Symmetric polynomials on the Cartesian power of $L_p$ on the semi-axis

Author
T. V. Vasylyshyn
Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract
The paper deals with polynomials in the complex Banach space $(L_p[0,+\infty))^n,$ which are the $n$th Cartesian power of the complex Banach space of Lebesgue measurable integrable in a power $p$ complex-valued functions on $[0,+\infty),$ where $1\leq p \le +\infty.$ It is proved that if $p$ is an integer, then every continuous symmetric polynomial on $(L_p[0,+\infty))^n$ can be uniquely represented as an algebraic combination of some ``elementary'' $p$-homogeneous symmetric polynomials. It is also proved that if $p$ is not an integer, then every continuous symmetric polynomial on $(L_p[0,+\infty))^n$ is constant. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on $(L_p[0,+\infty))^n.$
Keywords
polynomial; symmetric polynomial; algebraic combination
DOI
doi:10.15330/ms.50.1.93-104
Reference
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Pages
93-104
Volume
50
Issue
1
Year
2018
Journal
Matematychni Studii
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