[1810.11984] On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:1810.11984 (math) [Submitted on 29 Oct 2018 ( v1 ), last revised 27 Apr 2026 (this version, v5)] Title: On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields Authors: Zhi Qi View a PDF of the paper titled On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields, by Zhi Qi View PDF HTML (experimental) Abstract: In this article, we prove certain Weber-Schafheitlin type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel functions on complex numbers. This is applied to extend the work of A. Venkatesh on Beyond Endoscopy for $\mathrm{Sym}^2$ on $\mathrm{GL}_2$ from totally real to arbitrary number fields. Comments: 26 pages. An appendix added. Small errors corrected. To appear in IMRN. Some errors in Appendix A corrected Subjects: Number Theory (math.NT) ; Classical Analysis and ODEs (math.CA) MSC classes: 33C10, 42B10, 33C05 Cite as: arXiv:1810.11984 [math.NT] (or arXiv:1810.11984v5 [math.NT] for this version) https://doi.org/10.48550/arXiv.1810.11984 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1093/imrn/rnz205 Focus to learn more DOI(s) linking to related resources Submission history From: Zhi Qi [ view email ] [v1] Mon, 29 Oct 2018 07:35:55 UTC (28 KB) [v2] Fri, 16 Nov 2018 23:05:19 UTC (29 KB) [v3] Sat, 22 Jun 2019 09:55:06 UTC (30 KB) [v4] Mon, 4 Nov 2019 01:36:00 UTC (30 KB) [v5] Mon, 27 Apr 2026 22:51:09 UTC (31 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields, by Zhi Qi View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2018-10 Change to browse by: math math.CA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from