně0 forms an orthogonal basis forP W_cand has the property to form an orthonormal sequence inL²p´1,1q.

We consider the N-dimensional space

(1.9) VN “spantψ^c₀, . . . , ψ^c_Nu ĂP Wc.

The Landau-Pollak-Slepian theory shows that this subspace provides an optimal approxi- mation of a bandlimited function that is concentrated on r´1,1s. More precisely, V “ V_N minimizes the approximation error

sup

fPP Wc

kfk2“1 gPVinf

ż1

´1

|fpxq ´gpxq|² dx, among all N-dimensional subspaces of P W_c.

Sparse sinc translates with free nodes. In this model, we let

(1.10) V_N “

# _N ÿ

n“1

c_nsinccpx´λ_nq :c₁, . . . , c_N PC, λ₁, . . . , λ_N PR +

be the class of linear combinations of N arbitrary translates of the sinc kernel sincpxq “ ^sin_x^x Note that V_N is not a linear space. However, V_N´V_N ĎV_2N. Therefore, (1.8) with V “V_2N implies

A}f ´g}²₂ ď ż₁

0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯

´g_t

´mπ c k

¯ˇ ˇ ˇ

2

dtďB}f´g}²₂, f, g PV_N,

which ensures the numerical stability of the sampling problemf ÞÑ tf_tpmπk{cq:k PZ: 0ď t ď 1u restricted non-linearly to the class VN. In other words, if (1.8) holds with V “ V2N

then any f PVN 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,

(1.11) V_N “

# _N ÿ

n“0

c_nDⁿsincc¨:c₀, . . . , c_N P C +

,

where D:P WcÑP Wc is the differential operator Df “f¹. Observe that the union of such VN, N PN, is dense in P 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 ě2 be an integer, Φ be given by Φpξq “p e^´σ2ξ2. Let V “VN be given by (1.9), (1.10), or (1.11). Then (1.8) holds with

A“ cκ0pcq pσcq²`mexp

´

´κ₁pcqN ´m²`

´κ₂pcqlnσ`κ<sub>3</sub>pcqσ<sup>2</sup>`lnm˘¯

, B “1, where the κ_j’s are positive constants that depend on conly.

We provide a more precise expression for the lower frame constant in Theorem 3.5. Note that the lower bound deteriorates when σ² Ñ 0 (no diffusion) and σ² Ñ `8 (very rapid diffusion). This agrees with the intuition and numerical experiments for (non-bandlimited) sparse initial conditions presented in [20]: ifσ²is very small, because of spatial undersampling, some components of f may be hidden from the sensors, while for large σ² the diffusion completely blurs out the signal and no information can be extracted.

Remark 1.7. To simplify the discussion we takec“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 V₁ given by (1.10) with λ₁ “1. The samples at timet “0 are not sufficient to identify each signal since sincp¨ ´1q PV_N vanishes on mZ, mě2. Similarly, for Theorem 1.5: the function sinpω¨qsincp_a^¨q, with an appropriately chosen a and ω, belongs to V_E and vanishes on mZ for mě 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 att “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 heat- flow of a bandlimited function on a grid (1.6) as follows [16]. The function with Fourier transform

fp:“δ_´T ´δ_T

is formally bandlimited to I “ r´c, cs if T ăc, and vanishes on the lattice _T^πZ. Moreover, f is an eigenfunction of the diffusion operator since

fpt“e^´tσ2p´Tq2δ´T ´e^´tσ2T2δT “e^´tσ2T2f ,p

see (1.2) and (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, Tu, a more formal argument can be given by regularization. If η : R Ñ R is continuous and supported on r´1,1s and η_εpxq “ε^´1ηpx{εq, then f ¨ηp_ε PP 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 kernelsφas in (1.1). Indeed, an analysis along the lines of the Papoulis sampling theorem [18]

shows that the diffusion samples (1.6) of a functionf PP W_c do not lead to a stable recovery of fp. However, these samples do allow for the stable recovery away from certain blind spots determined by φ; that is, one can effectively recover fp¨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 fp. The main tools, in this case, are Remez-Tur´an-like inequalities of the form:

kf1p _IkďC_Ekfp1_Ek, f PV.

For Fourier polynomials (1.11) the classical Remez-Tur´an inequality provides an explicit con- stant C_E, while the case of sparse sinc translates (1.10) is due to Nazarov [17]. The corre- sponding 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 dynam- ical samples at sub-Nyquist rate allow one to stably reconstruct the function fpaway 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-Tur´an 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 dynam- ical sampling from [4, 5] and adapt them for problems studied in this paper.

Forφ PL¹, consider the function φp_ppxq “

ÿ

kPZ

φppx´2ckq1_r´c,cqpx´2ckq,

that is, the 2c-periodization of the piece of φpsupported in r´c, cq. Recall that we consider kernels from the set Φ given by (1.1). Hence,

κ_φďφp_ppξq ď1, ξP R. We also write

fp_tpξq:“fpξqp φp^tpξq, f P P W_c.

Next, we introduce thesampled diffusion matrix, which is themˆmmatrix-valued function given by

(2.12) B_mpξq “

¨

˝ ż1

0

pφqp^t_p ˆ2c

mpξ`jq

˙ pφqp ^t_p

ˆ2c

mpξ`kq

˙ dt

˛

‚

0ďj,kďm´1

“ ż1

0

A^˚_mpξ, tqA_mpξ, tqdt,

where

A_mpξ, tq “ ˆ

pφpq^t_p ˆ2c

mpξ`kq

˙˙

k“0,...,m´1

“ ˆ

pφqp ^t_p ˆ2c

mξ

˙

¨ ¨ ¨ pφqp^t_p ˆ2c

mpξ`m´1q

˙˙

PM1,mpCq.

Remark 2.1. Observe that the matrix function B_m is m-periodic. Its eigenvalues, however, are 1-periodic because the matrices B_mpξq and B_mpξ`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

(2.13) fpξq “ ˆ

pfpqp

ˆ2c

mpξ`jq

˙˙

j“0,...,m´1

“

¨

˚

˚

˚

˚

˚

˝

pfpqp

ˆ2c mξ

˙

... pfpqp

ˆ2c

mpξ`m´1q

˙

˛

‹

‹

‹

‹

‹

‚

PM_m,1pCq.

Note that if we recoverfpξqfor ξ P r0,1s then we can recover f_p. Observe also that

ż1 0

}fpξq}²dξ“

m´1

ÿ

j“0

ż1 0

ˇ ˇ ˇ ˇpfpqp

ˆ2c

mpξ`jq

˙ˇ ˇ ˇ ˇ

2

dξ“ m 2c

m´1

ÿ

j“0

ż2cpj`1q{m 2cj{m

|pfpqppuq|²du

“ m 2c

ż2c 0

|pfqp_ppsq|²ds“ m 2c

żc

´c

|fpsq|p ²ds (2.14)

In other words, f ÞÑ b2c

mf :P W_cÑL²pr0,1s,M_m,1pCqqis an isometric isomorphism.

Lemma 2.2. For f PP W_c, (2.15)

ż1 0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dt“

´ c mπ

¯2ż1 0

fpξq^˚B_mpξqfpξqdξ.

Proof. Observe that it suffices to prove the result inP W_cXSpRq(the Schwarz class). Consider the function

bpξ, tq “ ÿ

kPZ

f_t

´mπ c k

¯

e^´2iπkξ.

Using the Poisson summation formula and the definition of f_t, we get bpξ, tq “ c

mπ ÿ

jPZ

fp_t ˆ2c

mpξ`jq

˙

“ c mπ

ÿ

´^m₂´ξďjă^m₂´ξ

φp^t ˆ2c

mpξ`jq

˙ fp

ˆ2c

mpξ`jq

˙

“ c

mπ

m´1

ÿ

j“0

pφqp ^t_p ˆ2c

mpξ`jq

˙ pfpqp

ˆ2c

mpξ`jq

˙ ,

Note that the functionsbp¨, tqare 1-periodic,

(2.16) bpξ, tq “ c

mπA_mpξ, tqfpξq, and thus

ż1 0

|bpξ, tq|²dt “

´ c mπ

¯2

fpξq^˚B_mpξqfpξq, ξ PR. Combining the last equation with the Parseval’s relation

ż1 0

|bpξ, tq|²dξ “ ÿ

kPZ

ˇ ˇ ˇft

´mπ c k

¯ˇ ˇ ˇ

2

. (2.17)

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_mpξq in (2.12).

For symmetric φPΦ and m ě2, however, inf

ξPr0,1sλ_min`

B_mpξq˘

“λ_min`

B_mp0q˘

“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 Er Ď r0,1s such that λ_min`

B_mpξq˘

ě κ for ξ P E. Then, repeating ther computation in (2.14), we get

ż1 0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dt “

´ c mπ

¯2ż1 0

fpξq^˚B_mpξqfpξqdξ.ěκ

´ c mπ

¯2ż

E˜

}fpξq}²dξ

“ cκ 2mπ²

ż

E

}fpξq}p ²dξ (2.18)

where E “ ˆ2c

mpE˜`Zq

˙

X r´c, cs.

In the following example, we offer some numerics. To simplify the computations, we repre- sentB_mpξqin (2.12) as a Pick matrix (see, e.g., [7, 10]). Forξ P r´c, cq, we writeφpξq “p e^´ψpξq, so that ψ ě0 and ψp0q “0, and obtain forj, k “0, . . . , m´1,

pBmqjkpξq “ ż1

0

φp^t ˆ2c

mpξ`j¹q

˙ φp^t

ˆ2c

mpξ`k¹q

˙ dt where the indices j¹, k¹ are in the set

(2.19) I_ξ “

"

nP Z: ξ`n

m P r´1{2,1{2q

* ,

m divides |j´j¹|and |k´k¹|, and j,k, and ξ are not 0 simultaneously. Thus pB_mqjkpξq “

ż1 0

e^´tp^ψp^2cmpξ`j<sup>1</sup>qq<sup>`ψ</sup>p^2cmpξ`k¹qqqdt

“ ˆ

ψ ˆ2c

mpξ`j¹q

˙

`ψ ˆ2c

mpξ`k¹q

˙˙´1

´

1´e^´p^ψp^2cmpξ`j<sup>1</sup>qq<sup>`ψ</sup>p^2cmpξ`k¹qqq¯ (2.20)

Observe thatpB_mq00p0q “1.

Example 2.4. Here, we choose φ to be the Gaussian function, i.e., φppξq “φp₁pξq “e^´σ2ξ2

for various values of σ ­“0. Hence, ψpξq “ σ²ξ², and we get pBmqjkpξq “ m²

4c²σ² ¨ 1´e^´

´σ2

m2p^{pξ`j1q2`pξ`k1q2}q

¯

pξ`j<sup>1</sup>q<sup>2</sup>` pξ`k¹q² with j¹, k¹, and pB_mq00p0q as above.

In Figure 1, we show the condition numbers of the matricesB_mpξq with ξ“0.45, c“1{2, mP t2,3,5u, and σ varying from 1 to 200.

(a) (b) (c)

Figure 1. Condition numbers of B_mpξq for m P t2,3,5u, c “ 1{2, ξ “ 0.45, and σ P r1,200s.

In Figure 2, we also show the condition numbers of the matricesB_mpξq. 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 eigenvalueλ^pmq_minpξq of the matricesB_mpξqin (2.12). We also present an upper estimate for λ^pmq_minpξq, which follows from the general theory of Pick matrices [7, 10].

We begin with the following auxiliary result.

Lemma 2.5. Letv₀, v₁, . . . v_m´1 bemdistinct non-zero real numbers and letv“ pv₀, . . . , v_m´1q.

For k P N, define a function Ψ_k : R Ñ R by Ψ_kptq “ _1´t^1´t2{k² if t ‰ 1 and ψ_kp1q “ k. For j “0, . . . , m´1, define

σ_j² “

m´1

ÿ

k“0

v_j^2k “Ψ_mpv_j^mq “

$

&

%

m if v_j “1

1´v^2m_j

1´v²_j otherwise.

(a) (b) (c)

Figure 2. Condition numbers of B_mpξq for m P t2,3,5u, c “ 1{2, σ “ 200, and ξP r0.35,0.49s.

Let σ “

´řm´1 j“0 σ_j²

¯1{2

, γ_´ “min_j|v_j| ą0, γ_` “max_j|v_j| and let α“

ˆm´1 σ²

˙^m´1₂

ź

0ďjăkďm´1

|v_j ´v_k|.

ForN PN, letW_N be thepmNqˆmVandermonde matrix associated tov_N “ pv

1 N

0 , v

1 N

1 , . . . v

1 N

m´1q, i.e.,

W_N “

” v

i´1 N

j

ı

1ďiďmN,0ďjďm´1

. Then for each xPC^m, we have

α²Ψ_Npγ_´q}x}² ď }W_Nx}² ďσ²Ψ_Npγ_`q}x}². Proof. letV be the mˆm Vandermonde matrix associated to v:

V “ rvⁱ_js0ďiďm´1,0ďjďm´1.

Note that the Frobenius norm ofV and its determinant are given by kVkF “σ and |detV| “

ź

0ďjăkďm´1

|vj ´vk|.

Recall from [23] an estimate for the minimal singular value of an mˆm matrix A:

(2.21) σminpAq ě

ˆm´1 }A}²_F

˙pm´1q{2

|detA|.

Specifying this to V we get σ_minpVq ěα. AskVkďkVk_F, it follows that, for all xPC^m, (2.22) α²}x}² ď }V x}² ďσ²}x}².

LetD_N be the diagonal matrix withv_N on the main diagonal. Since }W_Nx}² “ xW_N^˚W_Nx, xy “

N´1

ÿ

`“0

xpD^{`</sup><sub>N</sub>q<sup>˚</sup>V<sup>˚</sup>V D<sub>N</sub><sup>`} x, xy “

N´1

ÿ

`“0

}V D^`_Nx}²,

we deduce from (2.22) that

N´1

ÿ

`“0

α²}D^`_Nx}² ď }WNx}² ď

N´1

ÿ

`“0

σ²}D^`_Nx}².

Moreover, we have γ

2`

N

´ }x}² ď }D^`_Nx}² ď γ

2`

N

`}x}² by definition of D_N. The conclusion now

follows by summing the two geometric sequences.

Note that the function ΨN is increasing on p0,`8qand that, for t ‰1, tą0

(2.23) lim

NÑ8

1

NΨ_Nptq “ 1´t²

lim_NÑ8Np1´e^{2 lnt{N}q “ ˇ ˇ ˇ ˇ

1´t² 2 lnt

ˇ ˇ ˇ ˇ .

Corollary 2.6. With the notation of Lemma 2.5, assume further that 0 ă ν ďvj ď 1 and mě2. Let

(2.24) αr“e^´1{2m^´m´12 ź

0ďjăkďm´1

|v_j ´v_k|.

Then for each xPC^m, we have

αr²Ψ_Npνq}x}² ď }W_Nx}² ďm²N}x}².

Proof. Indeed, νďγ_´ ďγ_{`</sub> ď1 so Ψ<sub>N</sub>pνq ď Ψ<sub>N</sub>pγ<sub>´</sub>q and Ψ<sub>N</sub>pγ<sub>`}q ď Ψ_Np1q “N. Further, sincev_j ď1, σ² ďm². Moreover, the derivative of `_t´1

t

˘pt´1q{2

“`

1´ ¹_t˘pt´1q{2

is 1

2 ˆ

1´ 1 t

˙pt´1q{2ˆ 1

t `ln ˆ

1´ 1 t

˙˙

ď0 for tě1. Thus,

ˆm´1 m

˙pm´1q{2

ě lim

tÑ`8exp

„t´1 2 ln

ˆ 1´ 1

t

˙

“e^´1{2. It follows that α in the statement of Lemma 2.5 satisfies

αě ś

0ďjăkďm´1|vj ´vk|

?em^pm´1q{2 ,

and the result is established.

Proposition 2.7. Let φP Φ. Define

∆_mpξq “ ź

0ďjăkďm´1

ˇ ˇ ˇ ˇ

φp_p ˆ2c

mpξ`jq

˙

´φp_p ˆ2c

mpξ`kq

˙ˇ ˇ ˇ ˇ .

Then, for each xPC^m, we have 1

2em^m2∆mpξq²¨

1´κ^2{m_φ

|lnκ_φ| }x}² ď xBmpξqx, xy ďm}x}².

Proof. We fixξ and apply Corollary 2.6 tov_j “ pφqp _p ˆ2c

mpξ`jq

˙¹

m

. Withαr given by (2.24),

αr“e^´1{2m^´m´12 ź

0ďjăkďm´1

ˇ ˇ ˇ ˇ ˇ

φp_p ˆ2c

mpξ`jq

˙1{m

´φp_p ˆ2c

mpξ`kq

˙1{mˇ ˇ ˇ ˇ ˇ , we get

αr²Ψ_Npκ^1{m_φ q}x}² ď }W_Nx}² ďm²N}x}².

On the other hand, _mN¹ W_N^˚W_N equals the left-endmN-term Riemann sum for the integral defining B_mpξq. It follows that

xB_mpξqx, xy “ lim

NÑ8

1

mNxW_N^˚W_Nx, xy “ lim

NÑ8

1

mN}W_Nx}². Using (2.23), we get

αr² 1´κ^2{m_φ

2m|lnκ_φ|}x}² ď xB_mpξqx, xy ďm}x}².

Finally, note that if 0 ă a, b ď1, using the mean value theorem, there is an η P pa, bq such that

|a^1{m´b^1{m| “ 1

m|a´b|η^´1`1{m ě 1

m|a´b|.

Therefore

αr “ e^´1{2m^´m´12 ź

0ďjăkďm´1

ˇ ˇ ˇ ˇ ˇ

φp_p ˆ2c

mpξ`jq

˙1{m

´φp_p ˆ2c

mpξ`kq

˙1{mˇ ˇ ˇ ˇ ˇ ě e^´1{2m^{´m´12 ´mpm´1q2} ∆pξq “ e^´1{2m^´m

2´1 2 ∆pξq

establishing the postulated estimates.

For an upper estimate of the minimal eigenvalueλ^pmq_minpξqwe use the estimates of the singular values of Pick matrices by Beckerman-Townsend [7]. For p_j P C, j “1, . . . , m, and 0ă aď x₁ ăx₂ ă ¨ ¨ ¨ ăx_m ďb let

(2.25) pP_mqjk “ p_j `p_k

x_j `x_k, j, k “1, . . . , m,

be the corresponding Pick matrix. Then the smallest singular value s_min of P_m is bounded above by

(2.26) s_min ďmin

$

&

% 1,4

« exp

˜ π² 2 ln`_4b

a

˘

¸ff´2tm{2u, . -

s_max,

where s_max is the largest singular value.

IfpPr_mqjk “ ^1´c_x ^jck

j`xk, then Pr<sub>m</sub> is related to a Pick matrix of the form (2.25) via the diagonal matrix D“diagp1`cjq:

1

2D^´1Pr_mD^´1 “P_m with p_j “ ^1´c_1`c^j

j, c_j ‰ ´1.

In our case, see (2.20), x_j “ ψ`_2c

mpξ`jq˘

and c_j “ e^´xj P p0,1s, so Id ď D ď 2Id and the singular values of B_mpξq and the corresponding Pick matrix P_m differ at most by a factor 4. Therefore, (2.26) holds with apξq “ min ψ`_2c

mpξ`kq˘

:k PI_ξ(

and bpξq “ max ψ`_2c

mpξ`kq˘

:k PI_ξ(

, I_ξ defined in (2.19), and an additional factor 4 provided that apξq ‰0.

For our main examples, we haveψpξq “ |ξ|^α, αą0. This yields bpξq ďc^α and apξq “min

"ˇ ˇ ˇ ˇ

2c

mpξ´kq ˇ ˇ ˇ ˇ

α

: 2c

m|ξ´k| ďc, |ξ| ď 1 2

*

“ ˆ2c

m|ξ|

˙α

So for the smallest singular value of B_mpξq we obtain the estimate λ^pmq_minpξq ď 4²

» –exp

¨

˝

π² 2 ln 4

´m 2|ξ|

¯α

˛

‚ fi fl

´2tm{2u

m

ď16m exp

˜

´ pm´1qπ² ln 16`2αln_2|ξ|^m

¸ . (2.27)

Observe that the Beckerman-Townsend estimate (2.26) holds for all Pick matrices with the same values for a “ minx_j and b “ maxx_j and is completely independent of the par- ticular 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</sub> are close together (if x<sub>j</sub> “ x<sub>j`1}, then P_m is singular). Thus, (2.27) is an optimistic upper estimate forλ^pmq_minpξq. 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.

2.3. Partial recoverability.

Theorem 2.8. Let φ P Φ, m ě 2 an integer and Er Ď I “ r0,1s be a compact set. Assume that there exists δą0 such that, for every 0ďj ăk ďm´1 and every ξ PEr

ˇ ˇ ˇ ˇ

φp_p ˆ2c

mpξ`jq

˙

´φp_p ˆ2c

mpξ`kq

˙ˇ ˇ ˇ ˇěδ.

Let E “ ˆ2c

mpE˜`Zq

˙

X r´c, cs. Then for any f P P Wc, the function fp1E can be recovered from the samples

(2.28) M“

! f_t

´mπ c k

¯

:k PZ,0ďtď1 )

in a stable way. Moreover, we have

(2.29) Akf1p _Ek² ď

ż1 0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dt ď c 2π²}f}p²,

where

A“ c 4eπ²

δ^mpm´1q m^1`m2

κ^2{m_φ ´1 lnκ_φ . Proof. Recall from (2.15) that we need to estimate

ż1 0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dt“

´ c mπ

¯2ż1 0

fpξq^˚B_mpξqfpξqdξ.

The upper bound follows directly from Proposition 2.7, and (2.14):

ż1 0

fpξq^˚Bmpξqfpξqdξďm ż1

0

}fpξq}²dξ “ m² 2ckfpk².

Let us now prove the lower bound using (2.18). First, ∆_mpξq ě δ^mpm´1q2 . It follows from Proposition 2.7 that, if ξP Er then

fpξq^˚B_mpξqfpξq ě

κ^2{m_φ ´1

2em^m2lnκ_φδ^mpm´1q}fpξq}². Takingκ “

κ^2{m_φ ´1

2em^m2lnκ_φδ^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 theσ_minof 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 PΦ, φpis even and strictly decreasing on R`, and m ě2 is an integer. GivenηP p0,<sup>1</sup><sub>4</sub>q, letEr “ r´<sup>1</sup><sub>2</sub>`η,´ηs Y rη,¹₂´ηsandE “

ˆ2c

mpE˜`Zq

˙

X r´c, cs. Then there exists A ą0 such that, for any f PP W_c,

Akf1p _Ek² ď ż1

0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dt ď }f}p².

Proof. We look into the main condition of Theorem 2.8: there exists δ ą 0 such that, for every 0ďj ăkďm´1 and every ξPEr

(2.30)

ˇ ˇ ˇ ˇ

φp_p ˆ2c

mpξ`jq

˙

´φp_p ˆ2c

mpξ`kq

˙ˇ ˇ ˇ ˇěδ.

For a general φ P Φ, the function φp is continuous and, therefore, φp_p is continuous, except possibly on c` 2cZ where a jump discontinuity occurs if φpp´cq ‰ φppcq. Under current assumptions, however, φpis even and, therefore φp_p is continuous everywhere.

For 0ď`ďm´1 andξ PI, we have ´ 1

2m ď ξ``

m ď1´ 1 2m and

φp_p ˆ2c

mpξ``q

˙

“

$

’’

&

’’

% φp

ˆ2c

mpξ``q

˙

if ξ``

m ă1{2 φp

ˆ2c

mpξ``´mq

˙

if ξ``

m ě1{2 .

Thus, the condition of Theorem 2.8 would be satisfied with Er “I if |φ|p were one-to-one onI, that is, either strictly decreasing or strictly increasing. However, φpis even and strictly decreasing onR`, so thatφpp is continuous, strictly decreasing onr0, csand strictly increasing on r´c,0s. It follows that (2.30) may only fail in small intervals around the points ξ P I where φpp

ˆ2c

mpξ`jq

˙

´φpp

ˆ2c

mpξ`kq

˙

“0 for some j, k PZ. Such points must satisfy ξ`j

m “1´ξ``

m , 0ďj ď m´1

2 ă` ďm´1.

Thus, we needξ “ ¹₂pm´j´`q, i.e.ξP t0,˘<sup>1</sup><sub>2</sub>u. In view of the continuity of φp<sub>p</sub>, it follows that there exists ηą0 such that (2.30) holds for ξPEr“ r´<sup>1</sup><sub>2</sub> `η,´ηs Y rη,¹₂ ´ηs. It remains to observe that with any given ηP p0,¹₄q inequality (2.30) will hold for δ sufficiently small.

2.4. 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

ξPEr

ψpξq, where, as above, Er “ r´¹₂ `η,´ηs Y rη,¹₂ ´ηs, ηP p0,¹₄q, and the function ψ is given by

ωpξq “ min

j,kPZ

ˇ ˇ ˇ ˇ

φp_p ˆ2c

mpξ`jq

˙

´φp_p ˆ2c

mpξ`kq

˙ˇ ˇ ˇ ˇ .

Lemma 2.11. Let E and Er be as in Corollary 2.10. Assume that the kernel φ P Φ is such that φpis differentiable on E and

minξPE

ˇ ˇ ˇφp¹pξq

ˇ ˇ ˇ ěR.

Then

min

ξPEr

ωpξq ě 4cRη m . Proof. Observe that

min

ξPEr

min

j,kPZ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

2c

mpξ`jq ˇ ˇ ˇ ˇ´

ˇ ˇ ˇ ˇ

2c

mpξ`kq ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ“ 2c

m2η.

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.

Proposition 2.12. Let φpξq “p e^´σ2ξ2, σ­“0, and mě2 be an integer. Given ηP p0,¹₄q, let Er “ r´¹₂ `η,´ηs Y rη,¹₂ ´ηs and E “

ˆ2c

mpE˜`Zq

˙

X r´c, cs. Then, for any f PP W_c, we have

(2.31) Akf1p _Ek² ď

ż1 0

ÿ

kPZ

ˇ ˇ ˇf_t

´mπ c k

¯ˇ ˇ ˇ

2

dt ď }fp}², where

(2.32) A“ c

2eπ²p2pσcq²`mq

p4cRηq^mpm´1q

m^1´m`2m2 with R “2σ²min

!

ηe^´pσηq2, ce^´pσcq2 )

. Proof. Observe that Lemma 2.11 applies withRgiven by (2.32). It remains to apply Theorem 2.8 with κ_φ “e^´pσcq2 and δ “4cRη{m. We deduce that (2.31) holds with

A “ c 4eπ²

δ^mpm´1q m^1`m2

κ^2{m_φ ´1 lnκφ

“ c 2eπ²

1´e^´

2pσcq² m 2pσcq²

m

¨ p4cRηq^mpm´1q m^2´m`2m2 .

Using 1´e^´t

t ě 1

t`1, we obtain the claimed bound.

We remark that the estimate in the above proposition is quite pessimistic. Our numerical experiments showed that the true bound may be much better.

3. Remez-Tur´an Property and Fixing the Blind Spots

In Theorem 2.8, the main issue is that the lower bound is only in terms of kfp1_Ek and not kfpkso 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.

3.1. Remez-Tur´an Property.

Definition 3.1. LetV ĂP W_cand writeVp “ tfp: f P Vu ĂL²pr´c, csq. We will say thatVp has the Remez-Tur´an property if, for every E Ă r´c, cs of positive Lebesgue measure, there existsC “CpE, Vqsuch that, for every f PV,

(3.33) kf1p _Ek₂ ěCkf1p _r´c,csk₂.

WhenV is a finite dimensional subspace ofP W_c such thatVp consists of analytic functions (restricted to I), then Vp has the Remez-Tur´an property since kfp1_Ek₂ is then a norm on V which, by finite dimensionality of V, is equivalent to kfp1_r´c,csk₂. However, the previous argument does not provide any quantitative estimate on the constant CpE, Vq. Let us start with two fundamental examples of spaces that have the Remez-Tur´an property, and for which quantitative estimates are known.

3.2. Fourier polynomials. Let VN be given by (1.11), so thatVpN “ tP1_r´c,cs, P P C^Nrxsu is the space of polynomials of degree at most N, restricted to I. The quantitative form of the Remez-Tur´an property for VpN is then known as the Remez Inequality [9]: for every polynomial of degree at most N,

(3.34) kP1_r´c,csk₂ ď

ˆ8c

|E|

˙N`1{2

kP1_Ek₂.

3.3. Sparse sinc translates with free nodes. Let V_N be given by (1.10), so that Vp_N “

#

P1r´c,cs :Ppξq “

N

ÿ

n“1

cne^2iπλnξ +

. Recall that VpN is not a linear subspace. The fact that Vp_N has the Remez-Tur´an property is a deep result of Nazarov [17]: for every exponential polynomial of order at most N, i.e. every P of the formPpξq “ řN

n“1c_ne^2iπλnξ one has

(3.35) kP1_r´c,cskď

ˆγc

|E|

˙N`1{2

kP1_Ek, where γ is an absolute constant.

3.4. Prolate spheroidal wave functions (PSWF). The Prolate spheroidal wave functions (PSWFs) denoted by pψ_n,cp¨qqně0, are defined as the bounded eigenfunctions of the Sturm- Liouville differential operatorL_c, defined on C²pr´1,1sq, by

(3.36) L_cpψq “ ´p1´x²qd²ψ

d x² `2xdψ

d x `c²x²ψ.

They are also the eigenfunctions of the finite Fourier transform F_c, as well as the ones of the operator Qc “ c

2πF_c^˚Fc, which are defined onL²pr´1,1sq by (3.37) F_cpfqpxq “

ż1

´1

e^{i c x y}fpyqdy, and Q_cpfqpxq “ ż1

´1

sinpcpx´yqq

πpx´yq fpyqdy.

They are normalized so that }ψ_n,c}L²pr´1,1sq “ 1 and ψ_n,cp1q ą 0. We call pχ_npcqq_ně0 the corresponding eigenvalues of L_c, µ_npcq the eigenvalues of F_c

(3.38) µnpcqψn,cpxq “ ż1

´1

ψn,cpyqe^´icxydy, xP r´1,1s.

and λ_npcq the ones ofQ_c which are arranged in decreasing order. They are related by λ_npcq “ c

2π|µ_npcq|². A well known property is then that kψ_n,ck_L²_pRq “ ?¹

λnpcq. Further, their Fourier transform is given by

(3.39) ψyn,cpξq “ ż

R

ψn,cpxqe^´ixξdx“ p´1q^k2π c

µ_n

|µ_npcq|²ψn,c

ˆξ c

˙ 1|ξ|ďc

The crucial commuting property of L_c and Q_c has been first observed by Slepian and co- authors [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 ofL²pr´1,1sq and an orthogonal basis of P W_c.

The prolate spheroidal wave functions admit a good representation in terms of the or- thonormal basis of Legendre polynomials. In agreement with the standard practice, we will be denoting byP_k the classical Legendre polynomials, defined by the three-term recursion

Pk`1pxq “ 2k`1

k`1 xPkpxq ´ k

k`1Pk´1pxq, with the initial conditions

P₀pxq “1, P₁pxq “ x.

These polynomials are orthogonal in L²pr´c, csqand are normalized so that P_kp1q “ 1 and

ż1

´1

P_kpxq²dx“ 1 k`1{2. We will denote by P_k,c the normalized Legendre polynomial Pr_k,cpxq “

c2k`1 2c P_k

´x c

¯ and the P_k,c’s then form an orthonormal basis of L²pr´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 everyxPR,

(3.40)

ż1

´1

e^ixyP_kpyqdy“2i^kj_kpxq, kP N, where j_k is the spherical Bessel function defined by j_kpxq “ p´xq^k

ˆ1 x

d dx

˙k

sinx

x . Note that j_k has the same parity as k and recall that, for xě0, j_kpxq “a_π

2xJ<sub>k`1{2</sub>pxq where J<sub>α</sub> is the Bessel function of the first kind. In particular, from the well-known bound|J<sub>α</sub>pxq| ď <sub>2</sub>αΓpα`1q^|x|α , valid for all xPR, we deduce that

|j_kpxq| ď?

π |x|^k

2<sup>k`1</sup>Γpk`3{2q, kP N. Using the bound Γpxq ě ?

2πx^x´1{2e^´x we get

(3.41) |j_kpxq| ď e^k`3{2

?2p2k`3q<sup>k`1</sup>|x|^k, k PN. We have the following lemma.

Lemma 3.2. Write ψy_n,c “ř

kě0β_kⁿpcqP_k,c. Then for every k, `ě0

|β_kⁿ| ď 10 c^3{2|λ_npcq|

ˆ e 2k`3

˙k`1

This bound is an adaptation of techniques from [15] to improve the proof of the exponential decay from [22].

Proof. Using (3.39), we have

β_kⁿpcq “ xψ_n,c, P_k,cy_L2pIq “ żc

´c

ψy_n,cpxqP_k,cpxqdx

“ p´1q^k µ_npcq

|µnpcq|² 2π

c

c2k`1 2c

żc

´c

ψ_n,cpx{cqP_k

´x c

¯ dx

“ p´1q^kπ µ_npcq c^1{2|µnpcq|²

?4k`2 ż1

´1

ψ_n,cpxqP_kpxqdx

“ p´1q^kπ? 4k`2 c^1{2|µ_npcq|²

ż1

´1

ż1

´1

ψ_n,cpyqe^´icxydy P_kpxqdx with (3.38). Recalling that λ_npcq “ c

2π|µ_npcq|² and using Fubini, we get β_kⁿpcq “ p´1q^k2?

4k`2 c^3{2λ_npcq

ż1

´1

ż1

´1

P_kpxqe^´icxydx ψ_n,cpyqdy

“ p´iq^k4? 4k`2 c^3{2λnpcq

ż1

´1

ψ_`pyqj_kpyqdy

with (3.40). But then, from (3.41) and Cauchy-Schwarz, we deduce that

|β_kⁿpcq| ď 4? 4k`2 c^3{2λnpcq

ˆż1

´1

j_kpyq²dy

˙1{2

ď 4?

2k`1e<sup>k`3{2</sup> p2k`3q<sup>k`1</sup>c^3{2λ_npcq

ˆż1

´1

|y|^2kdy

˙1{2

“ 4?

2e 1

c^3{2λ_npcq ˆ e

2k`3

˙k`1

. As 4?

2eď10, the result follows.

We will also need the following estimate.

Lemma 3.3. The eigenvalues (3.38) of Q_c satisfy

(3.42) Λ_N :“

˜ _N ÿ

n“0

1 λ_npcq

¸1{2

ď

$

&

%

?3`ec if N ďmaxpec,2q

´2pN`1q ec

¯^2N`1₂

if N ěmaxpec,2q .

Proof. Precise pointwise estimates of the λ_npcq’s have been obtained in [15, Section 4 &

Appendix C] and have been further improved in [8] to λ_npcq ď

ˆ ec 2pn`1q

˙2n`1

for n ěmax

´ n,ec

2

¯ . while we always haveλ_npcq ă 1.