The q-derivative and applications to q-Szász Mirakyan operators
- 216 Downloads
By using the properties of the q-derivative, we show that q-Szász Mirakyan operators are convex, if the function involved is convex, generalizing well-known results for q = 1. We also show that q-derivatives of these operators converge to q-derivatives of approximated functions. Futhermore, we give a Voronovskaya-type theorem for monomials and provide a Stancu-type form for the remainder of the q-Szász Mirakyan operator. Lastly, we give an inequality for a convex function f, involving a connection between two nonconsecutive terms of a sequence of q-Szász Mirakyan operators.
KeywordsHypergeometric Series Bernstein Polynomial Divided Difference Positive Linear Operator Bernstein Operator
Unable to display preview. Download preview PDF.
- 1. I. Babuska. The finite element method with Lagrange multipliers. Numer. Math., 1973.Google Scholar
- 2. Alan Berger, Ridgway Scott, and Gilbert Strang. Approximate boundary conditions in the finite element method. In Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972), pages 295–313. Academic Press, London, 1972.Google Scholar
- 3. S. Bertoluzza. A relevant property of approximation spaces. C.R. Acad. Sci. Paris, t. 329, Série I:1097–1102, 1999.Google Scholar
- 4. S. Bertoluzza. Analysis of a stabilized three fields domain decomposition method. Technical Report 1175, I.A.N.-C.N.R., 2000. Numer. Math., to appear.Google Scholar
- 5. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, 1991.Google Scholar
- 6. F. Brezzi and D. Marini. A three-field domain decomposition method. In A. Quarteroni, J. Periaux, Y.A. Kuznetsov, and O.B. Widlund, editors, Domain Decomposition Methods in Science and Engineering, volume 157 of American Mathematical Society, Contemporary Mathematics, pages 27–34, 1994.Google Scholar
- 7. Tony F. Chan and Tarek P. Mathew. Domain decomposition algorithms. In Acta numerica, 1994, Acta Numer., pages 61–143. Cambridge Univ. Press, Cambridge, 1994.Google Scholar
- 8. P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1976.Google Scholar
- 9. P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, London, 1985.Google Scholar
- 10. J.A. Nitsche and A.H. Schatz. Interior estimates for Ritz-Galerkin method. Math. of Comp., 28(128):937–958, 1974.Google Scholar
- 11. Barry F. Smith, Petter E. Bjørstad, and William D. Gropp. Domain decomposition. Cambridge University Press, Cambridge, 1996. Parallel multilevel methods for elliptic partial differential equations.Google Scholar