Rate of convergence for Szász-Bézier operators
We estimate the rate of convergence for functions of bounded variation for the Bezier variant of the Szasz operators $S_{n, \alpha}(f, x)$. We study the rate of convergence of $S_{n, \alpha}(f, x)$ for the case $0 < \alpha < 1$.
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| Datum: | 2005 |
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| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509845790130176 |
|---|---|
| author | Gupta, Vijay Гупта, В. |
| author_facet | Gupta, Vijay Гупта, В. |
| author_sort | Gupta, Vijay |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:57Z |
| description | We estimate the rate of convergence for functions of bounded variation for the Bezier variant of the Szasz operators $S_{n, \alpha}(f, x)$. We study the rate of
convergence of $S_{n, \alpha}(f, x)$ for the case $0 < \alpha < 1$. |
| first_indexed | 2026-03-24T02:47:35Z |
| format | Article |
| fulltext |
UDC 517.5
V. Gupta (School Appl. Sci., Netaji Subhas Inst. Technol., India)
RATE OF CONVERGENCE
FOR THE SZÁSZ – BÉZIER OPERATORS
ÍVYDKIST| ZBIÛNOSTI OPERATORIV SASA – BEZ’{
We estimate the rate of convergence for functions of bounded variation for the Bézier variant of the
Szász operators S f xn, ( , )α . We study the rate of convergence of S f xn, ( , )α for the case 0 < α < 1.
Znajdeno ocinku ßvydkosti zbiΩnosti funkcij obmeΩeno] variaci] dlq versi] Bez’[ operatoriv
Sasa S f xn, ( , )α . Vyvçeno ßvydkosti zbiΩnosti S f xn, ( , )α dlq 0 < α < 1.
1. Introduction. For the case where α ≥ 1 or 0 < α < 1 and a function f is defined
on [ , )0 ∞ , the Szász – Bézier operator Sn,α is defined by
S f x Q x f k
nn n k
k
, ,
( )( , ) ( )α
α=
=
∞
∑
0
, (1),
where Q xn k,
( )( )α = J xn k, ( )α – J xn k, ( )+1
α and J xn k, ( ) = s xn jj k , ( )=
∞∑ with the Szász
basis function s xn k, ( ) = e nx
k
nx
k
− ( )
!
, k = 0, 1, 2, … . It is well known that for α = 1,
the operators (1) reduce to the well-known Szász – Mirakyan operators
S f x s x f
k
nn n k
k
( , ) ( ),=
=
∞
∑
0
.
The rate of convergence for the Szász – Mirakyan operators on functions of
bounded variation was first studied by Cheng [1]. Recently, Zeng [2] and Zeng and
Zhao [3] estimated the rates of convergence for the Szász – Bézier operators whenever
α ≥ 1. The rates of convergence for the other case of α ∈ (0, 1) for functions of
bounded variation were obtained in [4] and [5], respectively, for the Kantorovich –
Bézier operators and Bernstein – Bézier operators. Motivated by this, we extend the
results of [1, 3, 6] and study the rate of convergence for the Szász – Bézier operators
S f xn, ( , )α , 0 < α < 1 for functions of bounded variation.
Our main theorem is stated as follows:
Theorem. Let f be a function of bounded variation on every finite subinterval of
[ , )0 ∞ . Let f t( ) = O tr( ) for some r ∈ N as t → ∞ . Then for x ∈ (0, ∞ ), 0 <
< α < 1, there exists a positive constant M ( f, α, x, r ) such that, for n sufficiently
large, we have
S f x f x f xn, ( , ) ( ) ( )α α α− + − −
−1
2
1 1
2
≤
≤ Z x
nx
f x f x
enx
x f x f xn
( ) ( ) ( ) ( ) ( ) ( )+ − − + − −1
2
ε +
+ 5
1nx
f x
kx x
k
n
Ω ,
=
∑ +
M f x r
nm
( , , , )α
,
where Z x( ) = min { 0 8 1 3, ( )+ x + 0,5, 1 6 2, x + 1,3},
© V. GUPTA, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12 1619
1620 V. GUPTA
Ωx f( , )λ = sup ( ) ( )
[ , ]t x x
f t f x
∈ − +
−
λ λ
, εn x( ) =
1
0
, , ,
if
otherwise
x k
n
k N= ′ ′ ∈
and
f t
f t f x t x
t x
f t f x x t
x( )
( ) ( ), ,
, ,
( ) ( ), .
=
− − ≤ <
=
− + < < ∞
0
0
It is clear that:
(i) Ωx f h( , ) is monotone nondecreasing with respect to h;
(ii) lim ( , )
h
x f h
→ ∞
Ω = 0 if f is continuous at the point x;
(iii) if f is a function of bounded variation on [ , ]a b and V fa
b( ) denotes the total
variation of f on [ , ]a b , then Ωx f h( , ) ≤ V fx h
x h
−
+ ( ).
We recall the Lebesgue – Stieltjes integral representation
S f x f t d K x tn t n, ,( , ) ( ) ( ( , ))α α=
∞
∫
0
,
where
K x t
Q x t
t
n
n k
k nt,
,
( )
( , )
( ), ,
,
α
α
=
< < ∞
=
≤
∑
.
0
0 0
We also define
H x t
K x t t
tn
n
,
,( , )
( , ), ,
,α
α=
− < < ∞
=
1 0
0 0.
2. Lemmas. In the sequel, we shall need the following lemmas:
Lemma 1 [6]. For all x ∈ (0, ∞ ) and x ∈ N we have
s x
e nxn k, ( ) < 1
2
.
Lemma 2 [2]. For x ∈ (0, ∞ ) we have
s x
x
nx
x
nxn k
k nx
, ( ) min
, ( ) ,
,
, ,− ≤ + +
+
+
+
>
∑ 1
2
0 8 1 3 0 5
1
1 6 1 3
1
2
and, for 0 ≤ t < x, we have
Q x x
n t xn k
k nt
,
( )( )
( )
α
≤
∑ ≤
− 2 .
Lemma 3. For 0 < α < 1 and 0 < x < t < ∞, we have
H xn, ( )α ≤ E
n t xm m
( )
( )
α
−
,
where E ( )α is a positive constant depending only on α.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
RATE OF CONVERGENCE FOR THE SZÁSZ – BÉZIER OPERATORS 1621
Proof. Since 0 < x < t < ∞, we have
k n x
t x
/ −
−
≥ 1 for k ≥ nt. Thus,
H xn, ( )α = 1 – K x tn, ( , )α =
= 1 – Q xn k
k nt
,
( )( )α
≤
∑ ≤ Q xn k
k nt
,
( )( )α
≥
∑ ≤
k n x
t x
s x
m
m n k
k nt
/
( )
( )
/
/ ,
−
−
≥
∑
2
2
α
α
α
≤
≤ 1
2
2
0( )
( )
/
,t x
k
n
x s xm
m
n k
k−
−
=
∞
∑
α
α
.
Put l = 2m / α and suppose that [ l ] denotes the integral part of l. Following [4]
(Lemma 6), choose the numbers p =
2
2 2
[ ]
[ ]
l
l l+ −
, q =
2
2
[ ]l
l −
.
For each real, put ψ x t( ) = t – x . Note that 2
p
+
2 1( [ ])+ l
q
=
2 2[ ]
[ ]
l l
l
+ −
+
+
l
l
l
− +2
1
[ ]
( ) = l. The application of Hölder’s inequality yields
k
n
x s x
m
n k
k
−
=
∞
∑
2
0
/
, ( )
α
≤ k
n
x s x k
n
x s xn k
k
p
l
n k
k
q
−
−
=
∞ +
=
∞
∑ ∑
2
0
1
2 1
0
1
,
/
([ ] )
,
/
( ) ( ) =
= S x S xn x
p
n x
l q
,
/
,
([ ] ) /
( , ) ( , )1
2 1
1
2 1 1
ψ ψ( ) ( )+ .
By using the well-known result S xn x
r
, ( , )1 ψ = O n r( )− as n → ∞ ( r = 1, 2, 3, … ), we
obtain
k
n
x s x
m
n k
k
−
=
∞
∑
2
0
/
, ( )
α
α
≤ O n p l q− − +( )α α/ ([ ] ) /1 = O n m−( ),
since
– α
p
–
α ( [ ])1 + l
q
= –α –
α[ ]
[ ]
l
l
l
2
2−
= –α – α l
l
− 2 = −α l
2
= – m.
This completes the proof of Lemma 3.
3. Proof of Theorem. We have
f t( ) = 2 1 2 2− − −+ + − − + + + − −( )( )α α α αf x f x g t f x f x tx( ) ( ) ( ) ( ) ( ) ( ) ( )( )sign +
+ f x f x f x tx( ) ( ) ( ) ( ) ( )− + − − −( )− −2 1 2α α δ ,
where
sign
if
if
if
( )( ) :
,
,α
α
t x
t x
t x
t x
− =
− >
=
− <
2 1
0
1
and δx t
x t
x t
( )
,
.
=
=
≠
1
0
if
if
Therefore,
S f x f x f xn, ( , ) ( ) ( )α α α− + − −
−1
2
1 1
2
≤
≤ S f xn x, ( , )α +
f x f x
S t x xn
( ) ( )
( ),,
( )+ − − −( )
2α α
αsign +
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1622 V. GUPTA
+ f x f x f x S xn x( ) ( ) ( ) ( , ),− + − −
−
1
2
1 1
2α α α δ . (2)
We first estimate
S t x xn,
( )( ),α
αsign −( ) = 2α αQ xn k
k nx
,
( )( )
>
∑ – 1 + e x Q xn n k( ) ( ),
( )
′
α =
= 2 1
α α αJ x J xn k n k
k nx
, ,( ) ( )−( )+
>
∑ – 1 + ε α
n n kx Q x( ) ( ),
( )
′ =
= 2α
α
s xn k
k nx
, ( )
>
∑
– 1 + ε α
n n kx Q x( ) ( ),
( )
′
and
S x x Q xn x n n k, ,
( )( , ) ( ) ( )α
αδ ε= ′ .
Hence, we have
f x f x
S t x xn
( ) ( )
( ( ), ),
+ − − −
2α α sign +
+ f x f x f x S xn x( ) ( ) ( ) ( , ),− + − −
−
1
2
1 1
2α α α δ =
=
f x f x
s x f x f x Q xn k
k nx
n n k
( ) ( )
( ) ( ) ( ) ( ), ,
( )+ − −
−
+ − −[ ]
>
′∑
2
2 1α
α
α
αε . (3)
By mean value theorem, we have
s x x s xn j
j nx
n j n j
j nx
, , ,( ) ( ) ( )
>
−
>
∑ ∑
− = ( ) −
α
α
αα ζ1
2
1
2
1
,
where ζn j x, ( ) lies between 1
2
and s xn j
j nx
, ( )
>
∑ . In view of Lemma 2, it is observed
that, for n sufficiently large, the intermediate point ςn j, is arbitrary close to 1
2
, i.e.,
ς
εn j, =
+
1
2
with an arbitrary small ε . Then we have
α ζ α εα α
n j x, ( ) ( )( ) ≤ +− −1 12 .
The latter expression is positive and strictly increasing for α ∈ (0, 1), since
∂
∂
+ = + − +[ ] >− −
α
α ε ε α εα α( ) ( ) log( )2 2 1 2 01 1
for sufficiently small ε . Thus, it takes maximum value at α = 1. This implies
α ζ α
n j x, ( )( ) ≤−1
1.
Hence,
s x Z x
nxn j
j nx
, ( ) ( )
>
∑
− ≤
+
α
α
1
2 1
, Z x x x( ) min , ( ) , , , ,= + + +{ }0 8 1 3 0 5 1 6 1 32 .
(4)
We also have
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
RATE OF CONVERGENCE FOR THE SZÁSZ – BÉZIER OPERATORS 1623
Q x J x J x s xn k n k n k n k n k,
( )
, , , ,( ) ( ) ( ) ( )′ ′ ′ + ′
−
′= − = ( )α α α α
α ζ1
1
,
where J xn k, ( )′ +1 < ζn k x, ( )′ < J xn k, ( )′ . Thus, by Lemma 1, we have
Q x
enxn k,
( ) ( )′ ≤α 1
2
. (5)
Combining the estimates of (3) – (5), we have
f x f x
S t x xn
( ) ( )
( ( ), ),
+ − − −
2α α sign +
+ f x f x f x S xn x( ) ( ) ( ) ( , ),− + − −
−
1
2
1 1
2α α α δ ≤
≤ Z x
nx
f x f x
enx
x f x f xn
( ) ( ) ( ) ( ) ( ) ( )
1
1
2+
+ − − + − −ε .
We next estimate S g xn x, ( , )α as follows:
S f xn x, ( , )α = f t d K x tx t n( ) ( , ),
0
∞
∫ ( )α =
=
I I II
x t nf t d K x t
1 2 43
∫ ∫ ∫∫+ + +
( )( ) ( , ),α = E E E E1 2 3 4+ + + say, (6)
where I1 = 0, x x
n
−
, I2 = x x
n
x x
n
− +
, , I3 = x x
n
x+
, 2 , and I4 =
= 2x, ∞[ ). We first estimate E2 . Noting that f xx( ) = 0, we have
E2 ≤ g t g x d K x t f x
nx x t
x x n
x x n
n x x( ) ( ) ( ( , )) ,
/
/
,− ≤
−
+
∫ α Ω ≤
≤ x
nx
f x
kx x
k
n
Ω ,
=
∑
1
. (7)
We next estimate E1. Writing y = x – x
n
and using Lebesgue – Stieltjes integration
by parts, we have
E1 = f t d K x tx t
y
n( ) ( , ),
0
∫ ( )α ≤ Ωx x
y
t nf x t d K x t( , ) ( , ),
0
∫ − α =
= Ω Ωx x n n t x x
y
f x y K x y K x t d f x t( , ) ( , ) ˆ ( , ) ( , ), ,− + − −( )∫α α
0
,
where ˆ ( , ),K x tn α is the normalized form of K x tn, ( , )α . Since ˆ ( , ),K x tn α ≤ K x tn, ( , )α
on ( , )0 ∞ , by Lemma 2 it follows that
E1 ≤ Ω Ωx x t
y
x xf x y x
n x y
x
n x t
d f x t( , )
( ) ( )
( , )−
−
+
−
− −( )∫2 2
0
1 .
Integrating by parts the last term, we have
1 2
2
0
2
0 0
3( )
( , )
( , )
( )
( , )
( )x t
d f x t
f x t
x t
f x t dt
x t
t
y
x x
x x
y
x x
y
−
− −( ) = − −
−
+ −
−∫ ∫
+
Ω Ω Ω .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1624 V. GUPTA
Hence, by replacing the variable t in the last integral by x – x
u
, we get
E
nx
f x
kx
k
n
x1
1
2≤
=
∑ Ω , . (8)
Using the similar method for the estimation of E3, we get
E
nx
f x
kx
k
n
x3
1
2≤
=
∑ Ω , . (9)
Finally, by assumption, we have the estimate
f t Mt M
t x
xx
r
r
( ) ≤ ≤ −
for t ≥ 2x .
Now
E4 = f t K x tx n
x
( ) ( , ),α
2
∞
∫ ≤ f t d K x tx
x
t n( ) ( , ),
2
∞
∫ α ≤
≤ M x t x d K x tr r
t n
−
∞
−∫ ( ) ( , ),
0
α ≤ − − −( )−
∞
∫Mx t x d K x tr r
x
t n( ) ( , ),
2
1 α =
= − − ( )−
∞
∫Mx t x d H x tr r
x
t n( ) ( , ),
2
α ≤ − − −( )−
∞
∫Mx t x d K x tr r
t n( ) ( , ),
0
1 α =
= Mx t x H x t H x t d t xr
R
r
n x
R
n t
r
x
R
−
→∞
− − + −
∫lim ( ) ( , ) ( , ) ( ), ,α α2
2
=
= M x t x H x t H x t r t x dtr
R
r
n x
R
n
r
x
R
−
→ ∞
−− − + −
∫lim ( ) ( , ) ( , ) ( ), ,α α2
1
2
=
= Mx t x
E
n t x
rE
n
t x dtr
R
r
m m
x
R
m
r m
x
R
−
→∞
− −− −
−
+ −
∫lim ( )
( )
( )
( )
( )
α α
2
1
2
=
= M
E
n x
r E
n m r xm m m m r
( ) ( )
( )
α α+
− − , m r> . (10)
Combining the estimates of (2) – (10), we obtain the required result.
This completes the proof of the theorem.
1. Cheng F. On the rate of convergence of the Szász – Mirakyan operator for bounded variation // J.
Approxim. Theory. – 1984. – 40. – P. 226 – 241.
2. Zeng X. M. On the rate of convergence of generalized Szász type operators for bounded variation
functions // J. Math. Anal. and Appl. – 1998. – 226. – P. 309 – 325.
3. Zeng X. M., Yang J., Zuo. S. L. Approximation of pointwise of Szász – Bézier type operators for
bounded functions // Adv. Math. Res. – 2003. – 3. – P. 117–124.
4. Pych Taberska P. Some properties of the Bézier – Kantorovich type operators // J. Approxim.
Theory. – 2003. – 123. – P. 256 – 269.
5. Zeng X. M. On the rate of convergence of two Bernstein – Bézier type operators for bounded
variation functions II // J. Approxim. Theory. – 2000. – 104. – P. 330 – 344.
6. Zeng X. M., Zhao J. N. Exact bounds for some basis functions of approximation operators // J.
Inequal. Appl. – 2001. – 6, # 5. – P. 563 – 575.
Received 10.11.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
|
| id | umjimathkievua-article-3713 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:47:35Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/84/88f248d5dc3c415cf96eb598e4edb584.pdf |
| spelling | umjimathkievua-article-37132020-03-18T20:02:57Z Rate of convergence for Szász-Bézier operators Швидкість збіжності операторів Сaca - Без'є Gupta, Vijay Гупта, В. We estimate the rate of convergence for functions of bounded variation for the Bezier variant of the Szasz operators $S_{n, \alpha}(f, x)$. We study the rate of convergence of $S_{n, \alpha}(f, x)$ for the case $0 < \alpha < 1$. Знайдено оцінку швидкості збіжності функцій обмеженої варіації для версії Без'є операторів Caca $S_{n, \alpha}(f, x)$. Вивчено швидкості збіжності $S_{n, \alpha}(f, x)$ для $0 < \alpha < 1$. Institute of Mathematics, NAS of Ukraine 2005-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3713 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 12 (2005); 1619–1624 Український математичний журнал; Том 57 № 12 (2005); 1619–1624 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3713/4151 https://umj.imath.kiev.ua/index.php/umj/article/view/3713/4152 Copyright (c) 2005 Gupta Vijay |
| spellingShingle | Gupta, Vijay Гупта, В. Rate of convergence for Szász-Bézier operators |
| title | Rate of convergence for Szász-Bézier operators |
| title_alt | Швидкість збіжності операторів Сaca - Без'є |
| title_full | Rate of convergence for Szász-Bézier operators |
| title_fullStr | Rate of convergence for Szász-Bézier operators |
| title_full_unstemmed | Rate of convergence for Szász-Bézier operators |
| title_short | Rate of convergence for Szász-Bézier operators |
| title_sort | rate of convergence for szász-bézier operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3713 |
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