zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The saturation of convergence on the interval [0,1] for the $q$-Bernstein polynomials in the case $q>1$. (English) Zbl 1236.41011
Summary: We consider saturation of convergence on the interval $[0,1]$ for the $q$-Bernstein polynomials of a continuous function $f$ for arbitrary fixed $q>1$. We show that the rate of uniform convergence on $[0,1]$ is $o(q ^{- n})$ if and only if $f$ is linear. The result is sharp in the following sense: it ceases to be true if we replace “$o$” by “$O$”.

MSC:
41A10Approximation by polynomials
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
WorldCat.org
Full Text: DOI
References:
[1] Bernstein, S. N.: Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné, Mém. acad. Belg. 4, No. 2, 1-103 (1912) · Zbl 45.0633.03
[2] Kac, V.; Cheung, P.: Quantum calculus, (2002) · Zbl 0986.05001
[3] Ostrovska, S.: Q-Bernstein polynomials and their iterates, J. approx. Theory 123, No. 2, 232-255 (2003) · Zbl 1093.41013 · doi:10.1016/S0021-9045(03)00104-7
[4] Ostrovska, S.: The first decade of the q-Bernstein polynomials: results and perspectives, J. math. Anal. approx. Theory 2, No. 1, 35-51 (2007) · Zbl 1159.41301
[5] Ostrovska, S.: The sharpness of convergence results for the q-Bernstein polynomials in the case q>1, Czechoslovak math. J. 58, No. 4, 1195-1206 (2008) · Zbl 1174.41010 · doi:10.1007/s10587-008-0079-7
[6] Phillips, G. M.: Bernstein polynomials based on the q-integers, Ann. numer. Math. 4, 511-518 (1997) · Zbl 0881.41008
[7] Wang, H.: Voronovskaya type formulas and saturation of convergence for q-Bernstein polynomials for 0<q<1, J. approx. Theory 145, No. 2, 182-195 (2007) · Zbl 1112.41016 · doi:10.1016/j.jat.2006.08.005
[8] Wang, H.; Wu, X.: Saturation of convergence for q-Bernstein polynomials in the case q>1, J. math. Anal. appl. 337, No. 1, 744-750 (2008) · Zbl 1122.33014 · doi:10.1016/j.jmaa.2007.04.014