Supersymmetric 2-homogeneous polynomials on $L_2((-\infty, +\infty))$
DOI:
https://doi.org/10.31471/2304-7399-2026-22(83)-36-43Keywords:
polynomial, symmetric function, supersymmetric function, Hilbert space, Lebesgue integrable functionAbstract
The work is devoted to the study of supersymmetric continuous 2-homogeneous $\mathbb{K}$-valued, where $\mathbb{K}\in\{\mathbb{R}, \mathbb{C}\},$ polynomials on the Hilbert space $L_2((-\infty, +\infty))$ of all functions $x:(-\infty, +\infty) \to \mathbb{K}$ such that $x^2$ is Lebesgue integrable. We show that every such a polynomial $P$ can be represented as $P(x) = \alpha \bigg(
\int_0^{+\infty} x^2(t)dt\!-\!
\int_{-\infty}^0 x^2(t)dt\bigg),
$ where $\alpha\in \mathbb{K}$. Consequently, the vector space of all such polynomials is one-dimensional.
References
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