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Cohn–Kumar–Miller–Radchenko–Viazovska conjecture on hexagonal-lattice interpolation

Let r1,r2,…r_1,r_2,\ldots be the positive real numbers of the form

(4/3)1/4j2+jk+k2,(4/3)^{1/4}\sqrt{j^2+jk+k^2},

where j,kj,k are integers. A radial Schwartz function is a function f:R2→Rf:\mathbf{R}^2\to\mathbf{R} invariant under rotations. Cohn–Kumar–Miller–Radchenko–Viazovska conjecture. Radial Schwartz functions f:R2→Rf:\mathbf{R}^2\to\mathbf{R} are not uniquely determined by the values of f(rn)f(r_n), F(f)(rn)\mathcal{F}(f)(r_n), dfdu(rn)\frac{d f}{du}(r_n), and dF(f)du(rn)\frac{d\mathcal{F}(f)}{du}(r_n) for n≥1n\geq1, where u=∣x∣2u=|x|^2. This conjecture concerns the failure of uniqueness for higher Fourier interpolation on the discrete set associated with the hexagonal lattice; it was motivated by numerical experiments and is presented as an open problem.

Sources & referencesView supporting material

Primary source

Naser Talebizadeh Sardari, “Higher Fourier interpolation on the plane”, arXiv:2102.08753 (2021).

fourier-analysisfourier-interpolationhexagonal-lattice

Progress summary

Refreshed
Solved

A 2021 paper proved that the listed measurements do not uniquely determine the function, so the conjecture is resolved.

The conjecture, posed by Cohn, Kumar, Miller, Radchenko, and Viazovska, asserts nonuniqueness for radial Schwartz functions in the two-dimensional hexagonal-lattice interpolation problem. It concerns simultaneous data for a function, its Fourier transform, and their first derivatives at the lattice radii.

2021 proof

Naser Sardari’s paper Higher Fourier interpolation on the plane explicitly proves the conjecture and constructs infinitely many linearly independent radial Schwartz functions whose function and Fourier-transform data vanish to order 22 on the relevant radii. A later exposition independently records that the conjectured interpolation formula does not exist.

Current status (as of August 2026): The conjecture is resolved by Sardari’s 2021 paper, which proves nonuniqueness and constructs infinitely many independent examples.

Sources

Solutions 0

No solutions have been posted yet.