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Exponential approximation and meromorphic interpolation

Yurii Belov, Alexander Borichev, Alexander Kuznetsov

TL;DR

This work investigates how to represent $f\in L^2[-\pi,\pi]$ by exponentials whose frequencies lie in a sparse set controlled by Beurling–Malliavin density. By linking non-linear meromorphic interpolation at $\mathbb Z$ with Paley–Wiener reproducing-kernel approximations, the authors develop a probabilistic model for random Fourier coefficients and prove that typical $f$ admit a representation with $D_{BM}(\Lambda)<1-\varepsilon$, where the exponential system forms a Riesz basis in its span. A key technical advance is a deterministic interpolation result showing that a Gaussian sequence can be interpolated by a meromorphic function with poles on a real set of density $<1$; this underpins the probabilistic conclusions. The main theorem extends to complex Gaussians and yields a constructive framework using Cartwright functions: almost surely $f$ can be written as a convergent series of exponentials at the zeros of a Cartwright function $V$ with $D_{BM}(\mathcal Z(V))=D(\mathcal Z(V))<1-\varepsilon$, while sharp density bounds and limitations are discussed. The results illuminate how sparse spectral representations in $L^2$ can be achieved and inform interpolation theory, sampling, and spectral approximation through a probabilistic–analytic synthesis.

Abstract

We establish a relation between the approximation in $L^2[-π,π]$ by exponentials with the set of frequencies of Beurling--Malliavin density less than $1$ and the meromorphic interpolation at $\mathbb Z$. Furthermore, we show that typical $L^2[-π,π]$ functions admit such an approximation.

Exponential approximation and meromorphic interpolation

TL;DR

This work investigates how to represent by exponentials whose frequencies lie in a sparse set controlled by Beurling–Malliavin density. By linking non-linear meromorphic interpolation at with Paley–Wiener reproducing-kernel approximations, the authors develop a probabilistic model for random Fourier coefficients and prove that typical admit a representation with , where the exponential system forms a Riesz basis in its span. A key technical advance is a deterministic interpolation result showing that a Gaussian sequence can be interpolated by a meromorphic function with poles on a real set of density ; this underpins the probabilistic conclusions. The main theorem extends to complex Gaussians and yields a constructive framework using Cartwright functions: almost surely can be written as a convergent series of exponentials at the zeros of a Cartwright function with , while sharp density bounds and limitations are discussed. The results illuminate how sparse spectral representations in can be achieved and inform interpolation theory, sampling, and spectral approximation through a probabilistic–analytic synthesis.

Abstract

We establish a relation between the approximation in by exponentials with the set of frequencies of Beurling--Malliavin density less than and the meromorphic interpolation at . Furthermore, we show that typical functions admit such an approximation.
Paper Structure (3 sections, 8 theorems, 113 equations)

This paper contains 3 sections, 8 theorems, 113 equations.

Table of Contents

  1. Introduction and main results
  2. An interpolation result
  3. Approximation results

Key Result

Theorem 1

There exist $\varepsilon>0$ such that for almost all random $f\in L^2[-\pi,\pi]$, we can find $\Lambda=\Lambda(f)\subset\mathbb R$ such that $D_{BM}(\Lambda)<1-\varepsilon$ and Moreover, the system $\{e^{i\lambda t}: \lambda\in\Lambda\}$ is a Riesz basis in its closed linear span in $L^2[-\pi,\pi]$, and with convergence in $L^2[-\pi,\pi]$, for some coefficients $(a_\lambda)_{\lambda\in\Lambda}\i

Theorems & Definitions (20)

  • Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof : Proof of Lemma
  • Proposition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 10 more