Joint BCAM-UPV/EHU Analysis and PDE seminar: Gabor orthonormal bases, tiling and periodicity

Given a Gabor orthonormal basis of L2(ℝ)​​

𝒢(g,T,S):={ g(x-t) e2π is x: g∈ L2(ℝ), t∈ T, s∈ S},​​

we study periodicity properties of the translation and modulation sets T​​ and S​​. In particular, we show that if the window function g​​ is compactly supported, then T​​ and S​​ must be periodic sets, i.e., of the form

T = aℤ+ {t1,…,tn},     S = bℤ +  {s1,…,sm}.​​

To achieve this, we first obtain a result of independent interest: if the system 𝒢(g,T,S)​​ is an orthonormal basis of L2(ℝ)​​, then both |g|2​​ and |ĝ|2​​ tile ℝ​​ by translations (when translated along the sets T​​ and S​​, respectively), and moreover,

∑t∈ T |g(x-t)|2=D(T),     ∑s∈ S |ĝ(x-s)|2=D(S),     a.e. x∈ ℝ,​​

where D(Λ)​​ denotes the uniform density of a set Λ⊂ ℝ​​.

Partial results towards the Liu-Wang conjecture are also obtained.