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• W1665563195 abstract "The spectral envelope S(F) of a subset F of integers is the set of probability measures on the circle groupT=R/Z that are weak ∗ limits of squared moduli of norm one trigonometric polynomials having frequencies in F. Clearly, if µ ∈ S(F) then the support of its Fourier transform b is contained in the difference set F − F. However, the converse generally fails and an explicit characterization of the ele ments in S(F) is extremely elusive. We associate to F a symbolic dynamical system (Ω(F), σ), where σ is the shift homeo- morphism on the product space{0, 1} Z and Ω(F) is the closure of the orbit under σ of the characteristic function χF of F. Analytic properties of S(F) are related to dynamical prop- erties of the binary sequence χF. The Riemann-Lebesque lemma implies that if χF is recur- rent, or more specifically minimal, then S(F) is convex and hence, by the Krein-Milman theorem, S(F) equals the closure of the convex hull of its set of extreme points Se(F). In this paper we (i) review the relationship between these concepts and the special case of the still open 1959 Kadison-Singer problem called Feichtinger's conjecture for exponen- tial functions, (ii) derive partial characterizations of e lements in Se(F), for minimal χF, in terms of ergodic properties of (Ω(F), λ, σ), where λ is a σ− invariant probability measure whose existence is ensured by the 1937 Krylov-Bogolyubov theorem, (iii) refine previous numerical studies of the Morse-Thue minimal binary sequence by exploiting a new MAT- LAB algorithm for computing smallest eigenvalues of 4, 000, 000× 4, 000, 000 matrices, (iv) describe recent results characterizing S(F) for certain Bohr sets F related to quasicrys- tals, (v) extend these concepts to general discrete groups i ncluding those with Kazhdan's T-property, such as SL(n, Z), n≥ 3, which can be characterized by several equivalent prop- erties such as: any sequence of positive definite functions c onverging to 1 uniformly on compact subsets converges uniformly. This exotic property may be useful to construct a counterexample to the generalization of Feichtinger's con jecture and hence provide a no answer to the question of Kadison and Singer which they themselves tended to suspect." @default.
• W1665563195 created "2016-06-24" @default.
• W1665563195 creator A5063738402 @default.
• W1665563195 date "2012-04-22" @default.
• W1665563195 modified "2023-09-27" @default.
• W1665563195 title "Spectral Envelopes - A Preliminary Report" @default.
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