Horizontal Fourier transform of the polyanalytic Fock kernel

TL;DR

We study the commutant of the horizontal translation action on the $n$-dimensional polyanalytic Bargmann--Segal--Fock spaces $\mathcal{F}_{\alpha,m}$ and provide a complete spectral description. A Flattened RKHS $\mathcal{H}_m$ is introduced, the reproducing kernel is Fourier-analyzed in the horizontal direction, and the kernel is decomposed into a finite sum of rank-one Hermite factors; this yields a direct integral decomposition into $d=\binom{n+m-1}{n}$ copies of $L^2(\mathbb{R}^n)$ and an explicit correspondence between translation-invariant operators and matrix-valued multipliers in $L^\infty(\mathbb{R}^n)^{d\times d}$. Consequently, $\mathcal{F}_{\alpha,m}$ is isometrically isomorphic to $L^2(\mathbb{R}^n)^d$ and its commutant to $L^\infty(\mathbb{R}^n)^{d\times d}$; this noncommutative structure arises for $m\ge 2$. The approach unifies Laguerre–Hermite transform techniques with RKHS and direct-integral theory, and the results are supported by numerical tests validating the key identities in low dimensions.

Abstract

Let $n,m\ge 1$ and $α>0$. We denote by $\mathcal{F}_{α,m}$ 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 $\exp(-α|z|^2)$. We study the von Neumann algebra $\mathcal{A}$ of bounded linear operators acting in $\mathcal{F}_{α,m}$ 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 $d=\binom{n+m-1}{n}$. 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 $\mathcal{F}_{α,m}$ 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 $L^\infty(\mathbb{R}^n)^{d\times d}$.

Figures (3)

• Figure 1: Isometric isomorphism $U_{\mathcal{F}_{\alpha,m}}^{\mathcal{H}_m}$ and unitary operators $\rho_{\mathcal{F}_{\alpha,m}}(a)$, $\rho_{\mathcal{H}_m}(\sqrt{\alpha}\,a)$.
• Figure 2: Main objects of Example \ref{['example:diagonalization_of_Weyl_translations']}.
• Figure 3: Main objects of Proposition \ref{['prop:gammas_of_vertical_Toeplitz_operators']}.

Theorems & Definitions (50)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• Proposition 3.3
• proof
• Corollary 3.4