Horizontal Fourier Transform of the Polyanalytic Fock Kernel

Abstract

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 $\exp (-\alpha |z|^2)$. 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 $\mathcal {F}_{\alpha ,m}$ 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 $d=\left( {\begin{array}{c}n+m-1\\ n\end{array}}\right)$. 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 $L^\infty (\mathbb {R}^n)^{d\times d}$.