The basis functions of Fourier interpolation

TL;DR

This work analyzes the basis functions a_n arising from Radchenko–Viazovska Fourier interpolation, recasting them through weakly holomorphic modular forms on the Hecke theta group. It delivers precise size estimates for the associated f_n(x), a detailed description of their zero distribution (including numerous extraneous zeros on the real axis), and the implications for Fourier nonuniqueness and non-Riesz-basis behavior in the relevant Hilbert space. Central to the results are the Φ(z) function and its functional equation, Rademacher-type formulas for modular coefficients via Kloosterman sums, and a Carleson-measure framework that yields sharp average bounds for sums of |f_n|^2. Numerics are used to illustrate the sharpness of the asymptotics and to pose open questions about finer-scale properties and potential Ramanujan–Petersson-type phenomena. Overall, the paper reveals fundamental limits on stability and uniqueness in Fourier interpolation when using these modular-form-based basis functions, with implications for time–frequency analysis and sampling theory.

Abstract

The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency properties. We give precise size estimates and study the distribution of zeros of these functions. We give in particular asymptotic estimates for the location and the number of extraneous zeros on or close to the real line. This result reveals the surprising existence of Fourier nonuniqueness pairs whose apparent ``excess'' compared to the Fourier uniqueness pair of Radchenko and Viazovska may be made arbitrarily large. Our estimates also show that the basis functions fail to yield a Riesz basis in the Hilbert space used by Kulikov, Nazarov, and Sodin in their recent study of Fourier uniqueness pairs. Some numerical data are presented, suggesting additional fine scale properties.

Figures (6)

• Figure 1: Plots of $f_{500}(x)$ (blue) and $\frac{\Phi(\sqrt{x}-\sqrt{500})}{\sqrt{500}}$ (red).
• Figure 2: Fundamental domain $\mathcal{F}$, $J(z)$ as a conformal map, and a deformed integration path $\ell$.
• Figure 3: The domain $\mathcal{D}$.
• Figure 4: Plots of $f_{n}$ for $100\le n \le 150$.
• Figure 5: Histogram of $n^{1/4}f_{n}(0.63)$, $0\le n \le 50000$.

Theorems & Definitions (44)

• Theorem 1.1: Asymptotics for small real values
• Theorem 1.2: Asymptotics for large real values
• Theorem 1.3: Average of $L^2$ integrals with respect to $n$
• Theorem 1.4: Asymptotics in $\mathbb C$ and on the negative real line
• Theorem 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 4.1
• Lemma 4.2