On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu
In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation. By using the theory of polynomials of binomial type \((p_m)\) he has associated to a function \(f\in C[0,1]\) a linear operator defined by the formula \[ \left( T_m...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
2001-02-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Online Access: | https://ictp.acad.ro/jnaat/journal/article/view/687 |
| _version_ | 1811291727210741760 |
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| author | Dimitrie D. Stancu |
| author_facet | Dimitrie D. Stancu |
| author_sort | Dimitrie D. Stancu |
| collection | DOAJ |
| description |
In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation. By using the theory of polynomials of binomial type \((p_m)\) he has associated to a function \(f\in C[0,1]\) a linear operator defined by the formula
\[
\left( T_m f\right) (x) = \tfrac{1}{p_m(1)} \textstyle\sum\limits _{k=0} ^m \tbinom{m}{k}
p_k (x) p_{m-k} (1-x) f\big(\tfrac{k}{m}\big).
\]
Examples of such operators were considered in several subsequent papers.
In this paper we present a convergence theorem corresponding to the sequence \(\left( T_mf\right)\) and we also present a more general sequence of operators of approximation \(S_{m,r,s}\), where \(r\) and \(s\) are nonnegative integers such that \(2sr\leq m\).
We give an integral expression for the remainders, as well as a representation by using divided differences of second order.
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| first_indexed | 2024-04-13T04:33:41Z |
| format | Article |
| id | doaj.art-ebc208dbea7c4ebfadbe61bf60bead1f |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-04-13T04:33:41Z |
| publishDate | 2001-02-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-ebc208dbea7c4ebfadbe61bf60bead1f2022-12-22T03:02:14ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2001-02-01301On the approximation of functions by means of the operators of binomial type of Tiberiu PopoviciuDimitrie D. Stancu0Babeş-Bolyai University, Cluj-Napoca In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation. By using the theory of polynomials of binomial type \((p_m)\) he has associated to a function \(f\in C[0,1]\) a linear operator defined by the formula \[ \left( T_m f\right) (x) = \tfrac{1}{p_m(1)} \textstyle\sum\limits _{k=0} ^m \tbinom{m}{k} p_k (x) p_{m-k} (1-x) f\big(\tfrac{k}{m}\big). \] Examples of such operators were considered in several subsequent papers. In this paper we present a convergence theorem corresponding to the sequence \(\left( T_mf\right)\) and we also present a more general sequence of operators of approximation \(S_{m,r,s}\), where \(r\) and \(s\) are nonnegative integers such that \(2sr\leq m\). We give an integral expression for the remainders, as well as a representation by using divided differences of second order. https://ictp.acad.ro/jnaat/journal/article/view/687 |
| spellingShingle | Dimitrie D. Stancu On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu Journal of Numerical Analysis and Approximation Theory |
| title | On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu |
| title_full | On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu |
| title_fullStr | On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu |
| title_full_unstemmed | On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu |
| title_short | On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu |
| title_sort | on the approximation of functions by means of the operators of binomial type of tiberiu popoviciu |
| url | https://ictp.acad.ro/jnaat/journal/article/view/687 |
| work_keys_str_mv | AT dimitriedstancu ontheapproximationoffunctionsbymeansoftheoperatorsofbinomialtypeoftiberiupopoviciu |