## Riesz-haviland criterion for incomplete data.(English)Zbl 1214.30020

Summary: The aim of this paper is to analyze a “support-free” version of the Riesz-Haviland theorem proved recently by the present authors, which characterizes truncations of the complex moment problem via positivity condition on appropriate families of polynomials in $$z$$ and $$\tilde z$$. The attention is focused on modifications of the positivity condition as well as the assumption on admissible truncations. The former results in truncations for which the corresponding “support-free” Riesz-Haviland condition locates a representing measure on the distinguished subset of the complex plane, while the latter effects a non-integral variant of the Riesz-Haviland theorem.

### MSC:

 3e+06 Moment problems and interpolation problems in the complex plane
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### References:

 [1] Berg, C.; Christensen, J.P.R.; Ressel, P., Harmonic analysis on semigroups, (1984), Springer Berlin [2] Bochnak, J.; Coste, M.; Roy, M.-F., Géometrie algébrique réelle, Ergeb. math. grenzgeb., vol. 12, (1987), Springer-Verlag Berlin-New York · Zbl 0633.14016 [3] Chihara, T.S., A characterization and a class of distribution functions for the Stieltjes-Wigert polynomials, Canad. math. bull., 13, 529-532, (1970) · Zbl 0205.07604 [4] D. Cichoń, J. Stochel, F.H. Szafraniec, Naimark extensions for indeterminacy in the moment problem. An example, Indiana Univ. Math. J., in press. · Zbl 1239.44002 [5] Cichoń, D.; Stochel, J.; Szafraniec, F.H., Extending positive definiteness, Trans. amer. math. soc., 363, 545-577, (2011) · Zbl 1213.43006 [6] Curto, R.E.; Fialkow, L.A., An analogue of the Riesz-haviland theorem for the truncated moment problem, J. funct. anal., 255, 2709-2731, (2008) · Zbl 1158.44003 [7] Fuglede, B., The multidimensional moment problem, Expo. math., 1, 47-65, (1983) · Zbl 0514.44006 [8] Haviland, E.K., On the momentum problem for distributions in more than one dimension, Amer. J. math., 57, 562-568, (1935) · Zbl 0013.05904 [9] Haviland, E.K., On the momentum problem for distributions in more than one dimension, II, Amer. J. math., 58, 164-168, (1936) · Zbl 0015.10901 [10] Kilpi, Y., Über das komplexe momentenproblem, Ann. acad. sci. fenn. ser. A. I., 236, (1957), 32 pp · Zbl 0078.28504 [11] Riesz, M., Sur le probléme des moments, troisième note, Ark. mat. astronom. fyz., 17, 1-52, (1923) · JFM 49.0195.01 [12] Shohat, J.A.; Tamarkin, J.D., The problem of moments, Math. surveys, vol. 1, (1943), Amer. Math. Soc. Providence, RI · Zbl 0063.06973 [13] Simon, B., The classical moment problem as a self-adjoint finite difference operator, Adv. math., 137, 82-203, (1998) · Zbl 0910.44004 [14] Stochel, J.; Szafraniec, F.H., The complex moment problem and subnormality: A polar decomposition approach, J. funct. anal., 159, 432-491, (1998) · Zbl 1048.47500 [15] Szafraniec, F.H., On extending backwards positive definite sequences, Numer. algorithms, 3, 419-425, (1992) · Zbl 0784.44007
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