# zbMATH — the first resource for mathematics

Positive linear operators generated by analytic functions. (English) Zbl 1236.41025
Summary: Let $$\varphi$$ be a power series with positive Taylor coefficients $$\{ak\}_k=0^{\infty}$$ and non-zero radius of convergence $$r \leq \infty$$. Let $$\xi_x, 0 \leq x < r$$ be a random variable whose values $$\alpha k$$ , $$k = 0, 1, \ldots$$, are independent of $$x$$ and taken with probabilities $$a_k x^k /\varphi(x)$$, $$k = 0, 1, \ldots$$.
The positive linear operator $$(A_\varphi f)(x):= \mathbf E[f(\xi_x)]$$ is studied. It is proved that if $$\mathbf E(\xi_x) = x$$, $$\mathbf E(\xi_x ^{2}) = qx^{2} + bx + c,\quad q, b, c \in \mathbf R, q > 0$$, then $$A_\varphi$$ reduces to the Szász-Mirakyan operator in the case $$q = 1$$, to the limit $$q$$-Bernstein operator in the case $$0 < q < 1$$, and to a modification of the Lupaş operator in the case $$q > 1$$.

##### MSC:
 41A36 Approximation by positive operators
Full Text:
##### References:
  Altomare F and Campiti M, Korovkin-type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics, 17 (Berlin: Walter de Gruyter & Co.) (1994) · Zbl 0924.41001  Andrews G E, Askey R and Roy R, Special Functions (Cambridge: Cambridge Univ. Press) (1999)  Feller W, An Introduction to Probability Theory and Its Applications. 2nd ed. (New-York: Wiley) (1968) · Zbl 0155.23101  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  Karlin S, Total Positivity (Stanford, California: Stanford Univ. Press) (1968) vol. I  Lupaş A, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, Nr. 9, (1987)  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  Ostrovska S, On the limit q-Bernstein operator, Mathematica Balkanica 18 (2004) 165–172  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  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  Szász O, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Research. Nat. Bur. Standards 45 (1950) 239–245  Trif T, Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numér. Théor. Approx. 29(2) (2000) 221–229  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)  Videnskii V S, On some classes of q-parametric positive operators, Op. Theory, Advances and Appl. 158 (2005) 213–222 · Zbl 1088.41008 · doi:10.1007/3-7643-7340-7_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  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.