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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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