In this paper, we discuss binomial operators structure properties, such as moments representation, derivatives representation, and binary representation and introduce some applications in preservation.
In this article, the eigenvalues and eigenvectors of positive binomial operators are presented. The results generalize the previously obtained ones related to Bernstein operators. Illustrative examples are supplied.
Binomial 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
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1
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∑
k
=
0
n
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n
k
)
b
k
(
x
)
b
n
−
k
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1
−
x
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f
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k
n
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,
f
∈
C
[
0
,
1
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,
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.
This paper is devoted to exploring asymptotic behavior of the Szász–Mirakian-type operators including the Szász–Mirakian operators Sn(f;x) and its two kinds of integral modifications [Formula: see text], [Formula: see text]. For these operators, we obtain the complete asymptotic expansions in the form of [Formula: see text] all coefficients ak (f;x) of n-k (k=1,2,…) are calculated explicitly in terms of the Stirling numbers of the first and second kind.
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