For a sequence of King type operators which preserve the functions begin{document}$ e_0(x)=1 $end{document} and begin{document}$ e_j(x)=x^j $end{document}, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse result of Berens-Lorentz type.
For a sequence of King type operators which preserve the functions begin{document}$ e_0(x)=1 $end{document} and begin{document}$ e_j(x)=x^j $end{document}, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse result of Berens-Lorentz type.
{"title":"Direct and converse theorems for King type operators","authors":"Z. Finta","doi":"10.3934/mfc.2022015","DOIUrl":"https://doi.org/10.3934/mfc.2022015","url":null,"abstract":"<p style='text-indent:20px;'>For a sequence of King type operators which preserve the functions <inline-formula><tex-math id=\"M1\">begin{document}$ e_0(x)=1 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M2\">begin{document}$ e_j(x)=x^j $end{document}</tex-math></inline-formula>, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse result of Berens-Lorentz type.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"221 1","pages":"379-387"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79873385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Assuming that begin{document}$ K(x) $end{document} is in begin{document}$ L^1( {mathbb R}) $end{document}, begin{document}$ K_t(x) = t^{-1} K(x/t) $end{document}, and begin{document}$ f(x) $end{document} is in begin{document}$ L^infty( {mathbb R}) $end{document}, we study the behavior of the convolution begin{document}$ K_t*f(x) $end{document} as the parameter begin{document}$ t $end{document} tends to begin{document}$ infty $end{document}. It turns out that the limit need not exist and, if it does exist, the limit is a constant independent of begin{document}$ x $end{document}. Situations where the limit exists and those where it fails to exist are identified. Several issues related to this are addressed, including the multivariate case. As one application, these results provide an accessible description of the behavior of bounded solutions to the initial value problem for the heat equation.
Assuming that begin{document}$ K(x) $end{document} is in begin{document}$ L^1( {mathbb R}) $end{document}, begin{document}$ K_t(x) = t^{-1} K(x/t) $end{document}, and begin{document}$ f(x) $end{document} is in begin{document}$ L^infty( {mathbb R}) $end{document}, we study the behavior of the convolution begin{document}$ K_t*f(x) $end{document} as the parameter begin{document}$ t $end{document} tends to begin{document}$ infty $end{document}. It turns out that the limit need not exist and, if it does exist, the limit is a constant independent of begin{document}$ x $end{document}. Situations where the limit exists and those where it fails to exist are identified. Several issues related to this are addressed, including the multivariate case. As one application, these results provide an accessible description of the behavior of bounded solutions to the initial value problem for the heat equation.
{"title":"Behavior in $ L^infty $ of convolution transforms with dilated kernels","authors":"W. Madych","doi":"10.3934/mfc.2022005","DOIUrl":"https://doi.org/10.3934/mfc.2022005","url":null,"abstract":"<p style='text-indent:20px;'>Assuming that <inline-formula><tex-math id=\"M1\">begin{document}$ K(x) $end{document}</tex-math></inline-formula> is in <inline-formula><tex-math id=\"M2\">begin{document}$ L^1( {mathbb R}) $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M3\">begin{document}$ K_t(x) = t^{-1} K(x/t) $end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id=\"M4\">begin{document}$ f(x) $end{document}</tex-math></inline-formula> is in <inline-formula><tex-math id=\"M5\">begin{document}$ L^infty( {mathbb R}) $end{document}</tex-math></inline-formula>, we study the behavior of the convolution <inline-formula><tex-math id=\"M6\">begin{document}$ K_t*f(x) $end{document}</tex-math></inline-formula> as the parameter <inline-formula><tex-math id=\"M7\">begin{document}$ t $end{document}</tex-math></inline-formula> tends to <inline-formula><tex-math id=\"M8\">begin{document}$ infty $end{document}</tex-math></inline-formula>. It turns out that the limit need not exist and, if it does exist, the limit is a constant independent of <inline-formula><tex-math id=\"M9\">begin{document}$ x $end{document}</tex-math></inline-formula>. Situations where the limit exists and those where it fails to exist are identified. Several issues related to this are addressed, including the multivariate case. As one application, these results provide an accessible description of the behavior of bounded solutions to the initial value problem for the heat equation.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"44 1","pages":"94-107"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86246487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mingjie Wang, Juxiang Zhou, Jun Wang, Jianhou Gan, Zijie Li
{"title":"A knowledge representation learning model based on relation rotation in two-dimensional Minkowski space","authors":"Mingjie Wang, Juxiang Zhou, Jun Wang, Jianhou Gan, Zijie Li","doi":"10.3934/mfc.2023020","DOIUrl":"https://doi.org/10.3934/mfc.2023020","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotically deferred statistical equivalent functions of order $ alpha $ in amenable semigroups","authors":"M. Et, H. Dutta, N. Braha","doi":"10.3934/mfc.2023018","DOIUrl":"https://doi.org/10.3934/mfc.2023018","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators","authors":"R. K. Bhardwaj, Tirth Ram","doi":"10.3934/mfc.2023036","DOIUrl":"https://doi.org/10.3934/mfc.2023036","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalised subclasses of meromorphically $ q $-starlike function using the Janowski functions","authors":"Abdullah Alatawi, M. Darus, S. Sivasubramanian","doi":"10.3934/mfc.2023021","DOIUrl":"https://doi.org/10.3934/mfc.2023021","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximation properties of Riemann-Liouville type fractional Bernstein-Kantorovich operators of order $ alpha $","authors":"Erdem Baytunç, Hüseyin Aktuğlu, N. Mahmudov","doi":"10.3934/mfc.2023030","DOIUrl":"https://doi.org/10.3934/mfc.2023030","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ioan Cristian Buşcu, Gabriela Motronea, Vlad Paşca
{"title":"Modified positive linear operators, iterates and systems of linear equations","authors":"Ioan Cristian Buşcu, Gabriela Motronea, Vlad Paşca","doi":"10.3934/mfc.2023031","DOIUrl":"https://doi.org/10.3934/mfc.2023031","url":null,"abstract":"","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For real numbers begin{document}$ a,qgeq 0 $end{document} and a weight begin{document}$ varrho(x) = 1/(1+x)^q $end{document}, the author provides necessary and sufficient conditions for a function begin{document}$ fin C[0,infty) $end{document} in order to begin{document}$ sup_{xgeq 0}mid varrho(x)(B_n^a(f,x)-f(x))mid to 0 $end{document} as begin{document}$ nto infty $end{document}, where begin{document}$ B_n^a(f) $end{document} is the Mihesan operator. In particular, it is proved that Mihesan operators behave similar to the classical Baskakov operators.
For real numbers begin{document}$ a,qgeq 0 $end{document} and a weight begin{document}$ varrho(x) = 1/(1+x)^q $end{document}, the author provides necessary and sufficient conditions for a function begin{document}$ fin C[0,infty) $end{document} in order to begin{document}$ sup_{xgeq 0}mid varrho(x)(B_n^a(f,x)-f(x))mid to 0 $end{document} as begin{document}$ nto infty $end{document}, where begin{document}$ B_n^a(f) $end{document} is the Mihesan operator. In particular, it is proved that Mihesan operators behave similar to the classical Baskakov operators.
{"title":"Approximation of functions and Mihesan operators","authors":"J. Bustamante","doi":"10.3934/mfc.2022033","DOIUrl":"https://doi.org/10.3934/mfc.2022033","url":null,"abstract":"<p style='text-indent:20px;'>For real numbers <inline-formula><tex-math id=\"M1\">begin{document}$ a,qgeq 0 $end{document}</tex-math></inline-formula> and a weight <inline-formula><tex-math id=\"M2\">begin{document}$ varrho(x) = 1/(1+x)^q $end{document}</tex-math></inline-formula>, the author provides necessary and sufficient conditions for a function <inline-formula><tex-math id=\"M3\">begin{document}$ fin C[0,infty) $end{document}</tex-math></inline-formula> in order to <inline-formula><tex-math id=\"M4\">begin{document}$ sup_{xgeq 0}mid varrho(x)(B_n^a(f,x)-f(x))mid to 0 $end{document}</tex-math></inline-formula> as <inline-formula><tex-math id=\"M5\">begin{document}$ nto infty $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M6\">begin{document}$ B_n^a(f) $end{document}</tex-math></inline-formula> is the Mihesan operator. In particular, it is proved that Mihesan operators behave similar to the classical Baskakov operators.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"19 1","pages":"369-378"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78014088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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