Abstract
We describe an algebraic basis of the algebra of symmetric continuous polynomials on the nth Cartesian power of the complex Banach space , where \(1\leqslant p <+\infty \).
Similar content being viewed by others
References
Alencar, R., Aron, R., Galindo, P., Zagorodnyuk, A.: Algebras of symmetric holomorphic functions on \(\ell_p\). Bull. London Math. Soc. 35(2), 55–64 (2003)
Aron, R., Galindo, P., Pinasco, D., Zalduendo, I.: Group-symmetric holomorphic functions on a Banach space. Bull. London Math. Soc. 48(5), 779–796 (2016)
Chernega, I.: Symmetric polynomials and holomorphic functions on infinite dimensional spaces. J. Vasyl Stefanyk Precarpathian Natl. Univ. 2(4), 23–49 (2015)
Chernega, I., Galindo, P., Zagorodnyuk, A.: Some algebras of symmetric analytic functions and their spectra. Proc. Edinburgh Math. Soc. 55(1), 125–142 (2012)
Chernega, I., Galindo, P., Zagorodnyuk, A.: The convolution operation on the spectra of algebras of symmetric analytic functions. J. Math. Anal. Appl. 395(2), 569–577 (2012)
Chernega, I., Galindo, P., Zagorodnyuk, A.: A multiplicative convolution on the spectra of algebras of symmetric analytic functions. Rev. Mat. Complut. 27(2), 575–585 (2014)
Galindo, P., Vasylyshyn, T., Zagorodnyuk, A.: The algebra of symmetric analytic functions on \(L_\infty \). Proc. Roy. Soc. Edinburgh Sect. A 147(4), 743–761 (2017)
Galindo, P., Vasylyshyn, T., Zagorodnyuk, A.: Symmetric and finitely symmetric polynomials on the spaces \(\ell_\infty \) and \(L_\infty [0,+\infty )\). Math. Nachr. (2018). https://doi.org/10.1002/mana.201700314
González, M., Gonzalo, R., Jaramillo, J.A.: Symmetric polynomials on rearrangement invariant function spaces. J. London Math. Soc. 59(2), 681–697 (1999)
Kravtsiv, V.: The analogue of Newton’s formula for block-symmetric polynomials. Int. J. Math. Anal. 10(7), 323–327 (2016)
Kravtsiv, V.V., Zagorodnyuk, A.V.: Representation of spectra of algebras of block-symmetric analytic functions of bounded type. Carpathian Math. Publ. 8(2), 263–271 (2016)
Kravtsiv, V., Vasylyshyn, T., Zagorodnyuk, A.: On algebraic basis of the algebra of symmetric polynomials on \(\ell_p(\mathbb{C}^{n})\). J. Funct. Spaces 2017, # 4947925 (2017)
Mujica, J.: Complex Analysis in Banach Spaces. Dover, Mineola (2010)
Nemirovskii, A.S., Semenov, S.M.: On polynomial approximation of functions on Hilbert space. Mat. Sb. (N.S.) 92(134), 257–281 (1973)
Vasylyshyn, T.V.: Symmetric continuous linear functionals on complex space \(L_\infty [0,1]\). Carpathian Math. Publ. 6(1), 8–10 (2014)
Vasylyshyn, T.V.: Continuous block-symmetric polynomials of degree at most two on the space \((L_\infty )^2\). Carpathian Math. Publ. 8(1), 38–43 (2016)
Vasylyshyn, T.V.: Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex \(L_\infty \). Carpathian Math. Publ. 9(1), 22–27 (2017)
Vasylyshyn, T.V.: Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex \(L_\infty \). Carpathian Math. Publ. 9(2), 198–201 (2017)
Vasylyshyn, T.: Symmetric polynomials on the space of bounded integrable functions on the semi-axis. Int. J. Pure Appl. Math. 117(3), 425–430 (2017)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vasylyshyn, T. Symmetric polynomials on . European Journal of Mathematics 6, 164–178 (2020). https://doi.org/10.1007/s40879-018-0268-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-018-0268-3