Fourier Interpolation From Spheres

Martin Peter Stoller
AMER MATHEMATICAL SOC, 2021
Article

In every dimension d >= 2, we give an explicit formula that expresses the values of any Schwartz function on R-d only in terms of its restrictions, and the restrictions of its Fourier transform, to all origin-centered spheres whose radius is the square root of an integer. We thus generalize an interpolation theorem by Radchenko and Viazovska [Publ. Math. Inst. Hautes Etudes Sci. 129 (2019), pp. 51-81] to higher dimensions. We develop a general tool to translate Fourier uniqueness and interpolation results for radial functions in higher dimensions, to corresponding results for non-radial functions in a fixed dimension. In dimensions greater or equal to 5, we solve the radial problem using a construction closely related to classical Poincare series. In the remaining small dimensions, we combine this technique with a direct generalization of the Radchenko-Viazovska formula to higher-dimensional radial functions, which we deduce from general results by Bondarenko, Radchenko and Seip [Fourier interpolation with zeros of zeta and L-functions, arXiv:2005.02996, 2020]

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Martin Peter Stoller
AMER MATHEMATICAL SOC, 2021
Article

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https://infoscience.epfl.ch/record/290080?ln=fr

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Perturbed Fourier uniqueness and interpolation results in higher dimensions

Martin Peter Stoller

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radii are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an applica-tion, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Via-zovska [1].(c) 2022 Published by Elsevier Inc.

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

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We prove that every Schwartz function in Euclidean space can be completely recovered given only its restrictions and the restrictions of its Fourier transform to all origin-centered spheres whose radii are square roots of integers. In particular, the only Schwartz function which, together with its Fourier transform, vanishes on these spheres, is the zero function. We show that this remains true if we replace the spheres by surfaces or discrete sets of points which are sufficiently small perturbations of these spheres. In a complementary, opposite direction, we construct infinite dimensional spaces of Fourier eigenfunctions vanishing on on certain discrete subsets of those spheres. The proofs combine harmonic analysis, the theory of modular forms and algebraic number theory.

EPFL2022

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Fourier non-uniqueness sets from totally real number fields

Martin Peter Stoller

Let K be a totally real number field of degree n >= 2. The inverse different of K gives rise to a lattice in Rn. We prove that the space of Schwartz Fourier eigenfunctions on R-n which vanish on the "component-wise square root" of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres root mS(n-1) for integers m >= 0 and, as m -> infinity, there are similar to c(K)m(n-1) many points on the m-th sphere for some explicit constant c(K), proportional to the square root of the discriminant of K. This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of K, we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes "root Lambda" for general lattices Lambda subset of R-n. Using results about lattices in Lie groups of higher rank we prove that if n >= 2 and a certain group Gamma(Lambda) >= PSL2.(R)(n) is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n >= 5 and all real lambda >= 2, Fourier interpolation results for sequences of spheres root 2m/lambda Sn-1, where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincare type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).

EUROPEAN MATHEMATICAL SOC-EMS2022

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