Density of exponentials and Perron-Frobenius operators

Abstract

In this article, we study the weak-star density of the linear span of the trigonometric functions{em,n(x,y)=eπi(mx+ny),em,n<β>(x,y)=eπiβ(m/x+n/y);m,n∈Z} for a positive real β. We aim to extend the results of Hedenmalm and Montes-Rodríguez (2011) [18] and Canto-Martín, Hedenmalm, and Montes-Rodríguez (2014) [8] in the plane. They have extensively studied the weak-star completeness of the hyperbolic trigonometric system in L∞(R). This is the dual formulation of the Heisenberg uniqueness pair for (hyperbola, certain lattice-cross).As in their work, β=1 turns out to be the critical value. In particular, one of our main results asserts that the space spanned by the aforesaid trigonometric functions is weak-star dense in L∞ of the set Θ1,β=(−1,1]2∪(R∖(−β,β])2 if and only if 0<β≤1, and the corresponding pre-annihilator space has finite dimension whenever β>1. However, for β>1, the pre-annihilator space can be made infinite-dimensional by allowing functions with slightly bigger support than Θ1,β. To be precise, let Θβ″⊆R2∖Θ1,β be such that (−β,β]2∩Θβ″ has positive Lebesgue measure. We prove that the weak-star closure of the linear span of em,n and em,n<β> as m,n varies over Z, has infinite codimension in L∞(Θ1,β∪Θβ″) whenever β>1. Our proof goes via the analysis of a two-dimensional Gauss-type map and its corresponding Perron-Frobenius operator.