Fourier interpolation in dimensions 3 and 4 and real-variable Kloosterman sums

TL;DR

The paper constructs radial Fourier interpolation formulas in dimensions $d=3$ and $d=4$ using Maass--Poincaré type series, expressing the basis functions via real-variable Kloosterman sums and Hurwitz class numbers. It establishes a precise generating-function framework, showing that two families of $2$-periodic analytic functions satisfy functional equations that encode the interpolation identities. By developing both weight-2 and weight-3/2 analyses, it derives explicit infinite-sum representations for the basis functions $a_{d,n}$ and $ ilde{a}_{d,n}$ with sharp growth bounds, improving upon prior Hecke-type estimates. The results connect Fourier interpolation to spectral theory and modular objects, enabling new bounds and potential generalizations to lower dimensions, with implications for sphere packing, energy minimization, and harmonic analysis on $ ext{SL}_2$-related spaces.

Abstract

We give a construction of radial Fourier interpolation formulas in dimensions 3 and 4 using Maass--Poincaré type series. As a corollary we obtain explicit formulas for the basis functions of these interpolation formulas in terms of what we call real-variable Kloosterman sums, which were previously introduced by Stoller. We also improve the bounds on the corresponding basis functions $a_{n,d}(x)$, $d=3,4$, for fixed $x$, in terms of the index $n$.

Theorems & Definitions (81)

• Theorem 1.1: Corollary of BondarenkoRadchenkoSeip2022
• Theorem 1.2
• Remark
• Theorem 1.3
• Remark
• Corollary 1.4
• Theorem 1.5: Corollary of RadchenkoViazovska2019
• Remark
• Corollary 1.6
• Remark