Ibrahim Islam, Naved Alam, Aaqib Ayoub Bhat, Asif Khan
{"title":"Fractal approximation via Phillips-type (p, q)-Bernstein operators","authors":"Ibrahim Islam, Naved Alam, Aaqib Ayoub Bhat, Asif Khan","doi":"10.1007/s13540-026-00536-3","DOIUrl":null,"url":null,"abstract":"This paper presents a new class of <inline-formula><alternatives><mml:math><mml:mi>α</mml:mi></mml:math><tex-math>\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}</tex-math></alternatives></inline-formula>-fractal functions generated using Phillips-type (<italic>p</italic>, <italic>q</italic>)-Bernstein operators as base functions. For a continuous function <italic>f</italic> on a compact interval and a nonzero scaling vector <inline-formula><alternatives><mml:math><mml:mi>α</mml:mi></mml:math><tex-math>\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}</tex-math></alternatives></inline-formula>, a sequence <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math><tex-math>\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{f_n^{(p,q,\\alpha )}\\}$$\\end{document}</tex-math></alternatives></inline-formula> of <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><tex-math>\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p,q,\\alpha )$$\\end{document}</tex-math></alternatives></inline-formula>-fractal approximants is constructed, which converges uniformly to <italic>f</italic> without the conventional requirement of vanishing scaling vectors. The study also explores <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><tex-math>\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p,q,\\alpha )$$\\end{document}</tex-math></alternatives></inline-formula>-fractal functions within Hölder spaces, confirming the existence, uniqueness, and convergence of the associated approximants. Additionally, the constrained approximation problem is examined within this novel framework. The findings reveal that the deformation parameters <italic>p</italic> and <italic>q</italic> provide flexibility in approximation over both the geometric and analytical characteristics of the derived fractal functions, which is also supported by graphical demonstrations.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"15 1","pages":""},"PeriodicalIF":3.1000,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fractional Calculus and Applied Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13540-026-00536-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper presents a new class of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-fractal functions generated using Phillips-type (p, q)-Bernstein operators as base functions. For a continuous function f on a compact interval and a nonzero scaling vector α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, a sequence {fn(p,q,α)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f_n^{(p,q,\alpha )}\}$$\end{document} of (p,q,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q,\alpha )$$\end{document}-fractal approximants is constructed, which converges uniformly to f without the conventional requirement of vanishing scaling vectors. The study also explores (p,q,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q,\alpha )$$\end{document}-fractal functions within Hölder spaces, confirming the existence, uniqueness, and convergence of the associated approximants. Additionally, the constrained approximation problem is examined within this novel framework. The findings reveal that the deformation parameters p and q provide flexibility in approximation over both the geometric and analytical characteristics of the derived fractal functions, which is also supported by graphical demonstrations.
This paper presents a new class of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-fractal functions generated using Phillips-type (p, q)-Bernstein operators as base functions. For a continuous function f on a compact interval and a nonzero scaling vector α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, a sequence {fn(p,q,α)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f_n^{(p,q,\alpha )}\}$$\end{document} of (p,q,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q,\alpha )$$\end{document}-fractal approximants is constructed, which converges uniformly to f without the conventional requirement of vanishing scaling vectors. The study also explores (p,q,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q,\alpha )$$\end{document}-fractal functions within Hölder spaces, confirming the existence, uniqueness, and convergence of the associated approximants. Additionally, the constrained approximation problem is examined within this novel framework. The findings reveal that the deformation parameters p and q provide flexibility in approximation over both the geometric and analytical characteristics of the derived fractal functions, which is also supported by graphical demonstrations.
期刊介绍：
Fractional Calculus and Applied Analysis (FCAA, abbreviated in the World databases as Fract. Calc. Appl. Anal. or FRACT CALC APPL ANAL) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order. The high standards of its contents are guaranteed by the prominent members of Editorial Board and the expertise of invited external reviewers, and proven by the recently achieved high values of impact factor (JIF) and impact rang (SJR), launching the journal to top places of the ranking lists of Thomson Reuters and Scopus.
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