## Extreme positive definite double sequences which are not moment sequences.(English)Zbl 1029.43002

Let $$\left(S,+\right)$$ be a countable abelian semigroup with zero. A function $$\varphi: S\to{\mathbb R}$$ is positive definite if $\sum_{j,k=1}^n c_jc_k\varphi\left(s_j+s_k\right)\geq 0$ for any $$n\in{\mathbb N}$$, $$s_1,\dots, s_n\in S$$, and $$c_1,\dots,c_n\in{\mathbb R}$$. The set of all positive definite functions is denoted by $$\mathcal{P}\left(S\right)$$. A character on $$S$$ is a function $$\sigma: S\to{\mathbb R}$$ such that $$\sigma\left(0\right)=1$$ and $$\sigma\left(s+t\right)=\sigma\left(s\right)\sigma\left(t\right)$$ for all $$s,t\in S$$. $$S^*$$ denotes the set of all characters on $$S$$ and it is equipped with the trace topology of the topology of pointwise convergence on $${\mathbb R}^S$$. A function $$\varphi: S\to{\mathbb R}$$ is called a moment function if there is a Radon measure $$\mu$$ on $$S^*$$ such that $\varphi\left(s\right)=\int_{S^*}\sigma\left(s\right) d\mu\left(\sigma\right), \quad s\in S.$ $$\mathcal{H}\left(S\right)$$ denotes the set of all moment functions on $$S$$. It is obvious that $$\mathcal{H}\left(S\right)\subset\mathcal{P}\left(S\right)$$. If $$\mathcal{H}\left(S\right)=\mathcal{P}\left(S\right)$$, the semigroup $$S$$ is called semiperfect. As an example of such a group we can use $${\mathbb N}_0=\left\{0,1,2,\dots\right\}$$. However, the group $${\mathbb N}_0^k$$ is not semiperfect for any $$k\geq 2$$.
Denote by $$\mathcal{P}_e\left(S\right)$$ the set of all generators of extreme rays in $$\mathcal{P}\left(S\right)$$. The author remarks that if $$S$$ is not semiperfect then the set $$\mathcal{P}_e\left(S\right)\setminus\mathcal{H}\left(S\right)$$ must be nonempty.
A sequence $$\left\{s_n\right\}_{n=0}^\infty$$ of reals is positive definite if and only if it is a moment sequence, i.e., $s_n=\int_{\mathbb R}x^n d\mu,\quad n\in{\mathbb N}_0,$ for some measure $$\mu$$ on $${\mathbb R}$$. The moment sequence $$\left\{s_n\right\}_{n=0}^\infty$$ is determinate if there is only one such $$\mu$$; otherwise it is indeterminate. The measure $$\mu$$ is called determinate or indeterminate if $$\left\{s_n\right\}_{n=0}^\infty$$ is determinate or indeterminate. The measure $$\mu$$ is $$N$$-extremal if the polynomial algebra $${\mathbb R}\left[x\right]$$ is dense in $$L^2\left(\mu\right)$$.
The aim of the present article is to show that if one can give a specific example of an $$N$$-extremal indeterminate measure then one can also give a specific example of a function in $$\mathcal{P}_e\left({\mathbb N}_0^2\right)\setminus\mathcal{H}\left({\mathbb N}_0^2\right)$$. As the author notes, such an example was obtained by Konrad Schmüdgen.
The method developed by the author makes it possible to obtain some examples of functions from $$\mathcal{P}\left(S\right)\setminus\mathcal{H}\left(S\right)$$ for several semigroups $$S$$. Some useful examples are constructed in the article.
The article is very interesting, especially for specialists in moment problems. An extended bibliography reflects several results obtained in this area.

### MSC:

 43A35 Positive definite functions on groups, semigroups, etc. 42A82 Positive definite functions in one variable harmonic analysis 44A60 Moment problems
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