Fourier interpolation on the real line

Radchenko, Danylo; Viazovska, Maryna

doi:10.1007/s10240-018-0101-z

Cited by 36 publications

(71 citation statements)

References 23 publications

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“…Finally let us remark that our result admits a natural generalization to the space of even/odd functions with 4W T replaced by 2W T . This result also perfectly matches interpolation formulas from [2] and [7].…”

Section: Introductionsupporting

confidence: 87%

“…In the recent breakthrough paper [7] Radchenko and Viazovska showed that any Schwartz function can be effectively reconstructed from the values of it and its Fourier transform at the points ± √ n, n ∈ Z ≥0 and two more values f (0),f (0). If we consider the counting function n (R) = | ∩ [−R, R]|, which in the case = {± √ n} takes the form n (R) = 1 + 2[R 2 ], we see that it satisfies the inequality n (W ) + n (T ) ≥ 4W T − O(1) for all W , T .…”

Section: Introductionmentioning

confidence: 99%

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Fourier Interpolation and Time-Frequency Localization

Kulikov

2021

J Fourier Anal Appl

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We prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ∑ λ ∈ Λ f ( λ ) a λ ( x ) + ∑ μ ∈ M f ^ ( μ ) b μ ( x ) we have a lower bound for the counting functions $$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n Λ ( R 1 ) + n M ( R 2 ) ≥ 4 R 1 R 2 - C log 2 ( 4 R 1 R 2 ) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.

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Section: Introductionsupporting

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Fourier Interpolation and Time-Frequency Localization

Kulikov

2021

J Fourier Anal Appl

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“…For comparison, the equality conditions for h in Theorem 2.1 are identical, except that the conditions f (r 1 ) = 0 and f (0) = 0 are replaced with h(r 1 ) = 0. These conditions arise naturally in interpolation problems [44,82]. Specifically, Open Problem 7.3 from [44] raised the question of whether radial Schwartz functions g : R d → R are uniquely determined by the values and radial derivatives of g and g at the radii r n for n ≥ 1.…”

Section: Bounds For the Average Kissing Numbermentioning

confidence: 99%

High-dimensional sphere packing and the modular bootstrap

Afkhami-Jeddi

Cohn

Hartman

et al. 2020

J. High Energ. Phys.

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We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)c× U(1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. Furthermore, we study when these bounds can be tight. Besides the known cases c = 1/2, 4, and 12 and the conjectured case c = 1, our calculations numerically rule out sharp bounds for all other c < 90, by combining the modular bootstrap with linear programming bounds for spherical codes.

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“…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 that…”

mentioning

confidence: 99%

Fourier uniqueness in even dimensions

Bakan¹,

Hedenmalm²,

Montes-Rodrı́guez³

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

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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.

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Fourier interpolation on the real line

Cited by 36 publications

References 23 publications

Fourier Interpolation and Time-Frequency Localization

Fourier Interpolation and Time-Frequency Localization

High-dimensional sphere packing and the modular bootstrap

Fourier uniqueness in even dimensions

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