Fourier Series Extension in terms of Powers of Sine and Cosine Functions

Fourier Series Extension in terms of Powers of Sine and Cosine Functions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Fourier Series Extension in terms of Powers of Sine and Cosine Functions Dagnachew Jenber Negash This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9644948/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Fourier series play a fundamental role in mathematical analysis by providing a powerful tool for representing periodic functions as infinite sums of sine and cosine functions. This classical representation has been extensively studied and successfully applied in various areas of mathematics, physics, and engineering. Despite its wide applicability, the traditional Fourier series relies exclusively on first-order trigonometric basis functions, namely sine and cosine. This naturally raises the question of whether alternative families of trigonometric functions can be employed to construct meaningful and effective expansions of periodic functions. Motivated by this observation, the present paper investigates a new framework for expanding periodic functions using a special class of trigonometric basis functions consisting of odd positive integer powers of sine and cosine. By replacing the standard trigonometric functions with their higher-order odd powers, we introduce a generalized form of Fourier series that extends the classical theory. The proposed expansion preserves the periodic structure of the original function while offering additional flexibility in representation. New definitions of extended Fourier series coefficients associated with this generalized expansion are established. Conditions for convergence of the expansion are discussed, along with illustrative examples. MSC 2020:42A16, 42A10, 42A20. Periodic functions trigonometric basis functions odd powers of sine and cosine convergence analysis Series coefficients Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9644948","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":636346135,"identity":"e3e35070-f46f-48fe-b958-6bb1db180d3b","order_by":0,"name":"Dagnachew Jenber Negash","email":"data:image/png;base64,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","orcid":"","institution":"Addis Ababa Science and Technology University","correspondingAuthor":true,"prefix":"","firstName":"Dagnachew","middleName":"Jenber","lastName":"Negash","suffix":""}],"badges":[],"createdAt":"2026-05-07 15:59:07","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9644948/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9644948/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108806483,"identity":"a81fb031-141c-4269-a797-f7d2a543d89f","added_by":"auto","created_at":"2026-05-08 15:28:41","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":439543,"visible":true,"origin":"","legend":"","description":"","filename":"FourierSeriesExtensionIntermsOfPowersOfSineAndCosine.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9644948/v1_covered_7cf2c07f-9bf8-4fe7-abe8-a7bb3810715a.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eFourier Series Extension in terms of Powers of Sine and Cosine Functions\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":" Periodic functions, trigonometric basis functions, odd powers of sine and cosine, convergence analysis, Series coefficients","lastPublishedDoi":"10.21203/rs.3.rs-9644948/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9644948/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFourier series play a fundamental role in mathematical analysis by providing a powerful tool for representing periodic functions as infinite sums of sine and cosine functions. 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