Enhanced Approximation Techniques: Stancu-type (λ,μ)-Bernstein-Kantorovich Operators

Enhanced Approximation Techniques: Stancu-type (λ,μ)-Bernstein-Kantorovich Operators | 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 Enhanced Approximation Techniques: Stancu-type (λ,μ)-Bernstein-Kantorovich Operators Qing-Bo Cai, Resat Aslan, Faruk Ozger, Syed Abdul Mohiuddine This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4689585/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 The principal aim of this study is to investigate a range of approximation characteristics exhibited by Stancu-type (λ, μ)-Bernstein-Kantorovich operators denoted as F λ,μ r,α,β (ϕ; z). In the initial stage, we conduct a comprehensive examination of diverse moment estimates pertaining to operators F λ,μ r,α,β (ϕ; z). Subsequently, we delve into the exploration of several facets of direct results, including the order of convergence with respect to the usual modulus of continuity, Lipschitz-type continuous functions, and the Peetre’s K-functional. Furthermore, in order to gain insights into the asymptotic characteristics of the operators F λ,μ r,α,β (ϕ; z), we derive a Voronovskaya-type asymptotic theorem. Additionally, we provide an analysis of the A-statistical convergence behavior and pointwise estimates associated with the operators F λ,μ r,α,β (ϕ; z). Finally, we have included graphical representations and numerical error value tables to demonstrate the efficiency and accuracy of our proposed operator. Our analysis reveals that our operator yields significantly superior approximation results compared to certain other linear positive operators documented in the existing literature. Keywords and (λ μ)-Bernstein operators Stancu-type Bernstein-Kantorovich operators A-statistical convergence Voronovskaya type theorem Pointwise estimates. AMS Subject Classification (2020): 41A10 41A25 41A36 Full Text Additional Declarations No competing interests reported. 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. We do this by developing innovative software and high quality services for the global research community. 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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-4689585","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":327377345,"identity":"f81eba9a-5883-4dd4-aaba-ad5691aafb9d","order_by":0,"name":"Qing-Bo Cai","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAn0lEQVRIiWNgGAWjYDACCSBOqJCQ4ydRyxkLY8kGkrQwtlUkbiBaC9/t5ocfHs6TYNzAwPzw0Q1itEjeOWYskbhNgtmcgc3YOIcYLQY3chhAWtgsG3jYpInVwvwjcY4Ej8EBErSwSSQ2SEgQrwXoFzOLhGMSBpLNxPoFGGKPb/6oqavvZ29++JgoLQwHYAxmopSjaBkFo2AUjIJRgAsAAKVFL1gVUmtNAAAAAElFTkSuQmCC","orcid":"","institution":"Quanzhou Normal University","correspondingAuthor":true,"prefix":"","firstName":"Qing-Bo","middleName":"","lastName":"Cai","suffix":""},{"id":327377346,"identity":"fb94bf70-8a4d-4ec6-a838-53f61fba55bd","order_by":1,"name":"Resat Aslan","email":"","orcid":"","institution":"Van Yüzüncü Yıl University","correspondingAuthor":false,"prefix":"","firstName":"Resat","middleName":"","lastName":"Aslan","suffix":""},{"id":327377347,"identity":"3b12e867-eec1-47e1-bf7e-09e6f9e3ac0e","order_by":2,"name":"Faruk Ozger","email":"","orcid":"","institution":"Iğdır Üniversitesi","correspondingAuthor":false,"prefix":"","firstName":"Faruk","middleName":"","lastName":"Ozger","suffix":""},{"id":327377348,"identity":"39bc4750-3ba4-45f5-9b9e-b8cf42c51c83","order_by":3,"name":"Syed Abdul Mohiuddine","email":"","orcid":"","institution":"King Abdulaziz University","correspondingAuthor":false,"prefix":"","firstName":"Syed","middleName":"Abdul","lastName":"Mohiuddine","suffix":""}],"badges":[],"createdAt":"2024-07-05 04:31:59","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4689585/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4689585/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":84687998,"identity":"8baa3b13-c453-4db4-8967-01372ab9a38a","added_by":"auto","created_at":"2025-06-16 09:17:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":568474,"visible":true,"origin":"","legend":"","description":"","filename":"SK.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4689585/v1_covered_366aa023-418b-4cb1-991c-c38207d9b3da.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eEnhanced Approximation Techniques: Stancu-type (λ,μ)-Bernstein-Kantorovich Operators\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"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":"(λ, μ)-Bernstein operators, Stancu-type Bernstein-Kantorovich operators, A-statistical convergence, Voronovskaya type theorem, Pointwise estimates. 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