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Exponential Riesz bases in $L^2$ on two interval

Yurii Belov, Mikhail Mironov

TL;DR

This work develops a framework for exponential bases on the union of two intervals by linking Riesz bases in $L^2(E)$ to complete interpolating sequences for the Paley-Wiener space $PW_E$. It introduces a generating function $F$ and an auxiliary zero set $ ext{T}$ to formulate sufficient conditions (Theorem suff) that ensure $ ext{Λ}$ is complete interpolating for $PW_E$, with a crucial positivity condition (iii) and a near-necessity perspective. The authors prove several interlinked results about minimality and completeness of conjugate and mixed systems, derive necessary conditions for the zeros $ ext{T}$, and establish an \'extra point\' phenomenon when gluing intervals, including concrete examples. The paper also provides constructive methods and explicit examples (including two equal-length intervals) that generate wide classes of complete interpolating sequences for $PW_E$ and demonstrates how one-point adjustments can affect completeness for the glued set $PW_{E^g}$.

Abstract

We give sufficient conditions for the exponential system to be a Riesz basis in $L^2(E)$, where $E$ is a union of two intervals. We show that these conditions are close to be necessary. In addition, we demonstrate ``extra point effect'' for such systems, i.e. it may happen that the Riesz basis in $L^2(E)$ differs by one point from the Riesz basis on an interval.

Exponential Riesz bases in $L^2$ on two interval

TL;DR

This work develops a framework for exponential bases on the union of two intervals by linking Riesz bases in to complete interpolating sequences for the Paley-Wiener space . It introduces a generating function and an auxiliary zero set to formulate sufficient conditions (Theorem suff) that ensure is complete interpolating for , with a crucial positivity condition (iii) and a near-necessity perspective. The authors prove several interlinked results about minimality and completeness of conjugate and mixed systems, derive necessary conditions for the zeros , and establish an \'extra point\' phenomenon when gluing intervals, including concrete examples. The paper also provides constructive methods and explicit examples (including two equal-length intervals) that generate wide classes of complete interpolating sequences for and demonstrates how one-point adjustments can affect completeness for the glued set .

Abstract

We give sufficient conditions for the exponential system to be a Riesz basis in , where is a union of two intervals. We show that these conditions are close to be necessary. In addition, we demonstrate ``extra point effect'' for such systems, i.e. it may happen that the Riesz basis in differs by one point from the Riesz basis on an interval.
Paper Structure (16 sections, 16 theorems, 130 equations)

This paper contains 16 sections, 16 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Exponential systems
  3. Paley-Wiener spaces
  4. Main results
  5. Preliminary results
  6. Complete interpolating sequences on two intervals
  7. Basic properties
  8. Proof of Theorem \ref{['thrm: ness T']}
  9. Proof of Theorem \ref{['thrm: ness L T']}
  10. Proof of Theorem \ref{['thrm: ness F+']}
  11. Proof of Theorem \ref{['thrm: ness +-1']}
  12. Sufficient conditions for completeness and interpolation on two intervals
  13. Proof of Theorem \ref{['thrm: suff']}
  14. Examples
  15. Two intervals of equal length.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $F$ be a generating function for $\Lambda$ in $PW_E$, and $\mathrm{T}$ be the other zeros of $F$. Suppose that: Then $\Lambda$ is a complete interpolating sequence for $PW_E$.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Remark 1
  • Example 1
  • Theorem 2
  • Theorem 3
  • ...and 25 more