Fourier uniqueness in even dimensions

Andrew Bakan; Haakan Hedenmalm; Alfonso Montes-Rodríguez; Danylo Radchenko; Maryna Viazovska

doi:10.1073/pnas.2023227118

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved February 24, 2021 (received for review November 7, 2020)

Significance

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of D.R. and M.V. and the lattice-cross uniqueness set for the Klein–Gordon equation studied by H.H. and A.M.-R. With appropriate modifications, the approach applies in any even dimension ≥4 and is based on a sophisticated analysis of the iterates of a Gauss-type map.

Abstract

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d=1,8,24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d=8 and d=24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein–Gordon equation. Since the existing method for the Klein–Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein–Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.

1. Introduction

1.1. Basic Notation in the Plane

We write Z for the integers, Z+ for the positive integers, R for the real line, and C for the complex plane. We write H for the upper half-plane {τ∈C: Imτ>0}. Moreover, we let ⟨⋅,⋅⟩d denote the Euclidean inner product of Rd.

1.2. Motivation

Oscillatory processes are governed by hyperbolic equations, and little is known about interpolation problems for such equations. About 10 y ago, Hedenmalm and Montes-Rodríguez (1) considered bounded solutions of the Klein–Gordon equation ∂x∂yu+u=0 of the formuρ(x,y)=∫R⁡exp(ixt+iy/t) ρ(t)dt, ρ∈L1(R)and examined the Goursat problem for such solutions, where instead of prescribing their values on the two characteristics x=0 and y=0, they assumed that these values are known only along a discrete subset of the characteristics consisting of the equidistant points (an,0) and (0,bn), where n∈Z and a,b>0 are fixed. They obtained that the values of uρ at those points determine the function uρ uniquely if and only if ab≤π2. The most delicate case is when ab=π2, and then we may take a=b=π without loss of generality. An essential property of the solutions uρ is that the boundary values in any spacelike quarter-plane are connected across the boundary half-lines by a Hankel (or Fourier–Bessel) transform. In particular, the quadrant [0,+∞)×(−∞,0] is spacelike, and the boundary values of u=uρ relate according tou(0,−y)=u(0,0)−∫0+∞J1(y,t)u(t,0)dt, y∈[0,+∞).Here we write Jν(x,y)≔(x/y)ν/2Jν(2xy), where Jν denotes the standard Bessel function, so that this functionJν(x,y)=∑m=0+∞(−1)mm!Γ(m+ν+1) xm+νymbecomes well defined and solves the Klein–Gordon equation for x>0 and ν>−1. We associate to this extended Bessel function the corresponding Hankel operator Jνf(x)=∫0+∞Jν(t,x)f(t)dt. Later, the result on the discretized Goursat problem was extended to give local uniqueness on spacelike quarter-planes, from data along discrete sequences along the boundary half-lines (2, 3). This suggests the following general problem. Let F be a linear space of continuous functions on the half-line [0,+∞) with the property that if f∈F, then Jνf is well defined and continuous as well. For a given parameter value ν and a,b>0 as above, we would like to have the implication(∀n=0,1,2,…: f(bn)=Jνf(an)=0) ⇒ f=0for as wide as possible space of functions f∈F. Moreover, since it is well known that the Fourier transform of radial functions in Rd may be expressed in terms of the Hankel transform with ν=d2−1, d≥1,∫Rde−2πix,ydf(π|x|2) dvold(x)=Jd2−1f(π|y|2), dvold(x)≔dx1⋯dxd, |x|2=⟨x,x⟩dthis question can be recast in terms of the uniqueness of a radial function in terms of its values and those of its Fourier transform along a sequence that is a fixed positive multiple of the square roots of nonnegative integers. This program was carried out for radial test functions in the Schwartz class in dimension d=1 by Radchenko and Viazovska with explicit interpolation formulæ (4), and a variant with double zeros along the sequences was successful in dimensions d=8 and d=24 (5, 6). Recently, Stoller extended the Fourier uniqueness problem to nonradial test functions with the function and its Fourier transform given along spherical shells, with explicit interpolation formulæ (7) in all dimensions d≥2. The goal of the present paper is to show that the uniqueness property holds for a substantially wider class of radial functions in even-dimensional Rd, without further explicit use of Bessel functions.

1.3. The Fourier Transform of Radial Functions

For a function f∈L1(Rd). we consider its Fourier transform (with x=(x1,…,xd) and y=(y1,…,yd))f^(y)≔∫Rde−2πi⟨x,y⟩df(x)dvold(x),If f is radial, then f^ is radial too. A particular example of a radial function is the GaussianGτ(x)≔eiπτ|x|2,[1]which decays nicely provided that Imτ>0, that is, when τ∈H. The Fourier transform of a Gaussian is another Gaussian, in this caseĜτ(y)≔τi−d/2e−iπ|y|2/τ=τi−d/2G−1/τ(y).[2]Here it is important that τ↦−1/τ preserves hyperbolic space H. The Fourier transform extends to tempered distributions, and in this sense, the above relationship [2] extends to boundary points τ∈R as well. We now consider the relationshipΦ(x)≔∫RGτ(x)ϕ(τ)dτ=∫Reiπτ|x|2ϕ(τ)dτ, x∈Rd.[3]In terms of the Fourier transform, the relationship readsΦ(x)=ϕ^1−|x|22,where the subscript signifies that we are dealing with the Fourier transform on R1. This tells us that Φ is radial, but pretty arbitrary, if, say, ϕ∈L1(R). If, e.g., ϕ ranges over the Schwartz test functions on R, then Φ ranges over the radial Schwartz test functions on Rd [this is a consequence of the work of Hassler Whitney on the structure of smooth even functions (8)]. In view of the functional identity [1], the Fourier transform of the radial function Φ equalsΦ^(y)≔∫RĜτ(y)ϕ(τ)dτ=∫Rτi−d/2G−1/τ(y)ϕ(τ)dτ=∫Rτi−d/2e−iπ|y|2/τϕ(τ)dτ.[4]The function Φ^ may be thought of as a Gauss–Schrödinger transform of ϕ, given that the integral kernel Ĝτ is the fundamental solution of a Schrödinger equation without potential. We now rewrite the relationships [3] and [4] using integration by parts. If ϕ is a tempered test function, integration by parts applied to [3] gives thatΦ(x)=iπ|x|2∫Reiπτ|x|2ϕ′(τ)dτ, x∈Rd\{0}.[5]A similar application of integration by parts to [4] gives thatΦ^(y)=iπ|y|2∫Rτi(4−d)/2ϕ(τ)∂τe−iπ|y|2/τdτ=1iπ|y|2∫R∂ττi(4−d)/2ϕ(τ)e−iπ|y|2/τdτ,[6]where y∈Rd\{0}, and we need to be a little careful around τ=0 unless d∈{0,2,4}. For d=4, [6] simplifies toΦ^(y)=1iπ|y|2∫Rϕ′(τ)e−iπ|y|2/τdτ, y∈R4\{0}.[7]As for the test function ϕ, we could think of the relations [5] and [7] as the fundamental relationship in place of [3] and [4]. This allows us to place conditions on the derivative ϕ′ in place of ϕ. For our considerations, we need one more piece of information:∫Rϕ′(τ)dτ=0,[8]which is obvious for, e.g., Schwartz test functions ϕ.

2. Main Results

2.1. The Setup in Dimension 4

We focus on R4 only and consider for ψ∈L1(R) the associated functionΨ(x)=iπ|x|2∫Reiπτ|x|2ψ(τ)dτ, x∈R4\{0}.[9]This is the same as relation [5], only ψ replaces ϕ′, while Ψ replaces Φ. For real τ, let Hτ,4 denote the functionHτ,4(x)≔eiπ|x|2τ|x|2, x∈R4\{0},[10]which is locally integrable and decays at infinity. As such, it is a tempered distribution, and its Fourier transform equalsĤτ,4(y)=1−e−iπ|y|2/τ|y|2=1|y|2−H−1/τ,4(y).[11]This is the integrated version of the Fourier transformation law for Gaussians [2] in dimension d=4. Indeed, if we differentiate with respect to τ in [11], we recover [2]. In other words, differentiation with respect to τ gives us that Ĥτ,4+H−1/τ,4 is independent of τ. By letting τ tend to 0, the identification with the Newton kernel as in [11] follows from the Riemann–Lebesgue lemma. In view of [11], the Fourier transform of the function Ψ given by [9] is in the sense of distribution theoryΨ^(y)=iπ∫RĤτ,4(y)ψ(τ)dτ=iπ|y|2∫Rψ(τ)dτ−iπ|y|2∫Re−iπ|y|2/τψ(τ)dτ, y∈R4\{0}.[12]This formula extends [1.7].

2.2. Fourier Uniqueness Meets Heisenberg Uniqueness and the Klein–Gordon Equation

In ref. 1, in the context of the Klein–Gordon equation in 1+1 dimensions, Hedenmalm and Montes-Rodríguez found discrete uniqueness sets along characteristic directions, based on ideas from dynamical systems and ergodic theory. We apply the approach in refs. 1 ⇓–3 and 9 to obtain a uniqueness result for the pair ψ,Ψ connected by [9]. Let H+1(R) denote the Hardy space of the upper half-plane. It may be defined as the subspace of functions in L1(R) with Poisson harmonic extension to H which is holomorphic.

Theorem 1. Let ψ∈L1(R) and Ψ be as above. If Ψ(x)=Ψ^(y)=0 holds for all x,y∈Z4\{0}, and if Ψ(x)=o(|x|−2) as |x|→0, then ψ∈H+1(R) and, as a consequence, Ψ(x)≡0 on R4\{0}.

Proof: In view of the assumption that Ψ(x)=o(|x|−2) as |x|→0, it follows from [9] that ψ∈L1(R) annihilates the constant function 1. Moreover, by the Lagrange four squares theorem, each positive integer may be written as |x|2 for some x∈Z4\{0}. Consequently, we see from [5] and [7] that ψ also annihilates the subspace of L∞(R) spanned by the functions eiπmτ and e−iπn/τ, where m,n∈Z+ and τ denotes the real variable. By theorem 1.8.2 in ref. 2, which relies on methods developed in ref. 3 and is motivated by ref. 1, we may conclude that ψ∈H+1(R). Finally, in view of the standard Fourier analysis characterization of H+1(R), it follows from this and [5] that Ψ=0 on R4\{0}. This finishes the proof of the theorem.

We return to the initial setup with ϕ and Φ and think of ϕ′=ψ and Φ=Ψ. In terms of notation, let C0(R) denote the space of continuous functions on R with limit value 0 at infinity. Then the condition at the origin in Theorem 1 may be replaced by ϕ∈C0(R).

Corollary 2. Let Φ be given by [5], where ϕ∈C0(R) with ϕ′∈L1(R) and d=4. If Φ(x)=Φ^(y)=0 for all x,y∈Z4\{0}, then ϕ′∈H+1(R) and, as a consequence, Φ(x)≡0 on R4\{0}.

Remark. The above theorem is a four-dimensional analogue of the uniqueness part of the Fourier interpolation formula found by Radchenko and Viazovska (4). That work is based on a method invented by Viazovska to realize the Cohn–Elkies upper bound for sphere packing (5, 6).

3. Modifications in Higher Even Dimensions

3.1. The Higher-Dimensional Kernel and Its Fourier Transform

We consider even dimensions d≥4 only and write d=2d0 with d0≥2. We are interested in the kernelHτ,d(x)≔eiπ|x|2τ|x|d−2, x∈Rd\{0}.[13]Its Fourier transform as a tempered distribution is given byĤτ,d(y)=(iτ)d0−2|y|d−2∑j=0d0−21j!(−iπ|y|2/τ)j−e−iπ|y|2/τ, y∈Rd\{0}.For fixed y, the function τ↦Ĥτ,d(y) is a bounded holomorphic function in H. Moreover, (τ,y)↦Ĥτ,d(y) solves a Schrödinger equation without potential and initial datum Ĥ0,d(y)=cd|y|−2. Here cd=πd0−2/(d0−2)! is the volume of the ball in dimension d−4.

3.2. The Fourier Uniqueness Theorem in Higher Even Dimensions

Next, for ψ∈L1(R), we letΨ(x)=iπd0−1∫RHτ,d(x)ψ(τ)dτ=iπ|x|2d0−1∫Reiπτ|x|2ψ(τ)dτ, x∈Rd\{0}.[14]If, e.g., ψ=ϕ(d0−1) for a function ϕ∈C0(R) with ϕ(j)∈C0(R) for j≤d0−2, we may make sense of [14] as arising from [3] by successive integration by parts, with Φ=Ψ. The Fourier transform of Ψ is given byΨ^(y)=iπd0−1∫RĤτ,d(x)ψ(τ)dτ, y∈Rd\{0}.We may now formulate the analogue of Theorem 1 which applies in arbitrary even dimension d≥4.

Theorem 2. Suppose the dimension d≥4 is even, and let ψ∈L1(R) and Ψ be as above. If Ψ(x)=Ψ^(y)=0 holds for all x,y∈Zd\{0}, and if Ψ(x)=o(|x|−d+2) as |x|→0, then ψ∈H+1(R), and as a consequence, Ψ(x)≡0 on Rd\{0}.

The proof of this theorem for d≥6 is based on an extension of the methods developed in refs. 2, 3 and will be presented elsewhere. Here we only mention that the method relies on a sophisticated analysis of the iterates of the weighted transfer operatorsTd0f(t)=∑k∈Z\{0}(2k−t)−d0f12k−tfor the even Gauss map x↦−1/x−2⌊(x−1)/(2x)⌋ on the punctured interval [−1,1)\{0}, which requires a new blend of ideas involving, e.g., methods of totally positive matrices. The needed results about the iterates of the weighted transfer operator cannot be obtained by application of the standard methods of ergodic theory. Here ⌊x⌋ denotes the usual integer part of x∈R. As a side remark, we mention that higher-dimensional uniqueness results for radial functions lead to Fourier uniqueness results along concentric shells for nonradial functions; see Stoller (7) for details.

Data Availability

There are no data underlying this work.

Acknowledgments

We are greatly indebted to associate editor Kenneth Ribet and the anonymous referee for insightful comments and suggestions. The research of H.H. is supported by Vetenskapsrådet (VR) grant 2016-04912. The research of A.M.-R. is supported by Knut and Alice Wallenberg grant 2015.0342 and by Plan Nacional MTM2015-70531-P as well as by Junta de Andalucía P12-FQM-633, FQM-260. The research of M.V. is supported by Swiss National Science Foundation project 184927. A.B. thanks Anatolii Romanyuk and Igor Shevchuk for inviting D.R. to Kyiv in May 2016, and the German Academic Exchange Service (DAAD) grant 57210233 as well as VR grant 2016-04912 for supporting his visit to Stockholm in November 2016.

Footnotes

↵¹A.B., H.H., A.M.-R., D.R., and M.V. contributed equally to this work.
↵²To whom correspondence may be addressed. Email: haakanh{at}kth.se.

Author contributions: A.B., H.H., A.M.-R., D.R., and M.V. performed research; and H.H. wrote the paper with contributions from all authors.
The authors declare no competing interest.
This article is a PNAS Direct Submission.

Published under the PNAS license.

References

↵
1. H. Hedenmalm,
2. A. Montes-Rodríguez
, Heisenberg uniqueness pairs and the Klein-Gordon equation. Ann. Math. 173, 1507–1527 (2011).
OpenUrl
↵
1. H. Hedenmalm,
2. A. Montes-Rodríguez
, The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps. J. Eur. Math. Soc. 22, 1703–1757 (2020).
OpenUrl
↵
1. H. Hedenmalm,
2. A. Montes-Rodríguez
, The Klein-Gordon equation, the Hilbert transform, and Gauss-type maps: H∞ approximation. J. Anal. Math., in press.
↵
1. D. Radchenko,
2. M. Viazovska
, Fourier interpolation on the real line. Publ. Math. Inst. Hautes Études Sci. 129, 51–81 (2019).
OpenUrl
↵
1. M. Viazovska
, The sphere packing problem in dimension 8. Ann. Math. 185, 991–1015 (2017).
OpenUrl
↵
1. H. Cohn,
2. A. Kumar,
3. S. D. Miller,
4. D. Radchenko,
5. M. Viazovska
, The sphere packing problem in dimension 24. Ann. Math. 185, 1017–1033 (2017).
OpenUrl
↵
1. M. Stoller
, Fourier interpolation from spheres. arXiv:2002.11627 (26 February 2020).
↵
1. H. Whitney
, Differentiable even functions. Duke Math. J. 10, 159–160 (1943).
OpenUrl
↵
1. F. Canto-Martín,
2. H. Hedenmalm,
3. A. Montes-Rodlríguez
, Perron-Frobenius operators and the Klein-Gordon equation. J. Eur. Math. Soc. 16, 31–66 (2014).
OpenUrl