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Oleh Lopushansky

Publications and source records attributed to Oleh Lopushansky.

5 recordsLinked to original sources
arXiv2024

Some exact constants for bilateral approximations in quasi-normed groups

We establish inverse and direct theorems on best approximations in quasi-normed Abelian groups through bilateral Bernstein-Jackson inequalities with exact constants. Using integral representations for quasi-norms of functions $f$ in Lebesgue's spaces by decreasing rearrangements $f^*$ with the help of approximation $E$-functionals, error estimates are found. Examples of numerical calculations and spectral approximations of self-adjoint operators obtained by obtained estimates are given.

math.FA↗
arXiv2020

Weyl-Schrödinger representations of infinite-dimensional Heisenberg groups on symmetric Wiener spaces

We investigate the group $\mathcal{H}_\mathbb{C}$ of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space $H$. Irreducible representations of the Weyl--Schr{ö}dinger type on the space $L^2_χ$ of quadratically integrable $\mathbb{C}$-valued functions are described. Integrability is understood with respect to the projective limit $χ=\varprojlimχ_i$ of probability Haar measures $χ_i$ defined on groups of unitary $i\times i$-matrices $U(i)$. The measure $χ$ is invariant under the infinite-dimensional group $U(\infty)=\bigcup U(i)$ and satisfies the abstract Kolmogorov consistency conditions. The space $L^2_χ$ is generated by Schur polynomials on Paley--Wiener maps. The Fourier-image of $L^2_χ$ coincides with the Hardy space ${H}^2_β$ of Hilbert--Schmidt analytic functions on $H$ generated by the correspondingly weighted Fock space $Γ_β(H)$. An application to heat equation over $\mathcal{H}_\mathbb{C}$ is considered.

math.FA↗
arXiv2020

Weyl-Schrödinger representations of Heisenberg groups in infinite dimensions

A complexified Heisenberg matrix group $\mathrm{H}_\mathbb{C}$ with entries from an infinite-dimensional Hilbert space $H$ is investigated. The Weyl--Schrödinger type irreducible representations of $\mathrm{H}_\mathbb{C}$ on the space $L^2_χ$ of square-integrable scalar functions is described. The integrability is understood under the invariant probability measure $χ$ which satisfies an abstract Kolmogorov consistency conditions over the infinite-dimensional unitary group $U(\infty)$ irreducible acted on $H$. The space $L^2_χ$ is generated by Schur polynomials in variables on Paley--Wiener maps over $U(\infty)$. Therewith, the Fourier-image of $L^2_χ$ coincides with a space of Hilbert--Schmidt entire analytic functions on $H$ generated by suitable Fock space. Applications to linear and nonlinear heat equations over the group $\mathrm{H}_\mathbb{C}$ are considered.

math.FA↗
arXiv2017

Paley-Wiener isomorphism over infinite-dimensional unitary groups

An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl-Schrödinger irreducible representations of complexified infinite-dimensional Heisenberg groups.

math.FA↗