From the Abstract: “We employ functional analysis techniques in order to deduce some versions of classical and recent interpolation results in Fourier analysis with perturbed nodes”.
For the ‘classical’ they refer to Kadec’s \(\frac{1}{4}\)-theorem for interpolation formulae in the Paley-Wiener space both in the real and complex case ([M. I. Kadets, Sov. Math., Dokl. 5, 559–561 (1964; Zbl 0196.42602); translation from Dokl. Akad. Nauk SSSR 155, 1253–1254 (1964)]) and for the ‘recent’ results to the papers [D. Radchenko and M. Viazovska, Publ. Math., Inst. Hautes Étud. Sci. 129, 51–81 (2019; Zbl 1455.11075)] and [H. Cohn et al., Ann. Math. (2) 185, No. 3, 1017–1033 (2017; Zbl 1370.52037)].
The layout of the paper is as follows:

§1. Introduction

(\(9\frac{1}{2}\) pages)
Contains the results in the form of 6 Theorems.

§2. Preliminaries

(\(3\frac{1}{2}\) pages)
Subsections: Band-limited functions, Modular forms and Functional analysis.

§3. Perturbed interpolation formulae for band-limited functions

(\(9\frac{1}{2}\) pages)
Subsections: Perturbed interpolation formulae for band-limited functions, From Shannon to Vaaler (the proof of Theorem 1.2), Perturbed interplation formulae with derivatives

§4. Perturbed Fourier interpolation on the real line

(\(14\) pages)
Contains the proofs of results (specifically Theorem 1.6).

§5. Applications of the main results and techniques

(\(15\) pages)
This long section contains the subsections Interpolation formulae perturbing the origin (with the proof of theorem 5.3), Uniqueness for small powers of integers, Annihilating pairs (proof of Theorem1.7), The Cohn-Kummer- Miller- Radchenko-Viazovska result and perturbed interpolation formulae with derivatives (proof of Theorem 5.11), Perturbed interpolation for odd functions.

§6. Comments and remarks

(\(6\) pages)
Addresses: Asymmetric perturbations, Maximal perturbed interpolation formulae for band-limited functions, Theorem 1.6 and optimal decay rates for interpolating functions and maximal perturbations.

References

(\(41\) items)

This paper brings the reader to the edge of research on Fourier analysis with perturbed nodes.