{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:38:17Z","timestamp":1760240297829,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T00:00:00Z","timestamp":1556668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we further extend the Filon-type method to the Bessel function expansion for calculating Fourier integral. By means of complex analysis, this expansion is effective for all the oscillation frequencies. Namely, the errors of the expansion not only decrease as the order of the derivative increases, but also decrease rapidly as the frequency increases. Some numerical experiments are also presented to verify the effectiveness of the method.<\/jats:p>","DOI":"10.3390\/sym11050607","type":"journal-article","created":{"date-parts":[[2019,5,2]],"date-time":"2019-05-02T03:15:22Z","timestamp":1556766922000},"page":"607","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Bessel Expansion of Fourier Integral on Finite Interval"],"prefix":"10.3390","volume":"11","author":[{"given":"Yongxiong","family":"Zhou","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524088, China"}]},{"given":"Zhenyu","family":"Zhao","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524088, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,1]]},"reference":[{"key":"ref_1","unstructured":"Born, M., and Wolf, E. (2001). Principles of Optics, Cambridge University Press."},{"key":"ref_2","unstructured":"Watson, G.N. (1966). A Treatise on the Theory of Bessel Function, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dea\u00f1o, A., Huybrechs, D., and Iserles, A. (2018). Computing Highly Oscillatory Integrals, SIAM.","DOI":"10.1137\/1.9781611975123"},{"key":"ref_4","first-page":"755","article-title":"From high oscillation to rapid approximation V: The equilateral triangle","volume":"30","author":"Huybrechs","year":"2010","journal-title":"IMA J. Numer. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1007\/s00211-006-0051-0","article-title":"Efficient Filon-type methods for \u222babf(x)ei\u03c9g(x)dx","volume":"105","author":"Xiang","year":"2007","journal-title":"Numer. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s10543-012-0399-8","article-title":"Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels","volume":"53","author":"Xiang","year":"2013","journal-title":"BIT Numer. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.amc.2018.09.066","article-title":"Efficient calculation and asymptotic expansions of many different oscillatory infinite integrals","volume":"346","author":"Kang","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1017\/S0370164600026262","article-title":"On a quadrature formula for trigonometric integrals","volume":"49","author":"Filon","year":"1928","journal-title":"Proc. R. Soc. Edinb."},{"key":"ref_9","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, Elsevier\/Academic Press. [7th ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications Inc.","DOI":"10.1115\/1.3625776"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wang, Z.X., and Guo, D.R. (1989). Special Functions, World Scientific.","DOI":"10.1142\/0653"},{"key":"ref_12","unstructured":"Ahlfors, L.V. (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, McGraw-Hill Book Company, Inc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/607\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:48:29Z","timestamp":1760186909000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/607"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,1]]},"references-count":12,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050607"],"URL":"https:\/\/doi.org\/10.3390\/sym11050607","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,5,1]]}}}