Let s(0), s(l), … be a given sequence, and define holds for all finite sequences (ξn n ε z, then it is known that there is a positive Borel measure μ. on the circle T such that sequence (s(n) that the measure μ. may be chosen to be smooth. A measure μ is said to be smooth if it has the same spectral type as the operator id/dt acting on L2(ð) with respect to Haarmeasure dt on ð :
... [Show full abstract] Equivalently, μ is a superposition (possibly infinite) of measures of the form ω(i)dt with ωL2(ð) such that dw/dt e L2(T). The condition is stated purely in terms of the initially given sequence (s(n)): We show that a smooth representation exists if and only if, for the a priori estimate.