Mediterr. J. Math.
Springer International Publishing AG 2017
On Approximation Properties of Phillips
Operators Preserving Exponential Functions
Vijay Gupta and Gancho Tachev
Abstract. In the present paper, we study a modiﬁcation of the Phillips
operators, which reproduces constant and the exponential functions. We
obtain the moments using the concept of moment-generating function
for the Phillips operators. Here we discuss a uniform convergence esti-
mate for this modiﬁed forms. Also some direct estimates, which also
involve the asymptotic-type result are established.
Mathematics Subject Classiﬁcation. 41A25, 41A36.
Keywords. Phillips operators, moment generating function, exponential
functions, moments, quantitative results.
The Phillips operator  is deﬁned as
(k − 1)!
f (t)dt + e
These operators preserve constant as well linear functions. Some approx-
imation results on these operators have been discussed in [4,11,12]. After the
work of King  on the well-known Bernstein polynomials, in the year 2010
the author  modiﬁed the Phillips operators so as to preserve the test func-
. It was observed that the modiﬁed form provides better approximation
over the usual Phillips operators. By simple computation, we have
n − θ
which is the moment generating-function (abbrev. m.g.f.) of the operators S
and it may be utilized to ﬁnd the moments of the Phillips operators. Using
the software Mathematica, we ﬁnd ﬁrst few moments as below: