Abstract
Abstract This paper deals with Durrmeyer type generalization of q-Baskakov type operators using the concept of q-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [ 0 , ∞ ) . An estimate for the rate of convergence and weighted approximation properties are also obtained. MSC:41A25, 41A36.
Highlights
In the year Agrawal and Mohammad [ ] introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as ∞Dn(f, x) = n pn,k(x) sn,k– (t)f (t) dt + pn, (x)f ( ), x ∈ [, ∞), ( . ) k= where n+k– xk pn,k(x) =k ( + x)n+k, sn,k (t) = e–ntk k!It is observed in [ ] that these operators reproduce constant as well as linear functions
Proof The operators Dnq are well defined on the function, t, t
Remark If we put q =, we get the moments of a new sequence Dn(f, x) considered in [ ] as operators as
Summary
It is observed in [ ] that these operators reproduce constant as well as linear functions. For f ∈ C[ , ∞), q > and each positive integer n, the q-Baskakov operators [ ] are defined as Remark The first three moments of the q-Baskakov operators are given by
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