Classical Kantorovich operators revisited
UDC 517.5 The main object of this paper is to improve some known estimates for the classical Kantorovich operators. We obtain a quantitative Voronovskaya-type result in terms of the second moduli of continuity, which improves some previous results. In order to explain the nonmultiplicativity of the...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507252725645312 |
|---|---|
| author | Acu, A.-M. Gonska, H. Асу, А.-М. Гонська, Х. |
| author_facet | Acu, A.-M. Gonska, H. Асу, А.-М. Гонська, Х. |
| author_sort | Acu, A.-M. |
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| collection | OJS |
| datestamp_date | 2019-12-05T08:56:42Z |
| description | UDC 517.5
The main object of this paper is to improve some known estimates for the classical Kantorovich operators. We obtain a
quantitative Voronovskaya-type result in terms of the second moduli of continuity, which improves some previous results.
In order to explain the nonmultiplicativity of the Kantorovich operators, we present a Chebyshev – Gr¨uss inequality. Two
Gr¨uss –Voronovskaya theorems for Kantorovich operators are also considered. |
| first_indexed | 2026-03-24T02:06:22Z |
| format | Article |
| fulltext |
UDC 517.5
A.-M. Acu (Lucian Blaga Univ. Sibiu, Romania),
H. Gonska (Univ. Duisburg-Essen, Germany)
CLASSICAL KANTOROVICH OPERATORS REVISITED*
ЗНОВУ ПРО КЛАСИЧНI ОПЕРАТОРИ КАНТОРОВИЧА
The main object of this paper is to improve some known estimates for the classical Kantorovich operators. We obtain a
quantitative Voronovskaya-type result in terms of the second moduli of continuity, which improves some previous results.
In order to explain the nonmultiplicativity of the Kantorovich operators, we present a Chebyshev – Grüss inequality. Two
Grüss – Voronovskaya theorems for Kantorovich operators are also considered.
Основним об’єктом дослiдження є полiпшення деяких вiдомих оцiнок для класичних операторiв Канторовича.
Отримано кiлькiсний результат типу Вороновської в термiнах других модулiв неперервностi, що полiпшує де-
якi попереднi результати. Щоб пояснити немультиплiкативнiсть операторiв Канторовича, ми наводимо нерiвнiсть
Чебишова – Грюсса. Також розглянуто теореми Грюсса – Вороновської для операторiв Канторовича.
1. Introduction. In 1930 L. V. Kantorovich [11] introduced a significant modification of the classical
Bernstein operators given by
Kn(f ;x) = (n+ 1)
n\sum
k=0
pn,k(x)
k+1
n+1\int
k
n+1
f(t)dt.
Here n \geq 1, f \in L1[0, 1], x \in [0, 1] and
pn,k(x) =
\biggl(
n
k
\biggr)
xk(1 - x)n - k, 0 \leq k \leq n,
pn,k \equiv 0, if k < 0 or k > n.
These mappings are relevant since they provide a constructive tool to approximate any function in
Lp[0, 1], 1 \leq p < \infty , in the Lp-norm. For p = \infty , C[0, 1] has to be used instead of L\infty [0, 1].
These classical Kantorovich operators have been attracting a lot of attention since then, but
results on them are somehow scattered in the literature. They share this with other relevant variations
of the Bernstein-type: Durrmeyer, genuine Bernstein – Durrmeyer and, last but not least, variation-
diminishing Schoenberg splines.
In the present note we first collect and improve some of the known estimates by giving quite a
precise inequality for f \in Cr[0, 1], r \in \BbbN \cup \{ 0\} , a new Voronovskaya result in terms of \omega 2 and a
Chebyshev – Grüss inequality giving an explanation of their nonmultiplicativity. The last part of this
article deals with two Grüss – Voronovskaya theorems for Kantorovich operators.
Most estimates in this article will be given in terms of moduli of smoothness of higher order. In
the background, but not explicitly mentioned, is always the K -functional technique. In this sense we
were very much influenced by the work of Zygmund (see, e.g., [16]), a hardly accessible conference
contribution of Peetre [12] and also by the book of Dzyadyk [4].
* The first author was supported by Lucian Blaga University of Sibiu research under grant LBUS-IRG-2018-04. The
second appreciates financial support of the University of Duisburg-Essen.
c\bigcirc A.-M. ACU, H. GONSKA, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6 739
740 A.-M. ACU, H. GONSKA
2. Some previous results. In this section we collect some results given earlier. Quite a strong
general result was given by the second author and Xin-long Zhou [10] in 1995.
Let \varphi (x) =
\sqrt{}
x(1 - x) and P (D) be the differential operator given by
P (D)f := (\varphi 2f \prime )\prime , f \in C2[0, 1].
For f \in Lp[0, 1], 1 \leq p \leq \infty , the functional K(f, t)p is defined as below
K(f, t)p := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
\| f - g\| p + t2\| P (D)g\| p : g \in C2[0, 1]
\bigr\}
.
Using the above functional in [10] the following theorem was proved.
Theorem 2.1. There exists an absolute positive constant C such that for all f \in Lp[0, 1], 1 \leq
\leq p \leq \infty , there holds
C - 1K(f, n - 1/2)p \leq \| f - Knf\| p \leq CK(f, n - 1/2)p.
Also, in order to characterize the K -functional used in Theorem 2.1, the next result was given
in [10].
Theorem 2.2. We have
K(f, t)p \sim \omega 2
\varphi (f, t)p + t2E0(f)p, 1 < p < \infty ,
and
K(f, t)\infty \sim \omega 2
\varphi (f, t)\infty + \omega (f, t2)\infty .
Here \omega (f, t)p is the classical modulus, \omega 2
\varphi (f, t)\infty denotes the second order modulus of smoothness
with weight function \varphi and E0(f)p is the best approximation constant of f defined by
E0(f)p = \mathrm{i}\mathrm{n}\mathrm{f}
c
\| f - c\| p.
Moreover, all quantities subscripted by \infty are taken with respect to the uniform norm in C[0, 1].
The following theorem of Păltănea [13] is the key to give a more explicit result in terms of classical
moduli for continous functions. (See [8] for details.)
Theorem 2.3 [13]. If L : C[0, 1] \rightarrow C[0, 1] is a positive linear operator, then for f \in C[0, 1],
x \in [0, 1] and each 0 < h \leq 1
2
the following holds:
| L(f ;x) - f(x)| \leq | L(e0;x) - 1| | f(x)| + 1
h
| L(e1 - x;x)| \omega (f ;h)+
+
\biggl[
(Le0)(x) +
1
2h2
L((e1 - x)2;x)
\biggr]
\omega 2(f ;h).
The condition h \leq 1/2 can be eliminated for operators L reproducing linear functions.
Theorem 2.4. For all f \in C[0, 1] and all n \geq 4,
\| Knf - f\| \infty \leq 1
2
\surd
n
\omega 1
\biggl(
f ;
1\surd
n
\biggr)
+
9
8
\omega 2
\biggl(
f ;
1\surd
n
\biggr)
.
This result can be extended to simultaneous approximation, see again [8].
Theorem 2.5. Let r \in \BbbN 0, n \geq 4, f \in Cr[0, 1]. Then
\| DrKnf - Drf\| \infty \leq (r + 1)r
2n
\| Drf\| \infty +
r + 1
2
\surd
n
\omega 1
\biggl(
Drf ;
1\surd
n
\biggr)
+
9
8
\omega 2
\biggl(
Drf ;
1\surd
n
\biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
CLASSICAL KANTOROVICH OPERATORS REVISITED 741
3. A quantitative Voronovskaya result. This part has its predecessor in a hardly known
booklet of Videnskij in which a quantitative version of the well-known Voronovskaya theorem for
the classical Bernstein operators can be found (see [15]). This estimate was generalized and improved
in [9]. An application for Kantorovich operators was given in [8]. Here we improve it as follows:
Theorem 3.1. For n \geq 1 and f \in C2[0, 1], one has\bigm\| \bigm\| \bigm\| \bigm\| n (Knf - f) - 1
2
\bigl(
Xf \prime \bigr) \prime \bigm\| \bigm\| \bigm\| \bigm\|
\infty
\leq 2
3(n+ 1)
\biggl(
3
4
\| f \prime \| \infty + \| f \prime \prime \| \infty
\biggr)
+
+
9
32
\biggl\{
2\surd
n+ 1
\omega 1
\biggl(
f \prime \prime ;
1\surd
n+ 1
\biggr)
+ \omega 2
\biggl(
f \prime \prime ;
1\surd
n+ 1
\biggr) \biggr\}
, (1)
where X = x(1 - x) and X \prime = 1 - 2x, x \in [0, 1].
Proof. From [9] (Theorem 3) we get\bigm| \bigm| \bigm| \bigm| Kn(f ;x) - f(x) - Kn(t - x;x)f \prime (x) - 1
2
Kn
\bigl(
(e1 - x)2;x
\bigr)
f \prime \prime (x)
\bigm| \bigm| \bigm| \bigm| \leq
\leq Kn((e1 - x)2;x)
\biggl\{
| Kn((e1 - x)3;x)|
Kn((e1 - x)2;x)
5
6h
\omega 1(f
\prime \prime ;h)+
\biggl(
3
4
+
Kn((e1 - x)4;x)
Kn((e1 - x)2;x)
1
16h2
\biggr)
\omega 2(f
\prime \prime ;h)
\biggr\}
.
Using the central moments up to order 4 for Kantorovich operators, namely
Kn (t - x;x) =
1 - 2x
2(n+ 1)
,
Kn
\bigl(
(t - x)2;x
\bigr)
=
1
(n+ 1)2
\biggl\{
x(1 - x)(n - 1) +
1
3
\biggr\}
,
Kn
\bigl(
(t - x)3;x
\bigr)
=
1 - 2x
4(n+ 1)3
\bigl\{
10x(1 - x)n+ 2x2 - 2x+ 1
\bigr\}
,
Kn
\bigl(
(t - x)4;x
\bigr)
=
1
(n+ 1)4
\biggl\{
3x2(1 - x)2n2 + 5x(1 - x)(1 - 2x)2n+ x4 - 2x3 + 2x2 - x+
1
5
\biggr\}
,
we have
| Kn
\bigl(
(t - x)3;x
\bigr)
|
Kn ((t - x)2;x)
\leq 5
2(n+ 1)
,
| Kn
\bigl(
(t - x)4;x
\bigr)
|
Kn ((t - x)2;x)
\leq 3(n+ 2)
(n+ 1)2
.
Therefore, the following inequality holds:\bigm| \bigm| \bigm| \bigm| Kn(f ;x) - f(x) - 1 - 2x
2(n+ 1)
f \prime (x) - 1
2
\biggl[
x(1 - x)(n - 1)
(n+ 1)2
+
1
3(n+ 1)2
\biggr]
f \prime \prime (x)
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
x(1 - x)
n - 1
(n+ 1)2
+
1
3(n+ 1)2
\biggr] \biggl\{
25
12h(n+ 1)
\omega 1(f
\prime \prime ;h) +
\biggl(
3
4
+
3(n+ 2)
16h2(n+ 1)2
\biggr)
\omega 2(f
\prime \prime ;h)
\biggr\}
and for h =
1\surd
n+ 1
we obtain, after multiplying both sides by n,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
742 A.-M. ACU, H. GONSKA\bigm| \bigm| \bigm| \bigm| n [Kn(f ;x) - f(x)] - n
n+ 1
\biggl(
1
2
- x
\biggr)
f \prime (x) - 1
2
\biggl[
x(1 - x)
n(n - 1)
(n+ 1)2
+
n
3(n+ 1)2
\biggr]
f \prime \prime (x)
\bigm| \bigm| \bigm| \bigm| \leq
\leq 9
32
\biggl\{
2\surd
n+ 1
\omega 1
\biggl(
f \prime \prime ;
1\surd
n+ 1
\biggr)
+ \omega 2
\biggl(
f \prime \prime ;
1\surd
n+ 1
\biggr) \biggr\}
.
We can write \bigm| \bigm| \bigm| \bigm| n [Kn(f ;x) - f(x)] - 1 - 2x
2
f \prime (x) - 1
2
x(1 - x)f \prime \prime (x)
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| n [Kn(f ;x) - f(x)] - n
n+ 1
\biggl(
1
2
- x
\biggr)
f \prime (x) - 1
2
\biggl[
x(1 - x)
n(n - 1)
(n+ 1)2
+
n
3(n+ 1)2
\biggr]
f \prime \prime (x)
\bigm| \bigm| \bigm| \bigm| +
+
\bigm| \bigm| \bigm| \bigm| 1 - 2x
2
1
n+ 1
f \prime (x) +
1
2
x(1 - x)
3n+ 1
(n+ 1)2
f \prime \prime (x) - n
6(n+ 1)2
f \prime \prime (x)
\bigm| \bigm| \bigm| \bigm| \leq
\leq 9
32
\biggl\{
2\surd
n+ 1
\omega 1
\biggl(
f \prime \prime ;
1\surd
n+ 1
\biggr)
+ \omega 2
\biggl(
f \prime \prime ;
1\surd
n+ 1
\biggr) \biggr\}
+
2
3(n+ 1)
\biggl(
3
4
\| f \prime \| \infty + \| f \prime \prime \| \infty
\biggr)
.
4. Chebyshev – Grüss inequality for Kantorovich operators. In a 2011 paper Raşa and
the present authors [1] published the following Grüss-type inequality for positive linear operators
reproducing constant functions. We give below the improved form of Rusu given in [14]:
Theorem 4.1. Let H : C[a, b] \rightarrow C[a, b] be positive, linear and satisfy He0 = e0. Put
D(f, g;x) := H(fg;x) - H(f ;x)H(g;x).
Then, for f, g \in C[a, b] and x \in [a, b] fixed, one has
| D(f, g;x)| \leq 1
4
\~\omega
\Bigl(
f ; 2
\sqrt{}
H ((e1 - x)2;x)
\Bigr)
\~\omega
\Bigl(
g; 2
\sqrt{}
H ((e1 - x)2;x)
\Bigr)
.
Here \~\omega is the least concave majorant of the first order modulus \omega 1 given by
\~\omega (f ; t) = \mathrm{s}\mathrm{u}\mathrm{p}
\biggl\{
(t - x)\omega 1(f ; y) + (y - t)\omega 1(f ;x)
y - x
: 0 \leq x \leq t \leq y \leq b - a, x \not = y
\biggr\}
.
Remark 4.1. For an accesible proof of the equality between \~\omega and a certain K -functional used
in the proof of the above theorem see [13].
Hence the nonmultiplicativity of Kantorovich operators can be explained as in the following
theorem.
Theorem 4.2. For the classical Kantorovich operators Kn : C[0, 1] \rightarrow C[0, 1] one has the
uniform inequality
\| Kn(fg) - KnfKng\| \infty \leq 1
4
\~\omega
\Biggl(
f ; 2
\sqrt{}
1
2(n+ 1)
\Biggr)
\~\omega
\Biggl(
g; 2
\sqrt{}
1
2(n+ 1)
\Biggr)
, n \geq 1, (2)
for all f, g \in C[0, 1].
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
CLASSICAL KANTOROVICH OPERATORS REVISITED 743
Proof. The most precise upper bound is obtained if we use the exact representation
Kn
\bigl(
(t - x)2;x
\bigr)
=
1
(n+ 1)2
\biggl\{
(n - 1)x(1 - x) +
1
3
\biggr\}
.
Close to x = 0, 1 this shows the familiar endpoint improvement. For shortness we use the estimate
Kn
\bigl(
(t - x)2;x
\bigr)
\leq 1
2(n+ 1)
.
5. Grüss – Voronovskaya theorems. The first Grüss – Voronovskaya theorem for classical Bern-
stein operators was given by Gal and Gonska [5]. In Theorem 2.1 of this paper a quantitative form
was given (see also Theorem 2.5 there). The other examples in [5] deal with operators reproducing
linear functions; this is not the case for the Kantorovich mappings. The limit for Kn was identified
recently in [2] to be the same as in the Bernstein case, namely
f \prime (x)g\prime (x)x(1 - x) for f, g \in C2[0, 1].
Our first quantitative version is given in the following theorem.
Theorem 5.1. Let f, g \in C2[0, 1]. Then, for each x \in [0, 1],
\bigm\| \bigm\| n [Kn(fg) - Knf \cdot Kng] - Xf \prime g\prime
\bigm\| \bigm\|
\infty =
\left\{
o(1), f, g \in C2[0, 1],
\scrO
\biggl(
1\surd
n
\biggr)
, f, g \in C3[0, 1],
\scrO
\biggl(
1
n
\biggr)
, f, g \in C4[0, 1].
Proof. We proceed as in [5] by creating first three Voronovskaya-type expressions from the dif-
ference in question plus the remaining quantities. Recall that the Voronovskaya limit for Kantorovich
operators is
1
2
(Xf \prime )\prime =
1
2
Xf \prime \prime (x) +
1
2
X \prime f \prime (x),
where X := x(1 - x), so X \prime = 1 - 2x.
For f, g \in C2[0, 1], one has
Kn(fg;x) - Kn(f ;x)Kn(g;x) -
1
n
Xf \prime (x)g\prime (x) =
= Kn(fg;x) - (fg)(x) - 1
2n
\bigl(
X(fg)\prime
\bigr) \prime -
- f(x)
\biggl[
Kn(g;x) - g(x) - 1
2n
(Xg\prime )\prime
\biggr]
- g(x)
\biggl[
Kn(f ;x) - f(x) - 1
2n
(Xf \prime )\prime
\biggr]
+
+ [g(x) - Kn(g;x)] [Kn(f ;x) - f(x)] -
- Kn(f ;x)Kn(g;x) -
1
n
Xf \prime g\prime + (fg)(x) +
1
2n
(X(fg)\prime )\prime +
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
744 A.-M. ACU, H. GONSKA
+f(x)
\biggl[
Kn(g;x) - g(x) - 1
2n
(Xg\prime )\prime
\biggr]
+ g(x)
\biggl[
Kn(f ;x) - f(x) - 1
2n
(Xf \prime )\prime
\biggr]
-
- [g(x) - Kn(g;x)] [Kn(f ;x) - f(x)] .
The first three lines will be estimated below. First we will show that the sum of the following
three lines equals 0.
For the time being we will leave out the argument x. One has
- Knf \cdot Kng -
1
n
Xf \prime g\prime + fg +
1
2n
\bigl(
X \prime (fg)\prime +X(fg)\prime \prime
\bigr)
+
+fKng - fg - 1
2n
f(X \prime g\prime +Xg\prime \prime ) + gKnf - fg - 1
2n
g(X \prime f \prime +Xf \prime \prime ) -
- [g - Kng] [Knf - f ] =
= - Knf \cdot Kng -
1
n
Xf \prime g\prime + fg +
1
2n
\bigl(
X \prime f \prime g +X \prime fg\prime
\bigr)
+
1
2n
X
\bigl(
f \prime \prime g + 2f \prime g\prime + fg\prime \prime
\bigr)
+
+fKng - fg - 1
2n
(fX \prime g\prime + fXg\prime \prime ) + gKnf - fg - 1
2n
(gX \prime f \prime + gXf \prime \prime ) -
- gKnf +Kng \cdot Knf + fg - fKng = 0.
For the first two lines above we will use the Voronovskaya estimate given earlier, namely that for
h \in C2[0, 1] one has\bigm\| \bigm\| \bigm\| \bigm\| n (Knh - h) - 1
2
\bigl(
Xh\prime
\bigr) \prime \bigm\| \bigm\| \bigm\| \bigm\|
\infty
\leq 2
3(n+ 1)
\biggl(
3
4
\| h\prime \| \infty + \| h\prime \prime \| \infty
\biggr)
+
+
9
32
\biggl\{
2\surd
n+ 1
\omega 1
\biggl(
h\prime \prime ;
1\surd
n+ 1
\biggr)
+ \omega 2
\biggl(
h\prime \prime ;
1\surd
n+ 1
\biggr) \biggr\}
=: U(h, n).
For the third line we use Theorem 2.4 showing that for h \in C2[0, 1] we get
\| Knh - h\| \infty \leq 1
2n
\| h\prime \| \infty +
9
8n
\| h\prime \prime \| \infty = \scrO
\biggl(
1
n
\biggr)
.
Collecting these inequalities gives\bigm\| \bigm\| n [Kn(fg) - Knf \cdot Kng] - Xf \prime g\prime
\bigm\| \bigm\|
\infty \leq
\leq U(fg, n) + \| f\| \infty U(g, n) + \| g\| \infty U(f, n) +\scrO
\biggl(
1
n
\biggr)
=
=
\left\{
o(1), f, g \in C2[0, 1],
\scrO
\biggl(
1\surd
n
\biggr)
, f, g \in C3[0, 1],
\scrO
\biggl(
1
n
\biggr)
, f, g \in C4[0, 1].
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
CLASSICAL KANTOROVICH OPERATORS REVISITED 745
In the following we give a Grüss – Voronovskaya type theorem when f and g are only in C1[0, 1].
Theorem 5.2. Let f, g \in C1[0, 1] and n \geq 1. Then there is a constant C independent of n,
f, g and x, such that\bigm\| \bigm\| \bigm\| \bigm\| Kn(fg) - Knf \cdot Kng -
X
n
f \prime g\prime
\bigm\| \bigm\| \bigm\| \bigm\|
\infty
\leq C
n
\biggl\{
\omega 3
\Bigl(
f \prime , n - 1
6
\Bigr)
\omega 3
\Bigl(
g\prime , n - 1
6
\Bigr)
+
+ \| f \prime \| \infty \omega 3
\Bigl(
g\prime , n - 1
6
\Bigr)
+ \| g\prime \| \infty \omega 3
\Bigl(
f \prime , n - 1
6
\Bigr)
+
+ \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| f \prime \| \infty
n
1
2
, \omega 3
\Bigl(
f \prime , n - 1
6
\Bigr) \biggr\}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| g\prime \| \infty
n
1
2
, \omega 3
\Bigl(
g\prime , n - 1
6
\Bigr) \biggr\} \biggr\}
.
Proof. Let
En(f, g;x) = Kn(fg;x) - Kn(f ;x)Kn(g;x) -
x(1 - x)
n
f \prime (x)g\prime (x), (3)
and denote C a constant independent of n, f, g and x, which may change its values during the
course of the proof.
For f, g \in C1[0, 1] fixed and u, v \in C4[0, 1] arbitrary, one has
| En(f, g;x)| = | En(f - u+ u, g - v + v;x)| \leq
\leq | En(f - u, g - v;x)| + | En(u, g - v;x)| + | En(f - u, v;x)| + | En(u, v;x)| . (4)
Let h(x) = x, x \in [0, 1]. Applying [1] (Theorem 4) there exists \eta , \theta \in [0, 1] such that
Kn(fg;x) - Kn(f ;x)Kn(g;x) = f \prime (\eta )g\prime (\theta )
\bigl[
Kn(h
2;x) - (Kn(h;x))
2
\bigr]
=
= f \prime (\eta )g\prime (\theta )
\biggl\{
x(1 - x)
n
(n+ 1)2
+
1
12(n+ 1)2
\biggr\}
. (5)
From (3) and (5) we get
| nEn(f, g;x)| \leq
\biggl[
x(1 - x)
n2
(n+ 1)2
+
n
12(n+ 1)2
+ x(1 - x)
\biggr]
\| f \prime \| \infty \| g\prime \| \infty \leq
\leq 2
\biggl[
x(1 - x) +
1
24(n+ 1)
\biggr]
\| f \prime \| \infty \| g\prime \| \infty . (6)
Using Theorem 3.1, for f \in C4[0, 1], we have\bigm| \bigm| \bigm| \bigm| n [Kn(f ;x) - f(x)] - 1
2
\bigl(
Xf \prime \bigr) \prime (x)\bigm| \bigm| \bigm| \bigm| \leq C
1
n
\Bigl(
\| f \prime \| \infty + \| f \prime \prime \| \infty + \| f \prime \prime \prime \| \infty + \| f (4)\| \infty
\Bigr)
.
But, for f \in Cn[a, b], n \in \BbbN , one has (see [6], Remark 2.15)
\mathrm{m}\mathrm{a}\mathrm{x}
0\leq k\leq n
\Bigl\{
\| f (k)\|
\Bigr\}
\leq C\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| f\| \infty , \| f (n)\| \infty
\Bigr\}
.
Therefore,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
746 A.-M. ACU, H. GONSKA\bigm| \bigm| \bigm| \bigm| n [Kn(f ;x) - f(x)] - 1
2
\bigl(
Xf \prime \bigr) \prime (x)\bigm| \bigm| \bigm| \bigm| \leq C
n
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| f \prime \| \infty , \| f (4)\| \infty
\Bigr\}
. (7)
For u, v \in C4[0, 1] using the same decomposition as in proof of Theorem 5.1, the relation (7) and
Theorem 2.4, we get
| En(u, v;x)| \leq
\bigm| \bigm| \bigm| \bigm| Kn(uv;x) - (uv)(x) - 1
2n
\bigl(
X(uv)\prime
\bigr) \prime \bigm| \bigm| \bigm| \bigm| +
+| u(x)|
\bigm| \bigm| \bigm| \bigm| Kn(v;x) - v(x) - 1
2n
(Xv\prime )\prime
\bigm| \bigm| \bigm| \bigm| + | v(x)|
\bigm| \bigm| \bigm| \bigm| Kn(u;x) - u(x) - 1
2n
(Xu\prime )\prime
\bigm| \bigm| \bigm| \bigm| +
+ | v(x) - Kn(v;x)| | Kn(u;x) - u(x)| \leq
\leq C
n2
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| u\prime \| \infty , \| u(4)\| \infty
\Bigr\}
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| v\prime \| \infty , \| v(4)\| \infty
\Bigr\}
. (8)
From the relations (4), (6) and (8) we obtain
| En(f, g;x)| \leq
2
n
\biggl[
x(1 - x) +
1
24(n+ 1)
\biggr] \bigl\{
\| (f - u)\prime \| \infty \| (g - v)\prime \| \infty + \| u\prime \| \infty \| (g - v)\prime \| \infty +
+\| (f - u)\prime \| \infty \| v\prime \| \infty
\bigr\}
+
C
n2
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| u\prime \| \infty , \| u(4)\| \infty
\Bigr\}
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| v\prime \| \infty , \| v(4)\| \infty
\Bigr\}
.
Using [7] (Lemma 3.1) for r = 1, s = 2, fh,3 = u and gh,3 = v, for all h \in (0, 1] and n \in \BbbN , we
have
| En(f, g;x)| \leq
C
n
\biggl\{
\omega 3(f
\prime , h)\omega 3(g
\prime , h) +
1
h
\omega 1(f, h)\omega 3(g
\prime , h) +
1
h
\omega 1(g, h)\omega 3(f
\prime , h)
\biggr\}
+
+
C
n2
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1
h
\omega 1(f, h),
1
h3
\omega 3(f
\prime , h)
\biggr\}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1
h
\omega 1(g, h),
1
h3
\omega 3(g
\prime , h)
\biggr\}
\leq
\leq C
n
\bigl\{
\omega 3(f
\prime , h)\omega 3(g
\prime , h) + \| f \prime \| \infty \omega 3(g
\prime , h) + \| g\prime \| \infty \omega 3(f
\prime , h)
\bigr\}
+
+
C
n2
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| f \prime \| \infty ,
1
h3
\omega 3(f
\prime , h)
\biggr\}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| g\prime \| \infty ,
1
h3
\omega 3(g
\prime , h)
\biggr\}
.
Choosing h = n - 1
6 , we obtain
| En(f, g;x)| \leq
C
n
\biggl\{
\omega 3
\Bigl(
f \prime , n - 1
6
\Bigr)
\omega 3
\Bigl(
g\prime , n - 1
6
\Bigr)
+
+ \| f \prime \| \infty \omega 3
\Bigl(
g\prime , n - 1
6
\Bigr)
+ \| g\prime \| \infty \omega 3
\Bigl(
f \prime , n - 1
6
\Bigr)
+
+ \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| f \prime \| \infty
n
1
2
, \omega 3
\Bigl(
f \prime , n - 1
6
\Bigr) \biggr\}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\| g\prime \| \infty
n
1
2
, \omega 3
\Bigl(
g\prime , n - 1
6
\Bigr) \biggr\} \biggr\}
.
This implies the theorem.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
CLASSICAL KANTOROVICH OPERATORS REVISITED 747
References
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Received 28.08.18
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
|
| id | umjimathkievua-article-1472 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:22Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b5/e1ab27f489ab53e4fa2a9add3525f5b5.pdf |
| spelling | umjimathkievua-article-14722019-12-05T08:56:42Z Classical Kantorovich operators revisited Знову про класичнi оператори Канторовича Acu, A.-M. Gonska, H. Асу, А.-М. Гонська, Х. UDC 517.5 The main object of this paper is to improve some known estimates for the classical Kantorovich operators. We obtain a quantitative Voronovskaya-type result in terms of the second moduli of continuity, which improves some previous results. In order to explain the nonmultiplicativity of the Kantorovich operators, we present a Chebyshev – Gr¨uss inequality. Two Gr¨uss –Voronovskaya theorems for Kantorovich operators are also considered. УДК 517.5 Основним об’єктом дослiдження є полiпшення деяких вiдомих оцiнок для класичних операторiв Канторовича. Отримано кiлькiсний результат типу Вороновської в термiнах других модулiв неперервностi, що полiпшує деякi попереднi результати. Щоб пояснити немультиплiкативнiсть операторiв Канторовича, ми наводимо нерiвнiсть Чебишова – Грюсса. Також розглянуто теореми Грюсса – Вороновської для операторiв Канторовича. Institute of Mathematics, NAS of Ukraine 2019-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1472 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 6 (2019); 739-747 Український математичний журнал; Том 71 № 6 (2019); 739-747 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1472/456 Copyright (c) 2019 Acu A.-M.; Gonska H. |
| spellingShingle | Acu, A.-M. Gonska, H. Асу, А.-М. Гонська, Х. Classical Kantorovich operators revisited |
| title | Classical Kantorovich operators revisited |
| title_alt | Знову про класичнi оператори Канторовича |
| title_full | Classical Kantorovich operators revisited |
| title_fullStr | Classical Kantorovich operators revisited |
| title_full_unstemmed | Classical Kantorovich operators revisited |
| title_short | Classical Kantorovich operators revisited |
| title_sort | classical kantorovich operators revisited |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1472 |
| work_keys_str_mv | AT acuam classicalkantorovichoperatorsrevisited AT gonskah classicalkantorovichoperatorsrevisited AT asuam classicalkantorovichoperatorsrevisited AT gonsʹkah classicalkantorovichoperatorsrevisited AT acuam znovuproklasičnioperatorikantoroviča AT gonskah znovuproklasičnioperatorikantoroviča AT asuam znovuproklasičnioperatorikantoroviča AT gonsʹkah znovuproklasičnioperatorikantoroviča |