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# The investigation of a generalized moment problem associated with correlation measures

### Abstract

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case where the algebra is a special commutative algebra of functions on the space of multiple finite configurations.

If the above-mentioned functional is generated by a measure on the space of usual finite configurations then this measure is a correlation measure for a probability spectral measure on the space of infinite configurations. The latter measure is practically arbitrary, so that we have a connection between this complicated measure and its correlation measure defined on more simple objects that are finite configurations. The paper gives an answer to the following question: when this latter measure is a correlation measure for a complicated measure on infinite configurations? (Such measures are essential objects of statistical mechanics).

Key words: Convolution, positive functional, finite and infinite configurations, generalized joint eigenvector, correlation measure.

### Article Information

 Title The investigation of a generalized moment problem associated with correlation measures Source Methods Funct. Anal. Topology, Vol. 13 (2007), no. 2, 124-151 MathSciNet MR2336717 Copyright The Author(s) 2007 (CC BY-SA)

### Authors Information

Yu. M. Berezansky
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

D. A. Mierzejewski
Zhytomyr State University, Zhytomyr, Ukraine

### Citation Example

Yu. M. Berezansky and D. A. Mierzejewski, The investigation of a generalized moment problem associated with correlation measures, Methods Funct. Anal. Topology 13 (2007), no. 2, 124-151.

### BibTex

@article {MFAT402,
AUTHOR = {Berezansky, Yu. M. and Mierzejewski, D. A.},
TITLE = {The investigation of a generalized moment problem associated with correlation measures},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {13},
YEAR = {2007},
NUMBER = {2},
PAGES = {124-151},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=402},
}