Cheney–Sharma type operators on a triangle with two and three curved edges
UDC 517.5 We construct some Cheney–Sharma type operators de ned on a triangle with two and three curved edges, their product and Boolean sum. We study their interpolation properties and the degree of exactness.
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| author | Baboş, Alina Babos, Alina Babos, Alina |
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UDC 517.5
We construct some Cheney–Sharma type operators de ned on a triangle with two and three curved edges, their product and Boolean sum. We study their interpolation properties and the degree of exactness.
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| doi_str_mv | 10.37863/umzh.v72i5.6017 |
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DOI: 10.37863/umzh.v72i5.6017
UDC 517.5
A. Baboş (Dep. Tech. Sci., “Nicolae Balcescu” Land Forces Academy, Sibiu, Romania)
CHENEY – SHARMA TYPE OPERATORS ON A TRIANGLE
WITH TWO AND THREE CURVED EDGES
ОПЕРАТОРИ ТИПУ ЧЕЙНI – ШАРМИ НА ТРИКУТНИКУ
З ДВОМА ТА ТРЬОМА ВИГНУТИМИ СТОРОНАМИ
We construct some Cheney – Sharma type operators defined on a triangle with two and three curved edges, their product
and Boolean sum. We study their interpolation properties and the degree of exactness.
Побудовано деякi оператори типу Чейнi – Шарми, визначенi на трикутнику з двома та трьома вигнутими сторонами,
визначено їхнiй добуток i булеву суму. Також вивчено їхнi iнтерполяцiйнi властивостi та ступiнь точностi.
1. Introduction. There have been constructed interpolation operators of Lagrange, Hermite and
Birkhoff type on a triangle with all straight sides, starting with the paper [5] of R. E. Barnhil,
G. Birkhoff and W. J. Gordon, and in many others papers (see, e.g., [4, 6, 9, 14]). Then, were
considered interpolation operators on triangles with curved sides (one, two or all curved sides), many
of them in connection with their applications in computer aided geometric design and in finite element
analysis (see, e.g, [1 – 3, 7, 8, 15, 20]).
Also the Bernstein-type operators were used as interpolation operators both on triangles with
straight sides (see, e.g., [10, 13, 17 – 19]) and with curved sides (see, e.g., [11, 12]).
The aim of this paper is to construct some Cheney – Sharma type operators that have some
interpolatory properties on a triangle with two and three curved edges. They are extension of the
Cheney – Sharma type operators of second type, given by E. W. Cheney and A. Sharma in [16], to
the case of a curved side. There will be studied the interpolation properties and degree of exactness.
Let m \in \BbbN and \beta a nonnegative parameter. The Cheney – Sharma operators of second kind Qm :
C([0, 1]) \rightarrow C([0, 1]), introduced in [16], are given by
(Qmf)(x) =
m\sum
i=0
qm,i(x)f
\biggl(
i
m
\biggr)
, (1.1)
qm,i(x) =
\biggl(
m
i
\biggr)
x(x+ i\beta )i - 1(1 - x)[1 - x+ (m - i)\beta ]m - i - 1
(1 +m\beta )m - 1
. (1.2)
2. Triangle with all curved sides. 2.1. Univariate operators. In [12], we have the triangle \~Th
with all curved sides, which has the vertices V1 = (0, h), V2 = (h, 0) and V3 = (0, 0), and the three
curved sides \gamma 1, \gamma 2 (along the coordinate axis), and \gamma 3 (opposite to the vertex V3). \gamma 1 is defined
by (x, f1(x)), with f1(0) = f1(h) = 0, f1(x) \leq 0, for x \in [0, h]; \gamma 2 is defined by (g2(y), y) with
g2(0) = g2(h) = 0, g2(y) \leq 0, for y \in [0, h] and \gamma 3 is defined by the one-to-one functions f3 and
g3, where g3 is the inverse of the function f3, i.e., y = f3(x) and x = g3(y) with x, y \in [0, h] and
f3(0) = g3(0) = h (see Fig. 1).
Let F be a real-valued function defined on \~Th and (x, f1(x)), (x, f3(x)), respectively, (g2(y), y),
(g3(y), y) the points in which the parallel lines to the coordinate axes, passing through the point
c\bigcirc A. BABOŞ, 2020
600 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
CHENEY – SHARMA TYPE OPERATORS ON A TRIANGLE WITH TWO AND THREE CURVED EDGES 601
Fig. 1. Triangle \~Th.
(x, y) \in \~Th, intersect the sides \gamma 1, \gamma 2 and \gamma 3. We consider the uniform partitions of the intervals
[g2(y), g3(y)] and [f1(x), f3(x)], x, y \in [0, h]:
\Delta x
m =
\biggl\{
g2(y) + i
g3(y) - g2(y)
m
\bigm| \bigm| \bigm| \bigm| i = 0,m
\biggr\}
,
respectively,
\Delta y
n =
\biggl\{
f1(x) + j
f3(x) - f1(x)
n
\bigm| \bigm| \bigm| \bigm| j = 0, n
\biggr\}
.
For m,n \in \BbbN , \alpha , \beta \in \BbbR +, we consider the following extensions of the Cheney – Sharma operator
given in (1.1):
\bigl(
Qx
mF
\bigr)
(x, y) =
m\sum
i=0
qm,i(x, y)F
\biggl(
g2(y) + i
g3(y) - g2(y)
m
, y
\biggr)
(2.1)
with
qm,i(x, y) =
\biggl(
m
i
\biggr) x - g2(y)
g3(y) - g2(y)
\biggl(
x - g2(y)
g3(y) - g2(y)
+ i\beta
\biggr) i - 1
(1 +m\beta )m - 1
\times
\times
\biggl[
1 - x - g2(y)
g3(y) - g2(y)
\biggr] \biggl[
1 - x - g2(y)
g3(y) - g2(y)
+ (m - i)\beta
\biggr] m - i - 1
,
respectively,
\bigl(
Qy
nF
\bigr)
(x, y) =
n\sum
j=0
qn,j(x, y)F
\biggl(
x, f1(x) + j
f3(x) - f1(x)
n
\biggr)
(2.2)
with
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
602 A. BABOŞ
qn,j(x, y) =
\biggl(
n
j
\biggr) y - f1(x)
f3(x) - f1(x)
\biggl(
y - f1(x)
f3(x) - f1(x)
+ j\alpha
\biggr) j - 1
(1 + n\alpha )n - 1
\times
\times
\biggl[
1 - y - f1(x)
f3(x) - f1(x)
\biggr] \biggl[
1 - y - f1(x)
f3(x) - f1(x)
+ (n - j)\alpha
\biggr] n - j - 1
.
Theorem 2.1. If F is a real-valued function defined on \~Th, then:
1) Qx
mF = F on \gamma 2 \cup \gamma 3,
2) Qy
nF = F on \gamma 1 \cup \gamma 3,
3)
\bigl(
Qx
meij
\bigr)
(x, y) = xiyj , i = 0, 1, j \in \BbbN ,
4)
\bigl(
Qy
neij
\bigr)
(x, y) = xiyj , i \in \BbbN , j = 0, 1.
Proof. 1. We write
\bigl(
Qx
mF
\bigr)
(x, y) =
1
(1 +m\beta )m - 1
\Biggl\{ \biggl[
1 - x - g2(y)
g3(y) - g2(y)
\biggr]
\times
\times
\biggl[
1 - x - g2(y)
g3(y) - g2(y)
+m\beta
\biggr] m - 1
F (g2(y), y)+
+
x - g2(y)
g3(y) - g2(y)
\biggl[
1 - x - g2(y)
g3(y) - g2(y)
\biggr]
\times
\times
m\sum
i=0
\biggl(
m
i
\biggr) \biggl(
x - g2(y)
g3(y) - g2(y)
+ i\beta
\biggr) i - 1
\times
\times
\biggl[
1 - x - g2(y)
g3(y) - g2(y)
+ (m - i)\beta
\biggr] m - i - 1
\times
\times F
\biggl(
g2(y) + i
g3(y) - g2(y)
m
, y
\biggr)
+
+
x - g2(y)
g3(y) - g2(y)
\biggl[
x - g2(y)
g3(y) - g2(y)
+m\beta
\biggr] m - 1
F (g3(y), y)
\Biggr\}
.
So, \bigl(
Qx
mF
\bigr)
(g2(y), y) = F (g2(y), y),\bigl(
Qx
mF
\bigr)
(g3(y), y) = F (g3(y), y).
2. We have
\bigl(
Qy
nF
\bigr)
(x, y) =
1
(1 + n\alpha )n - 1
\Biggl\{ \biggl[
1 - y - f1(x)
f3(x) - f1(x)
\biggr]
\times
\times
\biggl[
1 - y - f1(x)
f3(x) - f1(x)
+ n\alpha
\biggr] n - 1
F (x, f1(x))+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
CHENEY – SHARMA TYPE OPERATORS ON A TRIANGLE WITH TWO AND THREE CURVED EDGES 603
+
y - f1(x)
f3(x) - f1(x)
\biggl[
1 - y - f1(x)
f3(x) - f1(x)
\biggr]
\times
\times
n\sum
j=0
\biggl(
n
j
\biggr) \biggl(
y - f1(x)
f3(x) - f1(x)
+ j\alpha
\biggr) j - 1
\times
\times
\biggl[
1 - y - f1(x)
f3(x) - f1(x)
+ (n - j)\alpha
\biggr] n - j - 1
\times
\times F
\biggl(
x, f1(x) + j
f3(x) - f1(x)
n
\biggr)
+
y - f1(x)
f3(x) - f1(x)
\times
\times
\biggl[
y - f1(x)
f3(x) - f1(x)
+ n\alpha
\biggr] n - 1
F (x, f3(x))
\Biggr\}
.
So, \bigl(
Qy
nF
\bigr)
(x, f1(x)) = F (x, f1(x)),\bigl(
Qy
nF
\bigr)
(x, f3(x)) = F (x, f3(x)).
The proof for conditions 3 and 4 follows by the property \mathrm{d}\mathrm{e}\mathrm{x}(Qm) = 1 (proved in [16]).
Theorem 2.1 is proved.
2.2. Product operators. Let P 1
mn = Qx
mQy
n, respectively, P 2
nm = Qy
nQx
m be the product of the
operators Qx
m and Qy
n.
We have
\bigl(
P 1
mnF
\bigr)
(x, y) =
m\sum
i=0
n\sum
j=0
qm,i(x, y)qn,j(xi, y)F
\biggl(
xi, f1(xi) + j
f3(xi) - f1(xi)
n
\biggr)
,
xi = g2(y) + i
g3(y) - g2(y)
m
,
and
\bigl(
P 2
nmF
\bigr)
(x, y) =
m\sum
i=0
n\sum
j=0
qm,i(x, yj)qn,j(x, y)F
\biggl(
g2(yj) + i
g3(yj) - g2(yj)
m
, yj
\biggr)
,
yj = f1(x) + j
f3(x) - f1(x)
n
.
Theorem 2.2. If F is a real-valued function defined on \~Th, then:
1)
\bigl(
P 1
mnF
\bigr)
(V3) = F (V3),
\bigl(
P 1
mnF
\bigr)
= F on \Gamma 3,
2)
\bigl(
P 2
nmF
\bigr)
(V3) = F (V3),
\bigl(
P 2
nmF
\bigr)
= F on \Gamma 3.
Proof. The proof follows from the properties\bigl(
P 1
mnF
\bigr)
(x, 0) =
\bigl(
Qx
mF
\bigr)
(x, 0),\bigl(
P 1
mnF
\bigr)
(0, y) =
\bigl(
Qy
nF
\bigr)
(0, y),
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
604 A. BABOŞ\bigl(
P 1
mnF
\bigr)
(x, f3(x)) = F (x, f3(x)), x, y \in [0, h],
and \bigl(
P 2
nmF
\bigr)
(x, 0) =
\bigl(
Qx
mF
\bigr)
(x, 0),\bigl(
P 2
nmF
\bigr)
(0, y) =
\bigl(
Qy
nF
\bigr)
(0, y),\bigl(
P 2
nmF
\bigr)
(g3(y), y) = F (g3(y), y), x, y \in [0, h],
which can be verified by a straightforward computation.
Theorem 2.2 is proved.
2.3. Boolean sum operators. We consider the Boolean sums of the operators Qx
m and Qy
n, i.e.,
S1
mn = Qx
m \oplus Qy
n = Qx
m +Qy
n - Qx
mQy
n,
respectively,
S2
nm = Qy
n \oplus Qx
m = Qy
n +Qx
m - Qy
nQ
x
m.
Theorem 2.3. If F is a real-valued function defined on \~Th, then
S1
mn
\bigm| \bigm|
\partial \~T
= F
\bigm| \bigm|
\partial \~T
and
S2
nm
\bigm| \bigm|
\partial \~T
= F
\bigm| \bigm|
\partial \~T
.
Proof. As \bigl(
S1
mnF
\bigr)
(x, f1(x)) =
\bigl(
Qx
mF
\bigr)
(x, f1(x)),\bigl(
S1
mnF
\bigr)
(g2(y), y) =
\bigl(
Qy
nF
\bigr)
(g2(y), y),\bigl(
S1
mnF
\bigr)
(x, f3(x)) = F (x, f3(x)),
the proof follows.
3. Triangle with two curved sides. 3.1. For f1(x) = 0, x \in [0, h], the triangle \~Th becomes
a triangle with two curved sides (see Fig. 2).
We suppose that F is a real-valued function defined on \~Th and (g2(y), y), (g3(y), y), respec-
tively, (x, 0), (x, f3(x)) the points in which the parallel lines to the coordinate axes, passing through
the point (x, y) \in \~Th, intersect the sides \gamma 1, \gamma 2, and \gamma 3.
We consider the uniform partitions of the intervals [g2(y), g3(y)] and [0, f3(x)], x, y \in [0, h]:
\Delta x
m =
\biggl\{
g2(y) + i
g3(y) - g2(y)
m
\bigm| \bigm| \bigm| \bigm| i = 0,m
\biggr\}
,
respectively,
\Delta y
n =
\biggl\{
j
n
f3(x)
\bigm| \bigm| \bigm| \bigm| j = 0, n
\biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
CHENEY – SHARMA TYPE OPERATORS ON A TRIANGLE WITH TWO AND THREE CURVED EDGES 605
Fig. 2. Triangle \~Th with two curved sides.
For m,n \in \BbbN , \alpha , \beta \in \BbbR +, we have the Cheney – Sharma operator Qx
m defined in (2.1) and,
respectively,
\bigl(
Qy
nF
\bigr)
(x, y) =
n\sum
j=0
qn,j(x, y)F
\biggl(
x,
j
n
f3(x)
\biggr)
with
qn,j(x, y) =
\biggl(
n
j
\biggr) y
f3(x)
\biggl(
y
f3(x)
+ j\alpha
\biggr) j - 1\biggl(
1 - y
f3(x)
\biggr) \biggl[
1 - y
f3(x)
+ (n - j)\alpha
\biggr] n - j - 1
(1 + n\alpha )n - 1
.
Theorem 3.1. If F is a real-valued function defined on \~Th, then:
1) Qx
mF = F on \gamma 2 \cup \gamma 3,
2) Qy
nF = F on \gamma 1 \cup \gamma 3,
3)
\bigl(
Qx
meij
\bigr)
(x, y) = xiyj , i = 0, 1, j \in \BbbN ,
4)
\bigl(
Qy
neij
\bigr)
(x, y) = xiyj , i \in \BbbN , j = 0, 1.
Proof. The proof for condition 1 is made in previous section.
2. We have\bigl(
Qy
nF
\bigr)
(x, y) =
1
(1 + n\alpha )n - 1
\Biggl\{ \biggl(
1 - y
f3(x)
\biggr) \biggl[
1 - y
f3(x)
+ n\alpha
\biggr] n - 1
F (x, 0)+
+
y
f3(x)
\biggl(
1 - y
f3(x)
\biggr) n\sum
j=0
\biggl(
n
j
\biggr) \biggl(
y
f3(x)
+ j\alpha
\biggr) j - 1
\times
\times
\biggl[
1 - y
f3(x)
+ (n - j)\alpha
\biggr] n - j - 1
F
\biggl(
x,
j
n
f3(x)
\biggr)
+
+
y
f3(x)
\biggl(
y
f3(x)
+ n\alpha
\biggr) n - 1
F (x, f3(x))
\Biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
606 A. BABOŞ
So, \bigl(
Qy
nF
\bigr)
(x, 0) = F (x, 0)),\bigl(
Qy
nF
\bigr)
(x, f3(x)) = F (x, f3(x)).
Theorem 3.1 is proved.
Let P 1
mn = Qx
mQy
n, respectively, P 2
nm = Qy
nQx
m be the product of the operators Qx
m and Qy
n.
We have \bigl(
P 1
mnF
\bigr)
(x, y) =
m\sum
i=0
n\sum
j=0
qm,i(x, y)qn,j(xi, y)F
\biggl(
xi,
j
n
f3(xi)
\biggr)
,
xi = g2(y) + i
g3(y) - g2(y)
m
,
and \bigl(
P 2
nmF
\bigr)
(x, y) =
m\sum
i=0
n\sum
j=0
qm,i
\biggl(
x,
j
n
f3(x)
\biggr)
qn,j(x, y)\times
\times F
\left( g2
\biggl(
j
n
f3(x)
\biggr)
+ i
g3
\biggl(
j
n
f3(x)
\biggr)
- g2
\biggl(
j
n
f3(x)
\biggr)
m
,
j
n
f3(x)
\right) .
Theorem 3.2. If F is a real-valued function defined on \~Th, then:
1)
\bigl(
P 1
mnF
\bigr)
(V3) = F (V3),
\bigl(
P 1
mnF
\bigr)
= F on \Gamma 3,
2)
\bigl(
P 1
nmF
\bigr)
(V3) = F (V3),
\bigl(
P 2
nmF
\bigr)
= F on \Gamma 3.
Proof. The proof follows from the properties\bigl(
P 1
mnF
\bigr)
(x, 0) =
\bigl(
Qx
mF
\bigr)
(x, 0),\bigl(
P 1
mnF
\bigr)
(g2(y), y) =
\bigl(
Qy
nF
\bigr)
(g2(y), y),\bigl(
P 1
mnF
\bigr)
(x, f3(x)) = F (x, f3(x)), x, y \in [0, h],
and \bigl(
P 2
nmF
\bigr)
(x, 0) =
\bigl(
Qx
mF
\bigr)
(x, 0),\bigl(
P 2
nmF
\bigr)
(g2(y), y) =
\bigl(
Qy
nF
\bigr)
(g2(y), y),\bigl(
P 2
nmF
\bigr)
(g3(y), y) = F (g3(y), y), x, y \in [0, h],
which can be verified by a straightforward computation.
Theorem 3.2 is proved.
We consider the Boolean sums of the operators Qx
m and Qy
n, i.e.,
S1
mn = Qx
m \oplus Qy
n = Qx
m +Qy
n - Qx
mQy
n,
respectively,
S2
nm = Qy
n \oplus Qx
m = Qy
n +Qx
m - Qy
nQ
x
m.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
CHENEY – SHARMA TYPE OPERATORS ON A TRIANGLE WITH TWO AND THREE CURVED EDGES 607
Fig. 3. Triangle \~Th with two curved sides.
Theorem 3.3. If F is a real-valued function defined on \~Th, then
S1
mn
\bigm| \bigm|
\partial \~T
= F | \partial \~T
and
S2
nm
\bigm| \bigm|
\partial \~T
= F | \partial \~T .
Proof. As \bigl(
S1
mnF
\bigr)
(x, 0) =
\bigl(
Qx
mF
\bigr)
(x, 0),\bigl(
S1
mnF
\bigr)
(g2(y), y) =
\bigl(
Qy
nF
\bigr)
(g2(y), y),\bigl(
S1
mnF
\bigr)
(x, f3(x)) = F (x, f3(x)),
the proof follows.
3.2. For g2(y) = 0, y \in [0, h], the triangle \~Th also becomes a triangle with two curved sides
(see Fig. 3).
Also, we suppose that F is a real-valued function defined on \~Th and (0, y), (g3(y), y), respec-
tively, (x, f1(x)), (x, f3(x)) are the points in which the parallel lines to the coordinate axes, passing
through the point (x, y) \in \~Th, intersect the sides \gamma 1, \gamma 2, and \gamma 3.
We consider the uniform partitions of the intervals [0, g3(y)] and [f1(x), f3(x)], x, y \in [0, h]:
\Delta x
m =
\biggl\{
i
m
g3(y)
\bigm| \bigm| \bigm| \bigm| i = 0,m
\biggr\}
,
respectively,
\Delta y
n =
\biggl\{
f1(x) + j
f3(x) - f1(x)
n
\bigm| \bigm| \bigm| \bigm| j = 0, n
\biggr\}
.
For m,n \in \BbbN , \alpha , \beta \in \BbbR +, we have
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
608 A. BABOŞ
\bigl(
Qx
mF
\bigr)
(x, y) =
m\sum
i=0
qm,i(x, y)F
\biggl(
i
m
g3(y), y
\biggr)
with
qm,i(x, y) =
\biggl(
m
i
\biggr) x
g3(y)
\biggl(
x
g3(y)
+ i\beta
\biggr) i - 1\biggl(
1 - x
g3(y)
\biggr) \biggl[
1 - x
g3(y)
+ (m - i)\beta
\biggr] m - i - 1
(1 +m\beta )m - 1
,
respectively, the operator Qy
n from (2.2).
Theorem 3.4. If F is a real-valued function defined on \~Th, then:
1) Qx
mF = F on \gamma 2 \cup \gamma 3,
2) Qy
nF = F on \gamma 1 \cup \gamma 3,
3)
\bigl(
Qx
meij
\bigr)
(x, y) = xiyj , i = 0, 1, j \in \BbbN ,
4)
\bigl(
Qy
neij
\bigr)
(x, y) = xiyj , i \in \BbbN , j = 0, 1.
Proof. 1. We have
\bigl(
Qx
mF
\bigr)
(x, y) =
1
(1 +m\beta )m - 1
\Biggl\{ \biggl(
1 - x
g3(y)
\biggr) \biggl[
1 - x
g3(y)
+m\beta
\biggr] m - 1
F (0, y)+
+
x
g3(y)
\biggl(
1 - x
g3(y)
\biggr) m\sum
i=0
\biggl(
m
i
\biggr) \biggl(
x
g3(y)
+ i\beta
\biggr) i - 1
\times
\times
\biggl[
1 - x
g3(y)
+ (m - 1)\beta
\biggr] m - i - 1
F
\biggl(
i
m
g3(y), y
\biggr)
+
+
x
g3(y)
\biggl(
x
g3(y)
+m\beta
\biggr) m - 1
F (g3(y), y)
\Biggr\}
.
So, \bigl(
Qx
mF
\bigr)
(0, y) = F (0, y)),\bigl(
Qx
mF
\bigr)
(g3(y), y) = F (g3(y), y).
The proof for condition 2 is made in previous section.
The product operators will be
\bigl(
P 1
mnF
\bigr)
(x, y) =
m\sum
i=0
n\sum
j=0
qm,i(x, y)qn,j
\biggl(
i
m
g3(y), y
\biggr)
\times
\times F
\left( i
m
g3(y), f1
\biggl(
i
m
g3(y)
\biggr)
+ j
f3
\biggl(
i
m
g3(y)
\biggr)
- f1
\biggl(
i
m
gy(x)
\biggr)
n
\right)
and
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
CHENEY – SHARMA TYPE OPERATORS ON A TRIANGLE WITH TWO AND THREE CURVED EDGES 609
\bigl(
P 2
nmF
\bigr)
(x, y) =
m\sum
i=0
n\sum
j=0
qm,i(x, yj)qn,j(x, y)F
\biggl(
i
m
g3(y), yj
\biggr)
,
yj = f1(x) + j
f3(x) - f1(x)
n
.
Theorem 3.5. If F is a real-valued function defined on \~Th, then:
1)
\bigl(
P 1
mnF
\bigr)
(V3) = F (V3),
\bigl(
P 1
mnF
\bigr)
= F on \Gamma 3,
2)
\bigl(
P 1
nmF
\bigr)
(V3) = F (V3),
\bigl(
P 2
nmF
\bigr)
= F on \Gamma 3.
Proof. The proof follows from the properties\bigl(
P 1
mnF
\bigr)
(x, f1(x)) =
\bigl(
Qx
mF
\bigr)
(x, f1(x)),\bigl(
P 1
mnF
\bigr)
(0, y) =
\bigl(
Qy
nF
\bigr)
(0, y),\bigl(
P 1
mnF
\bigr)
(x, f3(x)) = F (x, f3(x)), x, y \in [0, h],
and \bigl(
P 2
nmF
\bigr)
(x, f1(x)) =
\bigl(
Qx
mF
\bigr)
(x, f1(x)),\bigl(
P 2
nmF
\bigr)
(0, y) =
\bigl(
Qy
nF
\bigr)
(0, y),\bigl(
P 2
nmF
\bigr)
(g3(y), y) = F (g3(y), y), x, y \in [0, h],
which can be verified by a straightforward computation.
For the Boolean sums we have the following theorem.
Theorem 3.6. If F is a real-valued function defined on \~Th, then
S1
mn
\bigm| \bigm|
\partial \~T
= F | \partial \~T
and
S2
nm
\bigm| \bigm|
\partial \~T
= F | \partial \~T .
Proof. As \bigl(
S1
mnF
\bigr)
(x, f1(x)) =
\bigl(
Qx
mF
\bigr)
(x, f1(x)),\bigl(
S1
mnF
\bigr)
(0, y) =
\bigl(
Qy
nF
\bigr)
(0, y),\bigl(
S1
mnF
\bigr)
(x, f3(x)) = F (x, f3(x)),
the proof follows.
References
1. A. Baboş, Some interpolation operators on triangle, The 16th Int. Conf. the Knowledge-Based Organization, Appl.
Tech. Sci. and Adv. Military Technologies, Sibiu (2010), p. 28 – 34.
2. A. Baboş, Some interpolation schemes on a triangle with one curved side, Gen. Math., 21, № 1-2, 97 – 106 (2013).
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610 A. BABOŞ
3. A. Baboş, Interpolation operators on a triangle with two and three edges, Creat. Math. Inform., 22, № 2, 135 – 142
(2013).
4. R. E. Barnhill, J. A. Gregory, Polynomial interpolation to boundary data on triangles, Math. Comput., 29(131),
726 – 735 (1975).
5. R. E. Barnhill, G. Birkhoff, W. J. Gordon, Smooth interpolation in triangles, J. Approxim. Theory, 8, 114 – 128
(1973).
6. R. E. Barnhill, L. Mansfield, Error bounds for smooth interpolation, J. Approxim. Theory, 11, 306 – 318 (1974).
7. D. Bărbosu, I. Zelina, About some interpolation formulas over triangles, Rev. Anal. Numer. Theor. Approxim., 2,
117 – 123 (1999).
8. C. Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal., 26, 1212 – 1240 (1989).
9. G. Birkhoff, Interpolation to boundary data in triangles, J. Math. Anal. and Appl., 42, 474 – 484 (1973).
10. P. Blaga, Gh. Coman, Bernstein-type operators on triangle, Rev. Anal. Numer. Theor. Approxim., 37, № 1, 9 – 21
(2009).
11. P. Blaga, T. Cătinaş, Gh. Coman, Bernstein-type operators on a triangle with one curved side, Mediterr. J. Math.
(2011).
12. P. Blaga, T. Cătinaş, Gh. Coman, Bernstein-type operators on triangle with all curved sides, Appl. Math. and Comput.,
218, № 7, 3072 – 3082 (2011).
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№ 4, 3 – 19 (2009).
14. T. Cătinaş, Gh. Coman, Some interpolation operators on a simplex domain, Stud. Univ. Babeş Bolyai, 52, № 3,
25 – 34 (2007).
15. Gh. Coman, T. Cătinaş, Interpolation operators on a triangle with one curved side, BIT. Numer. Math., 47 (2010).
16. E. W. Cheney, A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5, 77 – 84 (1964).
17. D. D. Stancu, Evaluation of the remainder term in approximation formulas by Berstein polynomials, Math. Comput.,
17, 270 – 278 (1963).
18. D. D. Stancu, A method for obtaining polynomials of Berstein type of two variables, Amer. Math. Monthly, 70,
260 – 264 (1963).
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Received 25.04.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
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| id | umjimathkievua-article-6017 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:19Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/28/6f703c219298d5bbe733a259ca44c228.pdf |
| spelling | umjimathkievua-article-60172022-03-26T11:01:45Z Cheney–Sharma type operators on a triangle with two and three curved edges Cheney–Sharma type operators on a triangle with two and three curved edges Baboş, Alina Babos, Alina Babos, Alina UDC 517.5 We construct some Cheney–Sharma type operators de ned on a triangle with two and three curved edges, their product and Boolean sum. We study their interpolation properties and the degree of exactness. Побудовано деякi оператори типу Чейнi–Шарми, визначенi на трикутнику з двома та трьома вигнутими сторонами, визначено їхнiй добуток i булеву суму. Також вивчено їхнi iнтерполяцiйнi властивостi та ступiнь точностi. Institute of Mathematics, NAS of Ukraine 2020-04-29 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6017 10.37863/umzh.v72i5.6017 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 5 (2020); 600–610 Український математичний журнал; Том 72 № 5 (2020); 600–610 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6017/8690 |
| spellingShingle | Baboş, Alina Babos, Alina Babos, Alina Cheney–Sharma type operators on a triangle with two and three curved edges |
| title | Cheney–Sharma type operators on a triangle with two and three curved edges |
| title_alt | Cheney–Sharma type operators on a triangle with two and three curved edges |
| title_full | Cheney–Sharma type operators on a triangle with two and three curved edges |
| title_fullStr | Cheney–Sharma type operators on a triangle with two and three curved edges |
| title_full_unstemmed | Cheney–Sharma type operators on a triangle with two and three curved edges |
| title_short | Cheney–Sharma type operators on a triangle with two and three curved edges |
| title_sort | cheney–sharma type operators on a triangle with two and three curved edges |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6017 |
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