Trigonometric Moment Problems for Arbitrary Finite Subsets of Z

We consider finite subsets Λ ⊂ Zn satisfying the extension property, i.e. the property that every collection {ck}k∈Λ−Λ of complex numbers which is positive-definite on Λ is the restriction to Λ − Λ of the Fourier coefficients of some positive measure on Tn. A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to Λ is shown to imply the failure of the extension property for Λ. This condition is used to characterize the one-dimensional sets satisfying the extension property and to provide many examples of sets failing to satisfy it in higher dimensions. Another condition, in terms of unitary matrices, is investigated and is shown to be equivalent to the extension property. New two-dimensional examples of sets satisfying the extension property are given as well as explicit examples of collections for which the extension property fails. 1. Notation If Λ is a finite set contained in Z, we will denote by |Λ| its cardinality and by Λ− Λ the difference set defined by Λ−Λ = {λ−λ , λ, λ ∈ Λ}. T will be identified with the set of n-tuples of complex numbers of modulus 1, i.e. T = {(z1, . . . , zn), |zi| = 1, i = 1, . . . , n}. If z = (z1, . . . , zn) ∈ T and k = (k1, . . . , kn) ∈ Z, we will use the notation z for the complex number z1 1 . . . z kn n . For a fixed finite set Λ ⊂ Z, we will denote by ΠΛ the set of all trigonometric polynomials with non-zero Fourier coefficients restricted to Λ, i.e. ΠΛ = {P (z) = ∑ k∈Λ ak z , ak ∈ C,k ∈ Λ}, and we will always consider the elements of ΠΛ as functions defined on T. M(T) will be the set of Borel measures on T. Finally, Mk denotes the set of k × k matrices with complex entries.