Papers › Horizontal Fourier transform of the polyanalytic Fock kernel
Horizontal Fourier transform of the polyanalytic Fock kernel
Erick Lee-Guzmán, Egor A. Maximenko, Gerardo Ramos-Vazquez, Armando Sánchez-Nungaray
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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 ℂⁿ 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 ℝⁿ. The reproducing kernel of ℱ_(1,m) was computed by Youssfi [Polyanalytic reproducing kernels in ℂⁿ, 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-1n. Finally, applying the scheme proposed by Herrera-Ya\~{n}ez, 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 L²(ℝⁿ)ᵈ, and 𝒜 is isometrically isomorphic to the algebra of matrix-functions L^∞(ℝⁿ)^(d×d).
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections