{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/horizontal-fourier-transform-of-the","title":"Horizontal Fourier transform of the polyanalytic Fock kernel","arxiv_id":"2309.03410","date":"2023-09-07","proceeding":null,"authors":["Erick Lee-Guzmán","Egor A. Maximenko","Gerardo Ramos-Vazquez","Armando Sánchez-Nungaray"],"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 $d=\\binom{n+m-1}{n}$. 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 $\\mathcal{F}_{\\alpha,m}$ 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 $L^\\infty(\\mathbb{R}^n)^{d\\times d}$.","url_abs":"https://arxiv.org/abs/2309.03410v1","url_pdf":"https://arxiv.org/pdf/2309.03410v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"horizontal-fourier-transform-of-the","repo_url":"https://github.com/egormaximenko/horizontal-fourier-transform-of-polyanalytic-fock-kernel","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}