The sharpness of convergence results for q -Bernstein polynomials in the case q > 1

Sofiya Ostrovska

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1195-1206
  • ISSN: 0011-4642

Abstract

top
Due to the fact that in the case q > 1 the q -Bernstein polynomials are no longer positive linear operators on C [ 0 , 1 ] , the study of their convergence properties turns out to be essentially more difficult than that for q < 1 . In this paper, new saturation theorems related to the convergence of q -Bernstein polynomials in the case q > 1 are proved.

How to cite

top

Ostrovska, Sofiya. "The sharpness of convergence results for $q$-Bernstein polynomials in the case $q>1$." Czechoslovak Mathematical Journal 58.4 (2008): 1195-1206. <http://eudml.org/doc/37896>.

@article{Ostrovska2008,
abstract = {Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q<1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.},
author = {Ostrovska, Sofiya},
journal = {Czechoslovak Mathematical Journal},
keywords = {$q$-integers; $q$-binomial coefficients; $q$-Bernstein polynomials; uniform convergence; analytic function; Cauchy estimates; -integers; -binomial coefficients; -Bernstein polynomials; uniform convergence; analytic function; Cauchy estimates},
language = {eng},
number = {4},
pages = {1195-1206},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The sharpness of convergence results for $q$-Bernstein polynomials in the case $q>1$},
url = {http://eudml.org/doc/37896},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Ostrovska, Sofiya
TI - The sharpness of convergence results for $q$-Bernstein polynomials in the case $q>1$
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1195
EP - 1206
AB - Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q<1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.
LA - eng
KW - $q$-integers; $q$-binomial coefficients; $q$-Bernstein polynomials; uniform convergence; analytic function; Cauchy estimates; -integers; -binomial coefficients; -Bernstein polynomials; uniform convergence; analytic function; Cauchy estimates
UR - http://eudml.org/doc/37896
ER -

References

top
  1. Cao, J.-D., 10.1006/jmaa.1997.5349, J. Math. Anal. Appl. 209 (1997), 140-146. (1997) Zbl0879.41010MR1444517DOI10.1006/jmaa.1997.5349
  2. Cao, J.-D., Gonska, H., Kacsó, D., On the impossibility of certain lower estimates for sequences of linear operators, Math. Balkanica 19 (2005), 39-58. (2005) MR2119784
  3. Cheney, E. W., Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 5 (1964), 77-84. (1964) Zbl0146.08202MR0198074
  4. Cooper, S., Waldron, S., 10.1006/jath.2000.3464, J. Approx. Theory 105 (2000), 133-165. (2000) Zbl0963.41006MR1768528DOI10.1006/jath.2000.3464
  5. Derriennic, M.-M., Modified Bernstein polynomials and Jacobi polynomials in q -calculus, Rendiconti Del Circolo Matematico Di Palermo, Serie II, Suppl. 76 (2005), 269-290. (2005) Zbl1142.41002MR2178441
  6. Gonska, H., The rate of convergence of bounded linear processes on spaces of continuous functions, Automat. Comput. Appl. Math. 7 (1999), 38-97. (1999) MR1886377
  7. Gupta, V., 10.1016/j.amc.2007.07.056, Appl. Math. Comput. 197 (2008), 172-178. (2008) Zbl1142.41008MR2396302DOI10.1016/j.amc.2007.07.056
  8. Gupta, V., Wang, H., The rate of convergence of q -Durrmeyer operators for 0 < q < 1 , Math. Meth. Appl. Sci. (2008). (2008) MR2447215
  9. Habib, A., Umar, S., On generalized Bernstein polynomials, Indian J. Pure Appl. Math. 11 (1980), 177-189. (1980) Zbl0443.41015MR0571065
  10. Il'inskii, A., Ostrovska, S., 10.1006/jath.2001.3657, J. Approx. Theory 116 (2002), 100-112. (2002) Zbl0999.41007MR1909014DOI10.1006/jath.2001.3657
  11. Lorentz, G. G., Bernstein Polynomials, Chelsea, New York (1986). (1986) Zbl0989.41504MR0864976
  12. Lupaş, A., A q -analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, No. 9 (1987). (1987) 
  13. Novikov, I. Ya., 10.1023/A:1013959231555, Mathematical Notes 71 (2002), 217-229. (2002) Zbl1034.42040MR1900797DOI10.1023/A:1013959231555
  14. Ostrovska, S., 10.1016/S0021-9045(03)00104-7, J. Approx. Theory 123 (2003), 232-255. (2003) Zbl1093.41013MR1990098DOI10.1016/S0021-9045(03)00104-7
  15. Ostrovska, S., On the q -Bernstein polynomials, Advanced Studies in Contemporary Mathematics 11 (2005), 193-204. (2005) Zbl1116.41013MR2169894
  16. Ostrovska, S., 10.1016/j.jat.2005.09.015, J. Approx. Theory 138 (2006), 37-53. (2006) Zbl1098.41006MR2197601DOI10.1016/j.jat.2005.09.015
  17. Petrone, S., 10.1111/1467-9469.00155, Scan. J. Statist. 26 (1999), 373-393. (1999) Zbl0939.62046MR1712051DOI10.1111/1467-9469.00155
  18. Videnskii, V. S., 10.1007/3-7643-7340-7_15, Operator Theory, Advances and Applications, Vol. 158 (2005), 213-222. (2005) MR2147598DOI10.1007/3-7643-7340-7_15
  19. Videnskii, V. S., On the polynomials with respect to the generalized Bernstein basis, In: Problems of modern mathematics and mathematical education, Hertzen readings. St.-Petersburg (2005), 130-134 Russian. (2005) 
  20. Wang, H., 10.1016/j.jat.2004.12.010, J. Approx. Theory 132 (2005), 258-264. (2005) Zbl1118.41015MR2118520DOI10.1016/j.jat.2004.12.010
  21. Wang, H., 10.1016/j.jat.2006.08.005, J. Approx. Theory 145 (2007), 182-195. (2007) Zbl1112.41016MR2312464DOI10.1016/j.jat.2006.08.005
  22. Wang, H., Wu, X. Z., 10.1016/j.jmaa.2007.04.014, J. Math. Anal.Appl. 337 (2008), 744-750. (2008) MR2356108DOI10.1016/j.jmaa.2007.04.014
  23. Wang, H., 10.1016/j.jmaa.2007.09.004, J. Math. Anal. Appl. 340 (2008), 1096-1108. (2008) Zbl1144.41004MR2390913DOI10.1016/j.jmaa.2007.09.004

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.