×

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:

30E05 Moment problems and interpolation problems in the complex plane
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.