[Paper Review] Spectral measures generated by arbitrary and random convolutions

Dorin Ervin Dutkay, Chun‐Kit Lai|arXiv (Cornell University)|Sep 15, 2015

Mathematical Analysis and Transform Methods14 references3 citations

TL;DR

This paper generalizes Strichartz's criterion for spectral measures generated by infinite convolutions of discrete measures from Hadamard triples, proving that under certain conditions, such measures are spectral. It further shows that in one dimension or under specific higher-dimensional conditions, 'almost all' random convolutions of finite atomic measures with rescaling produce spectral measures or translational tiles.

ABSTRACT

We study spectral measures generated by infinite convolution products of discrete measures generated by Hadamard triples, and we present sufficient conditions for the measures to be spectral, generalizing a criterion by Strichartz. We then study the spectral measures generated by random convolutions of finite atomic measures and rescaling, where the digits are chosen from a finite collection of digit sets. We show that in dimension one, or in higher dimensions under certain conditions, "almost all" such measures generate spectral measures, or, in the case of complete digit sets, translational tiles. Our proofs are based on the study of self-affine spectral measures and tiles generated by Hadamard triples in quasi-product form.

Motivation & Objective

• To generalize Strichartz's criterion for spectral measures generated by infinite convolutions of discrete measures from Hadamard triples.
• To investigate the spectral properties of random convolutions formed by choosing from finitely many digit sets and rescaling.
• To establish conditions under which such random convolutions yield spectral measures or translational tiles in one or higher dimensions.
• To extend the theory of self-affine spectral measures and tiles using Hadamard triples in quasi-product form.
• To prove completeness of the spectrum via the Ruelle transfer operator and dynamical invariance of extreme cycles.

Proposed method

• Define infinite convolution measures μ as weak limits of products of scaled discrete measures derived from Hadamard triples (R_i, B_i, L_i).
• Construct an infinite mutually orthogonal set Λ via the formula Λ = L₁ + R₁ᵀL₂ + R₁ᵀR₂ᵀL₃ + ⋯, which serves as a candidate spectrum.
• Use the Ruelle transfer operator Rf(ξ) = ∑_{ℓ∈L} |m_B(τ_ℓ(ξ))|² f(τ_ℓ(ξ)) to analyze the completeness of the exponential system.
• Apply Lemma 4.3 to ensure that the maps τ_ℓ(ξ) = (Rᵀ)⁻¹(ξ + ℓ) map large balls into themselves, enabling compactness arguments.
• Prove completeness by showing Q_Λ(ξ) = ∑_λ∈Λ |μ̂(ξ+λ)|² ≥ 1 via invariance of minimal compact sets under the Ruelle operator and the dynamical simplicity of the system.
• Use the fact that extreme cycles are contained in Λ and that entire functions with discrete zero sets must be constant, to conclude Q_Λ ≡ 1.

Experimental results

Research questions

• RQ1 Under what conditions is the infinite convolution of discrete measures from Hadamard triples a spectral measure?
• RQ2 Can the spectral property be guaranteed for random convolutions of finitely many digit sets in one or higher dimensions?
• RQ3 How does the dynamical structure of extreme cycles relate to the completeness of the spectrum in self-affine measures?
• RQ4 What role does the Ruelle transfer operator play in verifying the completeness of the exponential system?
• RQ5 In what cases do random convolutions of atomic measures generate translational tiles or spectral measures?

Key findings

• The paper generalizes Strichartz's criterion for spectral measures to arbitrary sequences of Hadamard triples, providing a sufficient condition for the infinite convolution to be spectral.
• For random convolutions of finitely many digit sets in dimension one, 'almost all' such measures are spectral, under mild assumptions.
• In higher dimensions, under additional conditions (e.g., complete digit sets), 'almost all' such random convolutions generate translational tiles.
• The completeness of the spectrum is established via the Ruelle transfer operator and the invariance of minimal compact sets under the dynamics of τ_ℓ.
• The proof relies on the fact that any minimal invariant set in the zero set of Q_Λ must be discrete and hence an extreme cycle, forcing Q_Λ ≡ 1.
• The result shows that the spectral property holds generically for random convolutions, extending known results on self-affine spectral measures.