The norm estimates for the -Bernstein operator in the case

Authors:
Heping Wang and Sofiya Ostrovska

Journal:
Math. Comp. **79** (2010), 353-363

MSC (2000):
Primary 46E15, 26A12, 47A30; Secondary 26D05, 41A10

Published electronically:
July 2, 2009

MathSciNet review:
2552230

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The -Bernstein basis with emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on In the case the behavior of the -Bernstein basic polynomials on combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of -Bernstein polynomials in the case of

The aim of this paper is to present norm estimates in for the -Bernstein basic polynomials and the -Bernstein operator in the case While for for all in the case the norm increases rather rapidly as We prove here that with Such a fast growth of norms provides an explanation for the unpredictable behavior of -Bernstein polynomials with respect to convergence.

**[1]**G.E. Andrews, R. Askey, R. Roy,*Special Functions*, Cambridge Univ. Press., Cambridge, 1999. MR**1688958 (2000g:33001)****[2]**L. C. Biedenharn,*The quantum group and a -analogue of the boson operators*, J. Phys. A: Math. Gen.**22**(1989), L873-L878. MR**1015226 (90k:17027)****[3]**W. Boehm, A. Müller,*On de Casteljau's algorithm*, Computer Aided Geometric Design**16**(1999), 587-605. MR**1718051 (2000h:65036)****[4]**L. Castellani , J. Wess (eds.),*Quantum Groups and Their Applications in Physics*, IOS Press, Amsterdam, The Netherlands, 1996. MR**1415848 (97d:81004)****[5]**Ch. A. Charalambides,*The -Bernstein basis as a -binomial distribution*, Journal of Statistical Planning and Inference (in press).**[6]**H. Gonska,*The rate of convergence of bounded linear processes on spaces of continuous functions*, Automat. Comput. Appl. Math.**7 (1)**(1998), 38-97. MR**1886377 (2003a:41026)****[7]**A. Il'inskii,*A probabilistic approach to -polynomial coefficients, Euler and Stirling numbers I*, Matematicheskaya Fizika, Analiz, Geometriya**11 (4)**(2004), 434 - 448. MR**2114004 (2005h:05019)****[8]**A. II'inskii, S.Ostrovska,*Convergence of generalized Bernstein polynomials*, J. Approx. Theory**116 (1)**(2002), 100-112. MR**1909014 (2003e:41037)****[9]**S. Jing,*The q-deformed binomial distribution and its asymptotic behaviour*, J. Phys. A: Math. Gen.**27**(1994), 493-499. MR**1267428 (95g:81080)****[10]**S. Lewanowicz, P. Woźny,*Dual generalized Bernstein basis*, J. Approx. Theory**138 (2)**(2006), 129-150. MR**2201155 (2007b:41006)****[11]**V. Lomonosov,*A counterexample to the Bishop-Phelps theorem in complex spaces*, Israel J. Math.**115**(2000), 25-28. MR**1749671 (2000k:46016)****[12]**I. Ya. Novikov,*Asymptotics of the roots of Bernstein polynomials used in the construction of modified Daubechies wavelets*, Mathematical Notes**71 (1-2)**(2002), 217-229. MR**1900797 (2002m:42043)****[13]**S. Ostrovska,*-Bernstein polynomials and their iterates*, J. Approx. Theory**123 (2)**(2003), 232-255. MR**1990098 (2004k:41038)****[14]**S. Ostrovska,*On the -Bernstein polynomials*, Advanced Studies in Contemporary Mathematics**11 (2)**(2005), 193-204. MR**2169894 (2006m:41011)****[15]**S. Ostrovska,*The approximation of logarithmic function by -Bernstein polynomials in the case*, Numerical Algorithms**44**(2007), 69-82. MR**2322145 (2008d:41003)****[16]**M. I. Ostrovskii,*Regularizability of inverse linear operators in Banach spaces with bases*, Siberian Math. J.**330 (3)**(1992), 470-476. MR**1178464 (93i:47004)****[17]**V. S. Videnskii,*On -Bernstein polynomials and related positive linear operators*, In: Problems of modern mathematics and mathematical education, Hertzen readings, St.-Petersburg, 2004, pp. 118-126 (Russian).**[18]**V.S.Videnskii,*On some classes of -parametric positive operators*, Operator Theory: Advances and Applications**158**(2005), 213-222. MR**2147598 (2006b:41034)****[19]**Wang Heping,*Korovkin-type theorem and application*, J. Approx. Theory**132 (2)**(2005), 258-264. MR**2118520 (2005k:41084)****[20]**Wang Heping, Meng Fanjun,*The rate of convergence of -Bernstein polynomials for*, J. Approx. Theory**136 (2)**(2005), 151-158. MR**2171684 (2006h:41007)****[21]**Wang Heping,*Voronovskaya type formulas and saturation of convergence for -Bernstein polynomials for*, J. Approx. Theory**145 (2)**(2007), 182-195. MR**2312464 (2008m:41008)****[22]**H. Wang, X.Z. Wu,*Saturation of convergence for -Bernstein polynomials in the case*, J. Math. Anal. Appl.**337 (1)**(2008), 744-750. MR**2356108 (2008k:41031)**

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Additional Information

**Heping Wang**

Affiliation:
School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China

Email:
wanghp@yahoo.cn

**Sofiya Ostrovska**

Affiliation:
Atilim University, Department of Mathematics, Incek 06836, Ankara, Turkey

Email:
ostrovskasofiya@yahoo.com

DOI:
http://dx.doi.org/10.1090/S0025-5718-09-02273-X

Keywords:
$q$-integers,
$q$-binomial coefficients,
$q$-Bernstein polynomials,
$q$-Bernstein operator,
operator norm,
strong asymptotic order

Received by editor(s):
December 12, 2007

Received by editor(s) in revised form:
November 7, 2008

Published electronically:
July 2, 2009

Additional Notes:
The first author was supported by National Natural Science Foundation of China (Project no. 10871132), Beijing Natural Science Foundation (1062004), and by a grant from the Key Programs of Beijing Municipal Education Commission (KZ200810028013).

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.