Analytical properties of the Lupas q-transform

作者：Ostrovska Sofiya^*

来源：Journal of Mathematical Analysis and Applications, 2012, 394(1): 177-185.

DOI：10.1016/j.jmaa.2012.04.047

• q-integers
• q-binomial theorem
• Lupas q-analogue of the Bernstein operator
• Lupas q-transform
• Analytic function
• Meromorphic function

The Lupas q-transform emerges in the study of the limit q-Lupas operator. The latter comes out naturally as a limit for a sequence of the Lupas q-analogues of the Bernstein operator. Given q is an element of (0, 1), f is an element of C left perpendicular0, 1right perpendicular, the q-Lupas transform off is defined by
(Lambda(q)f) (z) := 1/(-z; q)(infinity) . Sigma(infinity)(k=0) f(1 - q(k))q(k(k -1)/2)/(q; q)(k)z(k).
The transform is closely related to both the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus, and to Valiron's method of summation for divergent series. In general, Lambda(q)f is a meromorphic function whose poles are contained in the set J(q) := {-q(-j)}(j=0)(infinity).
In this paper, we study the connection between the behaviour of f on leftperpendicular0, 1right perpendicular and the decay of Lambda(q)f as z -> infinity.

• 出版日期2012-10-1