Horizontal Fourier Transform of the Polyanalytic Fock Kernel

Let n , m ≥ 1 and α > 0 . We denote by F α , m the m -analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all m -analytic functions defined on C n and square integrables with respect to the Gaussian weight exp ( - α | z | 2 ) . We study the von Neumann algebra A of bounded linear operators acting in F α , m and commuting with all “horizontal” Weyl translations, i.e., Weyl unitary operators associated to the elements of R n . The reproducing kernel of F 1 , m was computed by Youssfi [Polyanalytic reproducing kernels in C n , Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of 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 = n + m - 1 n . 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 F α , m is isometrically isomorphic to the space of vector-functions L 2 ( R n ) d , and A is isometrically isomorphic to the algebra of matrix-functions L ∞ ( R n ) d × d .