Stieltjes Moment Sequences and Positive Definite Matrix Sequences

For a certain constant δ > 0 (a little less than 1/4), every function f : N0 → ]0,∞[ satisfying f(n)2 ≤ δf(n − 1)f(n + 1), n ∈ N, is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence f : N0 → R there is a positive definite matrix sequence (an) which is not of positive type and which satisfies tr(an+2) = f(n), n ∈ N0. For a certain constant ε > 0 (a little greater than 1/6), for every function φ : N0 → ]0,∞[ satisfying φ(n)2 ≤ εφ(n − 1)φ(n + 1), n ∈ N, there is a convolution semigroup (μt)t≥0 of measures on R+, with moments of all orders, such that φ(n) = ∫ xn dμ1(x), n ∈ N0, and for every such convolution semigroup (μt) the measure μt is Stieltjes indeterminate for all t > 0.