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On the regularity of Fourier interpolation formulas

Gabriele Cassese, João P. G. Ramos

TL;DR

This work investigates the regularity of Fourier interpolation formulas through the lens of scale calculus and Schauder-frame theory. By combining sc-Fredholm operators with Schauder frames, it establishes Schwartz-space regularity for perturbed interpolation bases in subcritical regimes and Schwartz-convergence for a broad supercritical class, while showing that small exponential perturbations preserve analyticity of the basis functions. The results connect modular-form tools with interpolation theory to yield order-2 entire analyticity of the basis and robust global analyticity for perturbed functions under suitable decay. The findings substantially improve prior RS2/KNS results by achieving Schwartz convergence and by clarifying when analyticity is preserved, thereby enhancing the applicability of Fourier interpolation formulas to time-frequency reconstruction problems.

Abstract

By applying new functional analysis tools in the framework of Fourier interpolation formulas, such as sc-Fredholm operators and Schauder frames, we are able to improve and refine several properties of these aforementioned formulas on the real line. As two examples of our main contributions, we highlight: (i) that we may upgrade perturbed interpolation bases all the way to the Schwartz space, which shows that even the perturbed interpolation formulas are as regular as the Radchenko-Viazovska case; (ii) that a certain subset of the interpolation formulae considered by Kulikov-Nazarov-Sodin may actually be upgraded to be convergent in the Schwartz class, giving a first partial answer to a question posed by those authors. As a final contribution of this work, we also show that, if the perturbations are sufficiently small, then even analyticity properties of the basis functions are preserved. This shows, in particular, that any function that vanishes on all but finitely many of the (perturbed) nodes is automatically analytic, a feature previously only known to hold in supercritical contexts besides the Radchenko-Viazovska case.

On the regularity of Fourier interpolation formulas

TL;DR

This work investigates the regularity of Fourier interpolation formulas through the lens of scale calculus and Schauder-frame theory. By combining sc-Fredholm operators with Schauder frames, it establishes Schwartz-space regularity for perturbed interpolation bases in subcritical regimes and Schwartz-convergence for a broad supercritical class, while showing that small exponential perturbations preserve analyticity of the basis functions. The results connect modular-form tools with interpolation theory to yield order-2 entire analyticity of the basis and robust global analyticity for perturbed functions under suitable decay. The findings substantially improve prior RS2/KNS results by achieving Schwartz convergence and by clarifying when analyticity is preserved, thereby enhancing the applicability of Fourier interpolation formulas to time-frequency reconstruction problems.

Abstract

By applying new functional analysis tools in the framework of Fourier interpolation formulas, such as sc-Fredholm operators and Schauder frames, we are able to improve and refine several properties of these aforementioned formulas on the real line. As two examples of our main contributions, we highlight: (i) that we may upgrade perturbed interpolation bases all the way to the Schwartz space, which shows that even the perturbed interpolation formulas are as regular as the Radchenko-Viazovska case; (ii) that a certain subset of the interpolation formulae considered by Kulikov-Nazarov-Sodin may actually be upgraded to be convergent in the Schwartz class, giving a first partial answer to a question posed by those authors. As a final contribution of this work, we also show that, if the perturbations are sufficiently small, then even analyticity properties of the basis functions are preserved. This shows, in particular, that any function that vanishes on all but finitely many of the (perturbed) nodes is automatically analytic, a feature previously only known to hold in supercritical contexts besides the Radchenko-Viazovska case.

Paper Structure

This paper contains 20 sections, 31 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context and background
  3. Main results
  4. Preliminaries
  5. Schauder frames and bases
  6. Fredholm Operators and sc-Fredholm Operators
  7. Neumann series and perturbation of frames
  8. Modular forms
  9. Bounds on Fourier Interpolation Bases
  10. Notation and conventions
  11. Schwartz regularity of perturbed basis functions
  12. Proof of Theorem \ref{['thm : pertSch']}
  13. Remarks
  14. Frames and Schwartz convergence of supercritical interpolation formulas
  15. Analyticity of Perturbed Basis Functions
  16. ...and 5 more sections

Key Result

Theorem A

Let $f\in L^2$ be such that $\widehat{f}(x)=0$ for a.e. $x\not \in [-1/2,1/2]$. Then $f$ is an entire function; moreover, we have

Figures (1)

  • Figure :

Theorems & Definitions (71)

  • Definition 1.1
  • Theorem A: Higgins
  • Theorem B: RS2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Bases
  • Definition 2.2: Schauder frame
  • Definition 2.3: $\Lambda$ frame
  • Proposition 2.4
  • ...and 61 more