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

Summary: Let φ be a power series with positive Taylor coefficients {ak} k =0 and non-zero radius of convergence r. Let ξ x ,0x<r be a random variable whose values αk , k=0,1,..., are independent of x and taken with probabilities a k x k /φ(x), k=0,1,....

The positive linear operator (A φ f)(x):=𝐄[f(ξ x )] is studied. It is proved that if 𝐄(ξ x )=x, 𝐄(ξ x 2 )=qx 2 +bx+c,q,b,c𝐑,q>0, then A φ 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.

41A36Approximation by positive operators
[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 Σ 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