Abstract
Let φ be a power series with positive Taylor coefficients {a k } ∞ k=0 and non-zero radius of convergence r ≤ ∞. Let ξ x , 0 ≤ x < r be a random variable whose values α k , 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 2 x ) = qx 2 + bx + c, q, b, c ∈ R, 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.
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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)
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)
Il’inskii A and Ostrovska S, Convergence of generalized Bernstein polynomials, J. Approx. Theory 116 (2002) 100–112
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 e−nx Σ m k=0 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
Ostrovska S, On the Lupaş q-analogue of the Bernstein operator, Rocky Mountain J. Math. 36(5) (2006) 1615–1629
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
Wang H, Korovkin-type theorem and application, J. Approx. Theory 132(2) (2005) 258–264
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
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Ostrovska, S. Positive linear operators generated by analytic functions. Proc Math Sci 117, 485–493 (2007). https://doi.org/10.1007/s12044-007-0040-y
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DOI: https://doi.org/10.1007/s12044-007-0040-y