Universal frame set for rational functions

2025/10/29 by Andrei V. Semenov, Semenov, Andrei V.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2510.25930

Abstract

Let g ∈ L²(ℝ) be a rational function of degree M, i.e. there exist polynomials P, Q such that g = P \over Q and deg(P) < deg(Q) ≤ M. We prove that for any ε>0 and any M ∈ ℕ there exists universal set Λ⊂ ℝ of density less than 1+ε such that the system \ e^{2πi λt} g(t-n) \colon (λ, n) ∈ Λ× ℤ \ is a frame in L²(ℝ) for any well-behaved rational function g.

Citations

• Frame redundancy and Beurling density
• Gabor frames for functions supported on a semi-axis
• Frame set for shifted sinc-function
• Gabor frames for rational functions
• Gabor frames for rational functions
• Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions
• The 𝑎𝑏𝑐-problem for Gabor systems
• Redundancy for localized frames
• Some Weyl-Heisenberg frame bound calculations

Related

• Gabor frames for rational functions
• Frames for compactly supported functions with irrational density
• Gabor frames with rational density
• Exponential frames for unbounded sets
• Gabor frames for functions supported on a semi-axis