On the Properties of the Limit <i>q</I>-bernstein Operator

Abstract

The limit q-Bernstein operator B-q = B-infinity,B-q : C [0, 1]. C [0, 1] emerges naturally as a q-version of the Szasz-Mirakyan operator related to the q-deformed Poisson distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. The limit q-Bernstein operator has been widely studied lately. It has been shown that B-q is a positive shape-preserving linear operator on Cinverted right perpendicular0, 1inverted left perpendicular with. parallel to B-q parallel to = 1. Its approximation properties, probabilistic interpretation, behavior of iterates, and the impact on the smoothness have been examined. In this paper, it is shown that the possibility of an analytic continuation of B(q)f into {z : vertical bar z vertical bar < R}, R > 1, implies the smoothness of f at 1, which is stronger when R is greater. If B(q)f can be extended to an entire function, then f is infinitely differentiable at 1, and a sufficiently slow growth of B(q)f implies analyticity of f in {z : vertical bar z-1 vertical bar < delta}, where delta is greater when the growth is slower. Finally, there is a bound for the growth of B(q)f which implies f to be an entire function.