{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:18Z","timestamp":1760145678957,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T00:00:00Z","timestamp":1724716800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00144\/2020","UIDP\/00144\/2020"],"award-info":[{"award-number":["UIDB\/00144\/2020","UIDP\/00144\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We investigate the properties of matrices that emerge from the application of Fourier series to Jacobi matrices. Specifically, we focus on functions defined by the coefficients of a Fourier series expressed in orthogonal polynomials. In the operational formulation of integro-differential problems, these infinite matrices play a fundamental role. We have derived precise calculation formulas for their elements, enabling exact computation of these operational matrices. Numerical results illustrate the effectiveness of our approach.<\/jats:p>","DOI":"10.3390\/axioms13090581","type":"journal-article","created":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T03:51:06Z","timestamp":1724730666000},"page":"581","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Fourier Series in the Context of Jacobi Matrices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3512-5930","authenticated-orcid":false,"given":"Jos\u00e9 M. A.","family":"Matos","sequence":"first","affiliation":[{"name":"Instituto Superior de Engenharia do Instituto Polit\u00e9cnico do Porto, Centro de Matem\u00e1tica da Universidade do Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7132-880X","authenticated-orcid":false,"given":"Paulo B.","family":"Vasconcelos","sequence":"additional","affiliation":[{"name":"Faculdade de Economia da Universidade do Porto, Centro de Matem\u00e1tica da Universidade do Porto, Rua Dr. Roberto Frias s\/n, 4200-464 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0570-7913","authenticated-orcid":false,"given":"Jos\u00e9 A. O.","family":"Matos","sequence":"additional","affiliation":[{"name":"Faculdade de Economia da Universidade do Porto, Centro de Matem\u00e1tica da Universidade do Porto, Rua Dr. Roberto Frias s\/n, 4200-464 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1017\/S0962492900002622","article-title":"Orthogonal polynomials: Applications and computation","volume":"5","author":"Gautschi","year":"1996","journal-title":"Acta Numer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0377-0427(95)00033-X","article-title":"Second-order recurrence relation for the linearization coefficients of the classical orthogonal polynomials","volume":"69","author":"Lewanowicz","year":"1996","journal-title":"J. Comput. Appl. 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