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 \(\exp (-\alpha |z|^2)\). 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 \(L^\infty (\mathbb {R}^n)^{d\times d}\).