{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T18:42:14Z","timestamp":1778611334395,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,6,25]],"date-time":"2018-06-25T00:00:00Z","timestamp":1529884800000},"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>Lubich\u2019s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential.<\/jats:p>","DOI":"10.3390\/sym10070239","type":"journal-article","created":{"date-parts":[[2018,6,25]],"date-time":"2018-06-25T11:03:25Z","timestamp":1529924605000},"page":"239","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels"],"prefix":"10.3390","volume":"10","author":[{"given":"Junjie","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"},{"name":"Guizhou Provincial Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huilan","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"},{"name":"Guizhou Provincial Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1405","DOI":"10.1109\/8.725271","article-title":"Extrapolation methods for sommerfeld integral tails","volume":"46","author":"Michalski","year":"1998","journal-title":"IEEE Trans. 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