Korovkin-type theorems and local approximation problems
Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and inf...
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| Published in: | Expositiones mathematicae Vol. 40; no. 4; pp. 1229 - 1243 |
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| Format: | Journal Article |
| Language: | English |
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Elsevier GmbH
01.12.2022
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| ISSN: | 0723-0869, 1878-0792 |
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| Abstract | Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and infinite dimensional settings (Hilbert spaces).
Furthermore, some applications are discussed which concern classical sequences of positive linear operators including (one dimensional and multidimensional) Bernstein operators, Kantorovich operators, Szász–Mirakyan operators, Gauss–Weierstrass operators and Bernstein–Schnabl operators on convex subsets of Hilbert spaces.
Finally the paper ends with a reassessment of a result of Korovkin concerning subspaces of bounded 2π− periodic functions on R and with an application related to sequences of convolution operators generated by positive approximate identities. |
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| AbstractList | Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and infinite dimensional settings (Hilbert spaces).
Furthermore, some applications are discussed which concern classical sequences of positive linear operators including (one dimensional and multidimensional) Bernstein operators, Kantorovich operators, Szász–Mirakyan operators, Gauss–Weierstrass operators and Bernstein–Schnabl operators on convex subsets of Hilbert spaces.
Finally the paper ends with a reassessment of a result of Korovkin concerning subspaces of bounded 2π− periodic functions on R and with an application related to sequences of convolution operators generated by positive approximate identities. |
| Author | Altomare, Francesco |
| Author_xml | – sequence: 1 givenname: Francesco orcidid: 0000-0003-3407-3040 surname: Altomare fullname: Altomare, Francesco email: francesco.altomare@uniba.it organization: Dipartimento di Matematica, Campus Universitario, Università degli Studi di Bari Aldo Moro, Via E. Orabona, 4, 70125 Bari, Italy |
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| Cites_doi | 10.1016/j.jmaa.2021.125278 10.1007/s00605-021-01549-1 10.1090/proc/15445 10.1007/BFb0059495 10.1007/s00025-019-1012-0 |
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| Keywords | secondary Local uniform approximation Korovkin-type theorem Positive linear operator Convolution operator Bounded 2π− periodic function Bernstein-type operator primary |
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| References | Altomare (b2) 2021; 502 Stromberg (b15) 1981 Korovkin (b12) 1953; 90 Braha, Srivastava, Mohiuddine (b8) 2014; 228 Altomare, Campiti (b4) 1994 Altomare, Raşa (b6) 2012; 9 Altomare, Cappelletti Montano, Leonessa, Raşa (b5) 2014 DeVore (b10) 1972 Popa (b14) 2019; 74 Engelking (b11) 1989 Korovkin (b13) 1960 Altomare (b1) 2010; 5 Altomare (b3) 2021; 149 Bauer, Kumar, Rajan (b7) 2022; 197 Butzer, Nessel (b9) 1971 Engelking (10.1016/j.exmath.2022.06.001_b11) 1989 Korovkin (10.1016/j.exmath.2022.06.001_b13) 1960 Popa (10.1016/j.exmath.2022.06.001_b14) 2019; 74 Stromberg (10.1016/j.exmath.2022.06.001_b15) 1981 Bauer (10.1016/j.exmath.2022.06.001_b7) 2022; 197 Altomare (10.1016/j.exmath.2022.06.001_b6) 2012; 9 Altomare (10.1016/j.exmath.2022.06.001_b2) 2021; 502 Braha (10.1016/j.exmath.2022.06.001_b8) 2014; 228 Altomare (10.1016/j.exmath.2022.06.001_b1) 2010; 5 DeVore (10.1016/j.exmath.2022.06.001_b10) 1972 Butzer (10.1016/j.exmath.2022.06.001_b9) 1971 Altomare (10.1016/j.exmath.2022.06.001_b5) 2014 Korovkin (10.1016/j.exmath.2022.06.001_b12) 1953; 90 Altomare (10.1016/j.exmath.2022.06.001_b4) 1994 Altomare (10.1016/j.exmath.2022.06.001_b3) 2021; 149 |
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| SubjectTerms | Bernstein-type operator Bounded [formula omitted] periodic function Convolution operator Korovkin-type theorem Local uniform approximation Positive linear operator |
| Title | Korovkin-type theorems and local approximation problems |
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