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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.

MSC:
41A36Approximation by positive operators
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