Horizontal Fourier transform of the polyanalytic Fock kernel

Let $n,m\ge 1$ and $\alpha >0$. We denote by ${\mathcal{F}}_{\alpha ,m}$ the $m$-analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all $m$-analytic functions defined on ${\mathbb{C}}^n$ and square integrables with respect to the Gaussian weight $\exp (-\alpha \mathrm{\mid }z{\mathrm{\mid }}^2)$. We study the von Neumann algebra $\mathcal{A}$ of bounded linear operators acting in ${\mathcal{F}}_{\alpha ,m}$ and commuting with all horizontal'' Weyl translations, i.e., Weyl unitary operators associated to the elements of $\mathbb{R}^n$. The reproducing kernel of $\mathcal{F}_{1,m}$ was computed by Youssfi [Polyanalytic reproducing kernels in $\mathbb{C}^n$, Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of $\mathcal{F}_{α,m}$ by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel $K$ is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of $K$ in the horizontal direction’’ and decompose it into the sum of $d$ products of Hermite functions, with $d=\left(\genfrac{}{}{0px}{}{n+m-1}{n}\right)$. Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that ${\mathcal{F}}_{\alpha ,m}$ is isometrically isomorphic to the space of vector-functions $L^2({\mathbb{R}}^n{)}^d$, and $\mathcal{A}$ is isometrically isomorphic to the algebra of matrix-functions $L^{\mathrm{\infty }}({\mathbb{R}}^n{)}^{d\times d}$.