{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:17:37Z","timestamp":1760145457244,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T00:00:00Z","timestamp":1721952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Funds through the Foundation for Science and Technology of Portugal","award":["UIDB\/00066\/2020","UACM CCYT-CON-03"],"award-info":[{"award-number":["UIDB\/00066\/2020","UACM CCYT-CON-03"]}]},{"name":"Autonomous University of Mexico City","award":["UIDB\/00066\/2020","UACM CCYT-CON-03"],"award-info":[{"award-number":["UIDB\/00066\/2020","UACM CCYT-CON-03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Simple formulae for Lebesgue constants arising in the classical Fourier series approximation are obtained. Both even and odd cases are addressed, extending Fej\u00e9r\u2019s results. Asymptotic formulae are also obtained.<\/jats:p>","DOI":"10.3390\/axioms13080505","type":"journal-article","created":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T13:04:59Z","timestamp":1721999099000},"page":"505","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Lebesgue Constants"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4270-3284","authenticated-orcid":false,"given":"Manuel Duarte","family":"Ortigueira","sequence":"first","affiliation":[{"name":"UNINOVA-CTS and LASI, NOVA School of Science and Technology, NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9587-2586","authenticated-orcid":false,"given":"Gabriel","family":"Bengochea","sequence":"additional","affiliation":[{"name":"Academia de Matem\u00e1ticas, Universidad Aut\u00f3noma de la Ciudad de M\u00e9xico, Prolongaci\u00f3n San Isidro 151, Col. San Lorenzo Tezonco, Del. Iztapalapa, Mexico City 09790, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Finch, S. (2003). Mathematical Constants, Cambridge University Press.","DOI":"10.1017\/CBO9780511550447"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"184","DOI":"10.24033\/bsmf.859","article-title":"Sur la repr\u00e9sentation trigonom\u00e9trique approch\u00e9e des fonctions satisfaisant \u00e0 une condition de Lipschitz","volume":"38","author":"Lebesgue","year":"1910","journal-title":"Soci\u00e9t\u00e9 Math\u00e9matique Fr."},{"key":"ref_3","first-page":"610","article-title":"Inequalities and asymptotic expansions for the constants of Landau and Lebesgue","volume":"248","author":"Chen","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"64","DOI":"10.3103\/S1066369X23110099","article-title":"Approximation of the Lebesgue constant of the Fourier operator by a logarithmic-fractional-rational function","volume":"67","author":"Shakirov","year":"2023","journal-title":"Russ. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"63","DOI":"10.24033\/asens.634","article-title":"Sur les singularit\u00e9s de la s\u00e9rie de Fourier des fonctions continues","volume":"Volume 28","year":"1911","journal-title":"Annales Scientifiques de L\u2019\u00c9cole Normale Sup\u00e9rieure"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BF01378345","article-title":"\u00dcber die Lebesgueschen konstanten bei den Fourierschen reihen","volume":"9","author":"Szego","year":"1921","journal-title":"Math. Z."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1093\/qmath\/os-1.1.310","article-title":"The constants of Landau and Lebesgue","volume":"1","author":"Watson","year":"1930","journal-title":"Quart. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1112\/jlms\/s1-17.1.4","article-title":"Note on Lebesgue\u2019s constants in the theory of Fourier series","volume":"1","author":"Hardy","year":"1942","journal-title":"J. Lond. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.jmaa.2008.08.035","article-title":"Some sharp estimates of the constants of Landau and Lebesgue","volume":"349","author":"Zhao","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"70","DOI":"10.3103\/S1066369X22050073","article-title":"Approximation of the Lebesgue constant of the Fourier operator by a logarithmic function","volume":"66","author":"Shakirov","year":"2022","journal-title":"Russ. Math."},{"key":"ref_11","first-page":"96","article-title":"Properties of the Dirichlet kernel","volume":"11","author":"Alvarez","year":"2023","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_12","unstructured":"Stein, E., and Shakarchi, R. (2011). Fourier Analysis: An Introduction, Princeton University Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/505\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:24:32Z","timestamp":1760109872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/505"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,26]]},"references-count":12,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080505"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080505","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,7,26]]}}}