Abstract
AbstractBinomial operators are the most important extension to Bernstein operators, defined by $$ \bigl(L^{Q}_{n} f\bigr) (x)=\frac{1}{b_{n}(1)} \sum ^{n}_{k=0}\binom { n}{k } b_{k}(x)b_{n-k}(1-x)f\biggl( \frac{k}{n}\biggr),\quad f\in C[0, 1], $$
(
L
n
Q
f
)
(
x
)
=
1
b
n
(
1
)
∑
k
=
0
n
(
n
k
)
b
k
(
x
)
b
n
−
k
(
1
−
x
)
f
(
k
n
)
,
f
∈
C
[
0
,
1
]
,
where $\{b_{n}\}$
{
b
n
}
is a sequence of binomial polynomials associated to a delta operator Q. In this paper, we discuss the binomial operators $\{L^{Q}_{n} f\}$
{
L
n
Q
f
}
preservation such as smoothness and semi-additivity by the aid of binary representation of the operators, and present several illustrative examples. The results obtained in this paper generalize what are known as the corresponding Bernstein operators.
Funder
Supported by the Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
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