Shape-Preserving and Convergence Properties for theq-Szász-Mirakjan Operators for Fixedq∈(0,1)
Keyword(s):
We introduce aq-generalization of Szász-Mirakjan operatorsSn,qand discuss their properties for fixedq∈(0,1). We show that theq-Szász-Mirakjan operatorsSn,qhave good shape-preserving properties. For example,Sn,qare variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixedq∈(0,1), we prove that the sequence{Sn,qf}converges toB∞,q(f)uniformly on[0,1]for eachf∈C[0, 1/(1-q)], whereB∞,qis the limitq-Bernstein operator. We obtain the estimates for the rate of convergence for{Sn,qf}by the modulus of continuity off, and the estimates are sharp in the sense of order for Lipschitz continuous functions.
2008 ◽
Vol 40
(03)
◽
pp. 651-672
◽
2001 ◽
Vol 131
(3)
◽
pp. 667-700
2008 ◽
Vol 52
(2)
◽
pp. 583-590
◽
1988 ◽
Vol 57
(2)
◽
pp. 307-322
◽