The Eigenstructure of Operators Linking the Bernstein and the Genuine Bernstein–Durrmeyer operators

详细信息    查看全文

• 作者：Heiner Gonska (1)
  Ioan Ra?a (2)
  Elena-Dorina St?nil? (1)
  
• 关键词：Primary 41A36 ; 15A18 ; Secondary 41A10 ; Bernstein operators ; genuine Bernstein–Durrmeyer operators ; diagonalisation ; eigenvalues ; eigenfunctions ; Stirling numbers ; iterates
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2014
• 出版时间：May 2014
• 年：2014
• 卷：11
• 期：2
• 页码：561-576
• 全文大小：
• 参考文献：1. Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and its Applications. Walter de Gruyter, Berlin (1994)
  2. Altomare F., Leonessa V., Ra?a I.: On Bernstein-Schnabl operators on the unit interval. Zeit. Anal. Anwend. 27, 353-79 (2008) CrossRef
  3. Comtet, L.: Advanced Combinatorics—the Art of Finite and Infinite Expansions. Reidel, Dordrecht (1974)
  4. Cooper S., Waldron S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105(1), 133-65 (2000) CrossRef
  5. Gonska H., Kacsó D., Ra?a I.: On genuine Bernstein–Durrmeyer operators. Results Math. 50(3-), 213-25 (2007)
  6. Gonska H., Kacsó D., Ra?a I.: The genuine Bernstein–Durrmeyer operators revisited. Results Math. 62(3-), 295-10 (2012) CrossRef
  7. Gonska, H., P?lt?nea, R.: Simultaneous approximation by a class of Bernstein–Durrmeyer operators preserving linear functions. Czechoslovak Math. J. 60(3), 783-99 (2010)
  8. Gonska H., P?lt?nea R.: Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions. Ukrainian Math. J. 62(7), 1061-072 (2010) CrossRef
  9. Goodman T.N.T., Sharma A.: A Bernstein-type operator on the simplex. Math. Balkanica (N.S.) 5(2), 129-45 (1991)
  10. P?lt?nea R.: A class of Durrmeyer type operators preserving linear functions. Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca) 5, 109-17 (2007)
  11. Ra?a I.: Estimates for the semigroup associated with Bernstein operators. Rev. Anal. Numér. Théor. Approx. 33(2), 243-45 (2004)
  12. Ra?a I.: Estimates for the semigroup associated with Bernstein-Schnabl operators. Carpathian J. Math. 20(1), 157-62 (2012)
  
• 作者单位：Heiner Gonska (1)
  Ioan Ra?a (2)
  Elena-Dorina St?nil? (1)
  
  1. Faculty of Mathematics, University of Duisburg-Essen, Forsthausweg 2, 47057, Duisburg, Germany
  2. Department of Mathematics, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, 400114, Cluj-Napoca, Romania
  
• ISSN：1660-5454

文摘

We study the eigenstructure of a one-parameter class of operators \({U_{n}^{\varrho}}\) of Bernstein–Durrmeyer type that preserve linear functions and constitute a link between the so-called genuine Bernstein–Durrmeyer operators U n and the classical Bernstein operators B n . In particular, for \({\varrho\rightarrow\infty}\) (respectively, \({\varrho=1}\) ) we recapture results well-known in the literature, concerning the eigenstructure of B n (respectively, U n ). The last section is devoted to applications involving the iterates of \({U_{n}^{\varrho}}\) .