# 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)
Positive linear operators generated by analytic functions. (English) Zbl 1236.41025

Summary: Let $\varphi$ be a power series with positive Taylor coefficients ${\left\{ak\right\}}_{k}={0}^{\infty }$ and non-zero radius of convergence $r\le \infty$. Let ${\xi }_{x},0\le x be a random variable whose values $\alpha k$ , $k=0,1,...$, are independent of $x$ and taken with probabilities ${a}_{k}{x}^{k}/\varphi \left(x\right)$, $k=0,1,...$.

The positive linear operator $\left({A}_{\varphi }f\right)\left(x\right):=𝐄\left[f\left({\xi }_{x}\right)\right]$ is studied. It is proved that if $𝐄\left({\xi }_{x}\right)=x$, $𝐄\left({\xi }_{x}^{2}\right)=q{x}^{2}+bx+c,\phantom{\rule{1.em}{0ex}}q,b,c\in 𝐑,q>0$, then ${A}_{\varphi }$ reduces to the Szász-Mirakyan operator in the case $q=1$, to the limit $q$-Bernstein operator in the case $0, and to a modification of the Lupaş operator in the case $q>1$.

##### MSC:
 41A36 Approximation by positive operators
##### References:
 [1] Altomare F and Campiti M, Korovkin-type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics, 17 (Berlin: Walter de Gruyter & Co.) (1994) [2] Andrews G E, Askey R and Roy R, Special Functions (Cambridge: Cambridge Univ. Press) (1999) [3] Feller W, An Introduction to Probability Theory and Its Applications. 2nd ed. (New-York: Wiley) (1968) [4] Il’inskii A and Ostrovska S, Convergence of generalized Bernstein polynomials, J. Approx. Theory 116 (2002) 100–112 · Zbl 0999.41007 · doi:10.1006/jath.2001.3657 [5] Karlin S, Total Positivity (Stanford, California: Stanford Univ. Press) (1968) vol. I [6] Lupaş A, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, Nr. 9, (1987) [7] Mirakyan G, Approximation des fonctions continues au moyen polynômes de la forme ex ${\Sigma }$ k=0 m C k,n x k, Dokl. Acad. Sci. USSR (N.S.) 31 (1941) 201–205 [8] Ostrovska S, On the limit q-Bernstein operator, Mathematica Balkanica 18 (2004) 165–172 [9] Ostrovska S, On the improvement of analytic properties under the limit q-Bernstein operator, J. Approx. Theory 138 (2006) 37–53 · Zbl 1098.41006 · doi:10.1016/j.jat.2005.09.015 [10] Ostrovska S, On the Lupaş q-analogue of the Bernstein operator, Rocky Mountain J. Math. 36(5) (2006) 1615–1629 · Zbl 1138.41008 · doi:10.1216/rmjm/1181069386 [11] Szász O, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Research. Nat. Bur. Standards 45 (1950) 239–245 [12] Trif T, Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numér. Théor. Approx. 29(2) (2000) 221–229 [13] Videnskii V S, On q-Bernstein polynomials and related positive linear operators, in: Problems of modern mathematics and mathematical education, Hertzen readings (St.-Petersburg) (2004) pp. 118–126 (Russian) [14] Videnskii V S, On some classes of q-parametric positive operators, Op. Theory, Advances and Appl. 158 (2005) 213–222 · doi:10.1007/3-7643-7340-7_15 [15] Wang H, Korovkin-type theorem and application, J. Approx. Theory 132(2) (2005) 258–264 · Zbl 1118.41015 · doi:10.1016/j.jat.2004.12.010 [16] Wang H and Meng F, The rate of convergence of q-Bernstein polynomials for 0 < q < 1, J. Approx. Theory 136(2) (2005) 151–158 · Zbl 1082.41007 · doi:10.1016/j.jat.2005.07.001