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

T. V. Vasylyshyn
2018 Matematychni Studii  
Математичнi Студiї. Т.50, №1 Matematychni Studii. V.50, No.1 УДК 517.98 T. V. Vasylyshyn SYMMETRIC POLYNOMIALS ON THE CARTESIAN POWER OF L p ON THE SEMI-AXIS T. V. Vasylyshyn. Symmetric polynomials on the Cartesian power of L p on the semi-axis, Mat. Stud. 50 (2018), 93-104. The paper deals with polynomials in the complex Banach space (L p [0, +∞)) n , which are the nth Cartesian power of the complex Banach space of Lebesgue measurable integrable in a power p complex-valued functions on [0,
more » ... unctions on [0, +∞), where 1 ≤ p < +∞. It is proved that if p is an integer, then every continuous symmetric polynomial on (L p [0, +∞)) 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, +∞)) 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, +∞)) n .
doi:10.15330/ms.50.1.93-104 fatcat:272evw3lnrg5doppg4zguakxdu