[Paper Review] Combining Riesz bases

Gady Kozma, Shahaf Nitzan|arXiv (Cornell University)|Oct 23, 2012

Mathematical Analysis and Transform Methods17 references4 citations

TL;DR

This paper proves that any finite union of intervals in ℝ admits a Riesz basis of exponentials, resolving a long-standing open problem in harmonic analysis. Using a novel reduction to a single interval and leveraging the Paley-Wiener stability theorem, the authors show that exponentials indexed by a discrete set Λ ⊂ ℝ form a Riesz basis for L²(S), with Λ ⊆ ℤ when S ⊆ [0, 2π].

ABSTRACT

We show that any finite union of intervals supports a Riesz basis of exponentials

Motivation & Objective

• To establish the existence of a Riesz basis of exponentials for any finite union of intervals in ℝ, a problem with long-standing open questions in harmonic analysis.
• To resolve the gap in understanding whether such sets admit Riesz bases, especially when orthonormal bases are impossible.
• To provide a constructive method based on stability and separation properties of exponential systems.
• To extend prior partial results (e.g., for two intervals, equal-length intervals, or integer endpoints) to the general case of arbitrary finite unions of intervals.
• To demonstrate that the union of two Riesz bases for separate intervals can form a Riesz basis for their union under suitable perturbations, despite known counterexamples to naive union constructions.

Proposed method

• Reduces the problem of constructing a Riesz basis for a finite union of intervals to constructing one for a single interval via a duality argument.
• Applies the Paley-Wiener stability theorem to perturb existing Riesz sequences into separated systems that remain Riesz bases.
• Uses a contradiction argument based on the asymptotic growth of the sum $ S_K = \sum_{N=1}^K |\sum e(Na_j) - \sum e(Nb_j)|^2 $, showing it grows linearly with K.
• Analyzes interlacing behavior of fractional parts $\{Na_j\}$ and $\{Nb_j\}$, proving that interlacing implies small $s_N = |\sum e(Na_j) - \sum e(Nb_j)|^2$.
• Establishes that interlacing cannot occur for almost all N by showing $S_K$ grows as $2LK + O(1)$, contradicting the bound $s_N \leq 4$ under interlacing.
• Handles the critical case $L=2$ (two intervals) by using Dirichlet approximation to find $n \approx N$ with $\|na_j\|$ and $\|nb_j\|$ small, leading to $s_n \leq C\epsilon$, which is $<1$ for small $\epsilon$, thus strengthening the bound.

Experimental results

Research questions

• RQ1 Does every finite union of intervals in ℝ admit a Riesz basis of exponentials?
• RQ2 Can the union of two Riesz bases for disjoint intervals form a Riesz basis for their union, even when the individual sets are not separated?
• RQ3 Is it possible for two sets of fractional parts $\{Na_j\}$ and $\{Nb_j\}$ to interlace for almost all N, given irrational ratios of lengths?
• RQ4 Can the Paley-Wiener stability theorem be used effectively to construct Riesz bases for complex sets by perturbing simpler systems?
• RQ5 What conditions ensure that a union of exponential systems remains a Riesz basis, and how can such systems be stabilized against perturbations?

Key findings

• For any finite union of intervals $S \subset \mathbb{R}$, there exists a set $\Lambda \subset \mathbb{R}$ such that $\{e^{i\lambda t}\}_{\lambda \in \Lambda}$ forms a Riesz basis in $L^2(S)$.
• When $S \subseteq [0, 2\pi]$, the Riesz basis can be chosen with $\Lambda \subseteq \mathbb{Z}$, providing a discrete, integer-indexed basis.
• The proof shows that interlacing of fractional parts $\{Na_j\}$ and $\{Nb_j\}$ cannot occur for almost all $N$, which is key to avoiding degeneracy in the exponential system.
• The sum $S_K = \sum_{N=1}^K |\sum e(Na_j) - \sum e(Nb_j)|^2$ grows linearly as $2LK + O(1)$, contradicting the bound $s_N \leq 4$ under interlacing, thus proving non-interlacing for a positive density of $N$.
• For $L=2$, the use of Dirichlet approximation allows the construction of $n \approx N$ such that $s_n \leq C\epsilon < 1$, strengthening the contradiction argument and closing the gap in the general case.
• The result establishes that Riesz bases of exponentials exist for all finite unions of intervals, even when orthonormal bases do not, and fills a major gap in the theory of exponential bases.