q-Bernstein polynomials and their iterates

AbstractLet Bn(f,q;x),n=1,2,… be q-Bernstein polynomials of a function f:[0,1]→C. The polynomials Bn(f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z:|z|<q+ε} the rate of convergence of {Bn(f,q;x)} to f(x) in the norm of C[0,1] has the order q−n (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn(f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q∈(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞

On the Lupaş q-transform

AbstractThe Lupaş q-transform emerges in the study of the limit q-Lupaş operator. The latter comes out naturally as a limit for a sequence of the Lupaş q-analogues of the Bernstein operator. Lately, it has been studied by several authors from different perspectives in mathematical analysis and approximation theory. This operator is closely related to the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus.Given q∈(0,1),f∈C[0,1], the q-Lupaş transform of f is defined by: (Λqf)(z)≔1(−z;q)∞⋅∑k=0∞f(1−qk)qk(k−1)/2(q;q)kzk. In this paper, we study some analytic properties of (Λqf)(z). In particular, we examine the conditions under which Λqf can either be an entire function, or a rational one