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Fourier interpolation on the real line

Danylo Radchenko, Maryna Viazovska

Abstract

We use weakly holomorphic modular forms for the Hecke theta group to construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set $\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}$.

Fourier interpolation on the real line

Abstract

We use weakly holomorphic modular forms for the Hecke theta group to construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set .

Paper Structure

This paper contains 11 sections, 16 theorems, 157 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The theta group
  3. Upper half-plane and the action of $\operatorname{SL}_{2}({\mathbb{R}})$
  4. Modular forms for the group $\Gamma_{\theta}$
  5. Asymptotic notation.
  6. Weakly holomorphic modular forms on $\Gamma_{\theta}$ of weight $3/2$
  7. Interpolation basis for even functions
  8. Growth estimate
  9. Proof of the main results
  10. Interpolation basis for odd functions
  11. Open questions and concluding remarks

Key Result

Theorem 1

There exists a collection of even Schwartz functions $a_n\colon{\mathbb{R}}\to{\mathbb{R}}$ with the property that for any even Schwartz function $f\colon{\mathbb{R}}\to{\mathbb{R}}$ and any $x \in {\mathbb{R}}$ we have where the right-hand side converges absolutely.

Figures (4)

  • Figure 1: Plots of $a_n$ and $\widehat{a_n}$ for $n=0,1,2$.
  • Figure 2: Fundamental domain for $\Gamma_{\theta}$.
  • Figure 3: Fundamental domain for $\Gamma_{\theta}$ and the contour of integration.
  • Figure 4: Deforming the contour of integration.

Theorems & Definitions (26)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 4
  • ...and 16 more