A note on mixed summation-integral-type operators
Very recently Deo, in the paper “Simultaneous approximation by Lupas operators with weighted function of Szasz operators” [J. Inequal. Pure Appl. Math., 5, No. 4 (2004)] claimed to introduce the integral modifications of Lupas operators. These operators were first introduced in 1993 by Gupta and Sri...
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irk-123456789-1724772020-11-03T01:26:23Z A note on mixed summation-integral-type operators Gupta, M.K. Manoj Kumar Rupen Pratap Singh Короткі повідомлення Very recently Deo, in the paper “Simultaneous approximation by Lupas operators with weighted function of Szasz operators” [J. Inequal. Pure Appl. Math., 5, No. 4 (2004)] claimed to introduce the integral modifications of Lupas operators. These operators were first introduced in 1993 by Gupta and Srivastava. They estimated the simultaneous approximation for these operators and called them Baskakov-Szasz operators. There are several misprints in the paper by Deo. This motivated us to perform subsequent investigations in this direction. We extend the study and estimate a saturation result in simultaneous approximation for the linear combinations of these summation-integral-type operators. Нещодавно Део у роботі "Simultaneous approximation by Lupas operators with weighted function of Szasz operators" (J. Inequal. Pure and Appl. Math., 2004, Vol. 5, № 4) заявив про введення ним інтегральних модифікацій операторів Лупаса. Вперше такі оператори ввели Гупта та Шрівастава у 1993 р. Вони оцінили одночасне наближення цих операторів та назвали їх операторами Васкакова - Шаша. У роботі Део є кілька неточностей. Це спонукало авторів продовжити дослідження у згаданому напрямі. У даній статті розширено коло досліджень та отримано оцінку результату щодо насичення при одночасному наближенні для лінійних комбінацій цих операторів сумовно-інтегрального типу. 2007 Article A note on mixed summation-integral-type operators / М.К. Gupta, Manoj Kumar, Rupen Pratap Singh // Український математичний журнал. — 2007. — Т. 59, № 8. — С. 1135–1139. — Бібліогр.: 6 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/172477 517.5 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Gupta, M.K. Manoj Kumar Rupen Pratap Singh A note on mixed summation-integral-type operators Український математичний журнал |
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Very recently Deo, in the paper “Simultaneous approximation by Lupas operators with weighted function of Szasz operators” [J. Inequal. Pure Appl. Math., 5, No. 4 (2004)] claimed to introduce the integral modifications of Lupas operators. These operators were first introduced in 1993 by Gupta and Srivastava. They estimated the simultaneous approximation for these operators and called them Baskakov-Szasz operators. There are several misprints in the paper by Deo. This motivated us to perform subsequent investigations in this direction. We extend the study and estimate a saturation result in simultaneous approximation for the linear combinations of these summation-integral-type operators. |
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Gupta, M.K. Manoj Kumar Rupen Pratap Singh |
| author_facet |
Gupta, M.K. Manoj Kumar Rupen Pratap Singh |
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Gupta, M.K. |
| title |
A note on mixed summation-integral-type operators |
| title_short |
A note on mixed summation-integral-type operators |
| title_full |
A note on mixed summation-integral-type operators |
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A note on mixed summation-integral-type operators |
| title_full_unstemmed |
A note on mixed summation-integral-type operators |
| title_sort |
note on mixed summation-integral-type operators |
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Інститут математики НАН України |
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2007 |
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Короткі повідомлення |
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A note on mixed summation-integral-type operators / М.К. Gupta, Manoj Kumar, Rupen Pratap Singh // Український математичний журнал. — 2007. — Т. 59, № 8. — С. 1135–1139. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний журнал |
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2025-07-15T08:47:02Z |
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2025-07-15T08:47:02Z |
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1837702025691791360 |
| fulltext |
UDC 517.5
M. K. Gupta, Manoj Kumar, Rupen Pratap Singh (Ch Charan Singh Univ., India)
A NOTE ON MIXED SUMMATION
INTEGRAL TYPE OPERATORS
PRO OPERATORY MIÍANOHO
SUMOVNO-INTEHRAL|NOHO TYPU
Very recently Deo in the paper ”Simultaneous approximation by Lupas operators with weighted function
of Szasz operators” (J. Inequal. Pure and Appl. Math., 2004, Vol. 5, # 4) claimed to introduce the
integral modifications of Lupas operators. These operators were first introduced in the year 1993 by
Gupta and Srivastava. They have estimated the simultaneous approximation for these operators and
termed these operators as Baskakov – Szasz operators. There are several misprints in the paper of Deo.
This motivated us to study further in this direction and, in the present paper, we extend the study and
estimate a saturation result in simultaneous approximation for the linear combinations of these
summation integral type operators.
Newodavno Deo u roboti “Simultaneous approximation by Lupas operators with weighted function of
Szasz operators” (J. Inequal. Pure and Appl. Math., 2004, Vol. 5, # 4) zaqvyv pro vvedennq nym in-
tehral\nyx modyfikacij operatoriv Lupasa. Vperße taki operatory vvely Hupta ta Írivastava
u 1993*r. Vony ocinyly odnoçasne nablyΩennq cyx operatoriv ta nazvaly ]x operatoramy Baska-
kova – Íaßa. U roboti Deo [ kil\ka netoçnostej. Ce sponukalo avtoriv prodovΩyty doslid-
Ωennq u zhadanomu naprqmi. U danij statti rozßyreno kolo doslidΩen\ ta otrymano ocinku re-
zul\tatu wodo nasyçennq pry odnoçasnomu nablyΩenni dlq linijnyx kombinacij cyx operatoriv
sumovno-intehral\noho typu.
1. Introduction. Gupta and Srivastava [1] introduced the sequence of linear positive
operators by combining the well-known Baskakov (Lupas) and Szasz basis functions in
summation and integration respectively, to approximate Lebesgue integrable functions
on the interval [ 0, ∞ ) as
Bn ( f, x ) = n p x s t f t dtn n, ,( ) ( ) ( )ν ν
ν 00
∞
=
∞
∫∑ , (1.1)
where f C f C f t Me M∈ ∞ ≡ ∈ ∞ ≤ > >γ γ
γ γ[ , ) [ , ) : ( ) ,{ }0 0 0 0for some and
p xn k, ( ) =
n x
x n
+ −
+ +
ν
ν
ν
ν
1
1( )
, s tn, ( )ν = e ntnt− ( )
!
ν
ν
.
The norm γ is defined as f γ = sup ( )
0< <∞
−
t
tf t e γ .
In [2] Deo also claimed to introduce these operators. In the same paper, Deo esti-
mated the direct theorems in simultaneous approximation for the operators (1.1).
Actually, the direct theorems in simultaneous approximation for a more general class
had already been obtained by Gupta and Srivastava in [1].
It turns out that the order of approximation for the operators (1.1) is at best
O n( )−1 . Thus, to improve the order of approximation, Gupta and Srivastava [3]
considered the linear combinations of operators (1.1), which are defined as follows:
For a fixed natural number k and arbitrary fixed distinct positive integers dj
, j =
= 0, 1, 2, … , k
, the linear conbinations Bn ( f, k, x ) of B f xd nj
( , ) are defined as
Bn ( f, k, x ) =
j
k
d nC j k B f x
j
=
∑
0
( , ) ( , ) , (1.2)
where
© M. K. GUPTA, MANOJ KUMAR, RUPEN PRATAP SINGH, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8 1135
1136 M. K. GUPTA, MANOJ KUMAR, RUPEN PRATAP SINGH
C j k( , ) =
i
i j
k
j
j i
d
d d=
≠
∏ −0
, k ≠ 0, C( , )0 0 = 1.
This type of linear combinations was first considered by May [4] to improve the order
of approximation for exponential type operators. Gupta and Srivastava [3] estimated a
Voronovskaja-type asymptotic formula and error estimation in simultaneous approxi-
mation for Bn ( f, k, x ) . In [5, 6], respectively, the corresponding direct estimate in
terms of higher order modulus of continuity and an inverse theorem were established.
Actually, a saturation result is a more curious phenomenon. The order of approximati-
on beyond a certain limit O n( ( ))φ , φ( )n → 0, n → ∞ , is possible only for a trivial
subspace. The function for which O n( ( ))φ approximation is attained form the Favard
class and those with o n( ( ))φ approximation forma trivial class. Thus, a saturation re-
sult consists of a determination of a saturation order φ( )n , the Favard class, and the
trivial class. In the present paper, we extend the study and estimate a saturation theo-
rem in simultaneous approximation for the linear combinations of the Baskakov –
Szasz operators defined by (1.2).
2. Auxiliary results. In this section, we mention certain lemmas and definitions,
which are necessary to prove the saturation theorem.
Lemma 2.1 [1]. Let the function µn m x, ( ), m N∈ 0 , be defined as
µn m x, ( ) = n p x b t t x dtn r n r
m
ν
ν ν
=
∞
+
∞
+∑ ∫ −
0 0
, ,( ) ( )( ) .
Then
µn x, ( )0 = 1, µn x, ( )1 =
1 1+ +r x
n
( )
,
µn x, ( )2 =
rx x r x nx x
n
( ) [ ( )] ( )1 1 1 1 22
2
+ + + + + +
,
and we also have the recurrence relation
n xn mµ , ( )+1 = x x x m r x x mx x xn m n m n m( ) ( ) [( ) ( )] ( ) ( ) ( ),
( )
, ,1 1 1 21
1+ + + + + + + −µ µ µ ,
m ≥ 1.
Consequently, for each x ∈ [ 0, ∞ ) , we have from this recurrence relation that
µn m x, ( ) = O n m( )[( )/ ]− +1 2 .
Remark 2.1. It is remarked here that the above Lemma 2.1 was not estimated
correctly by Deo [2]. In the recurrence relation, he missed the last term on the right-
hand side of the recurrence relation.
Remark 2.2. The main result, i.e., Theorem 3.1 of [2], is not correct. Actually,
the last term in the conclusion must be divided by 2.
Remark 2.3. In Theorem 3.2 of [2], the existence of f r( )+1 on [ 0, ∞ ) is used
while in the hypothesis its existence is assumed only on the interval [ 0, a] .
Lemma 2.2 [6]. Let f C∈ ∞γ [ , )0 . If f k r( )2 2+ + exists at a fixed point x ∈ ( 0,
∞ ) , then
lim ( , , ) ( ){ }( ) ( )
n
k
n
r rn B f k x f x
→∞
+ −1 =
i r
k r
iQ i k r x f x
=
+ +
∑
2 2
( , , , ) ( )( ) ,
where Q i k r x( , , , ) are certain polynomials in x of degree at most i.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
A NOTE ON MIXED SUMMATION INTEGRAL TYPE OPERATORS 1137
By C0 we denote the set of continuous functions on the interval [ a, b] having the
compact support and Ck
0 the subset of C0 of k-times continuously differentiable
functions.
Lemma 2.3. Let f C∈ ∞γ [ , )0 and g C∈ ∞
0 with supp g a b⊂ ( , ). Then
n B f k B f k gk
n
r
n
r+ ⋅ − ⋅1
2[ ( , , ) ( , , )],( ) ( ) ≤ M f γ ,
where the constant M is independent of f and n and 〈 〉h g, =
0
∞
∫ h t g t dt( ) ( ) .
A function f continuous in the interval [ a, b] is said to belong to generalized
Zygmund class Liz ( α, k, a, b ) , 0 < α < 2, k ∈ N , if there exist a constant M such
that
ω δ2k f a b( , , , ) ≤ M kδα , δ > 0,
where ω δ2k f a b( , , , ) denotes the modulus of continuity of 2k-th order of f on the
interval [ a, b] . In particular, we denote by Lip∗( , , )α a b the class Liz( , , , )α 1 a b .
Theorem 2.1 [6]. If 0 < α < 2, f C∈ ∞γ [ , )0 and 0 < a1 < a2 < a3 < b3 <
< b1 < b2 < ∞ , then the following statements (i) ⇒ (ii) ⇔ (iii) ⇒ (iv) hold true:
(i) f r( ) exist in the interval [ a1, b1] and
B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
1 1
= O n k( )( )/− +α 1 2 ;
(ii) f k a br( ) ( , , , )∈ +Liz α 1 2 2 ;
(iii) (a) for m < α( )k + 1 < m + 1, m = 0, 1, 2, … , 2k + 1, f r m( )+ exists and
belongs to the class Lip( ( ) , , )α k r a b+ −1 2 2 ;
(b) for α( )k + 1 = m + 1, m = 0, 1, 2, … , 2k, f r m( )+ exists and belongs
to the class Lip∗( , , )1 2 2a b ;
(iv) B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
3 3
= O n k( )( )/− +α 1 2 .
3. Saturation theorem. In this section, we shall prove the following main result:
Theorem 3.1. Let f C∈ ∞γ [ , )0 and 0 < a1 < a2 < a3 < b3 < b1 < b2 < ∞ .
Then in the following statements the implications (i) ⇒ (ii) ⇒ (iii) and (iv) ⇒ (v) ⇒
⇒ (vi) hold true:
(i) f r( ) exist in the interval [ a1, b1] and
B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
1 1
= O n k( )( )− +1 ;
(ii) f A C a bk r( ) . .[ , ]2 1
2 2
+ + ∈ and f L a bk r( ) [ , ]2 2
2 2
+ +
∞∈ ;
(iii) B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
3 3
= O n k( )( )− +1 ;
(iv) B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
1 1
= o n k( )( )− +1 ;
(v) f C a bk r∈ + +2 2
2 2[ , ] and
j k
k r
jQ j k r x f x
= +
+ +
∑
1
2 2
( , , , ) ( )( ) = 0, x a b∈[ , ]2 2 ,
where the polynomials Q j k r x( , , , ) are defined in Lemma 2.2;
(vi) B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
3 3
= o n k( )( )− +1 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1138 M. K. GUPTA, MANOJ KUMAR, RUPEN PRATAP SINGH
Proof. We first assume (i). Then in view of (i) ⇒ (iii) of Theorem 2.1, it fol-
lows that f k r( )2 1+ + exists and is continuous on ( , )a b1 1 . Moreover, it is obviously se-
en that the statements
B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
1 1
= O n k( )( )− +1 (3.1)
and
B f k B f kn
r
n
r
C a b2
1 1
( ) ( )
[ , ]
( , , ) ( , , )⋅ − ⋅ = O n k( )( )− +1 (3.2)
are equivalent. Obviously (3.1) ⇒ (3.2). We just have to show that (3.2) ⇒ (3.1).
Assuming (3.2), since lim ( , , )( )
n
n
rB f k x
→∞
= f xr( )( ) , we have
f xr( )( ) = B f k x B f k x B f k xn
r
n
r
n
r( ) ( ) ( )( , , ) ( , , ) ( , , )[ ]+ −2 +
+ [ ] [ ]( ) ( ) ( ) ( )( , , ) ( , , ) ( , , ) ( , , )B f k x B f k x B f k x B f k xn
r
n
r
n
r
n
r
m m4 2 2 2 1− + … + − − + … ,
B f k fn
r r
C a b
( ) ( )
[ , ]
( , , )⋅ −
1 1
≤ B f k B f kn
r
n
r
C a b2
1 1
( ) ( )
[ , ]
( , , ) ( , , )⋅ − ⋅ +
+ B f k B f k B f k B f kn
r
n
r
C a b n
r
n
r
C a b
m m4 2 2 21 1
1
1 1
( ) ( )
[ , ]
( ) ( )
[ , ]
( , , ) ( , , ) ( , , ) ( , , )⋅ − ⋅ + … + ⋅ − ⋅− + … .
Applying (3.2), in the above, (3.1) follows immediately.
We assume that { ( )}( ) ( )( , , ) ( , , )n B f k x B f k xk
n
r
n
r+ −1
2 is bounded as a sequence in
C a b[ , ]1 1 and hence in L a b∞[ , ]1 1 . Since L a b∞[ , ]1 1 is the dual space of L a b1 1 1[ , ],
by Alaoglu’s theorem there exists h L a b∈ ∞[ , ]1 1 such that for some sequence { }ni of
the natural numbers and for every g C∈ ∞
0 , with supp g a b⊂ ( , )1 1
lim ( , , ) ( , , ) ,( ) ( )
n
i
k
n
r
n
r
i
i i
n B f k B f k g
→∞
+ ⋅ − ⋅1
2 = 〈 〉h g, . (3.3)
Next, since C a b Ck r2 2
1 1 0+ + ∞[ , ] [ , )∩ γ is dense in Cγ [ , )0 ∞ , there exists a sequence
{ }fλ λ=
∞
1 in C a b Ck r2 2
1 1 0+ + ∞[ , ] [ , )∩ γ converging to f in ⋅ γ -norm. For any
g C∈ ∞
0 with supp g a b⊂ ( , )1 1 and each function fλ , making use of Lemma 2.2,
we get
lim ( , , ) ( , , ) ,( ) ( )
n
i
k
n
r
n
r
i
i i
n B f k B f k g
→∞
+ ⋅ − ⋅1
2 λ λ =
= − − ⋅ ⋅
+
=
+ +
∑1 1
2 1
1
2 2
k
j
k r
jQ j k x f g( , , ) ( ), ( )( )
λ =
= P D f gk r2 2+ + ⋅ ⋅( ) ( ), ( )λ = f P D gk rλ , ( ) ( )2 2+ +
∗ ⋅ , (3.4)
where P Dk2 2+
∗ ( ) is the dual operator of P Dk2 2+ ( ). Using Lemma 2.3, we have
lim ( , , ) ( , , ) ,( ) ( )
n
i
k
n
r
n
r
i
i i
n B f f k B f f k g
→∞
+ − ⋅ − − ⋅1
2 λ λ ≤ M f f− λ γ . (3.5)
Combining the estimates (3.3), (3.4), and (3.5), we obtain
f P D gk r, ( ) ( )2 2+ +
∗ ⋅ = lim , ( ) ( )
λ λ→∞
+ +
∗ ⋅f P D gk r2 2 =
= lim lim ( , , ) ( , , ) , ( ) ( )( ) ( )
λ λ λ λ→∞ →∞
+ +
∗− ⋅ − − ⋅ + ⋅
n
n
r
n
r
k r
i
i i
B f f k B f f k g f P D g2 2 2 =
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
A NOTE ON MIXED SUMMATION INTEGRAL TYPE OPERATORS 1139
= lim [ ( , , ) ( , , .)], ( )( ) ( )
n
i
k
n
r
n
r
i
i i
n B f k B f k g
→∞
+ ⋅ − ⋅1
2 = 〈 〉h g, , (3.6)
where
h ( x ) = P D f xk r2 2+ + ( ) ( )
as a generalized function. Since Q k r k x( , , )2 2+ + ≠ 0, by Lemma 2.2, regarding
(3.6) as a first order linear differential equation for f k r( )2 1+ + , we deduce that
f A C a bk r( ) . .[ , ]2 1
2 2
+ + ∈ and f L a bk r( ) [ , ]2 2
2 2
+ +
∞∈ . This completes the proof of (i) ⇒
⇒ (ii). Next, (ii) ⇒ (iii) follows from Lemma 2.2. The proof of (iv) ⇒ (v) is
similar to that of (i) ⇒ (ii) and (v) ⇒ (vi) follows from Lemma 2.2.
This completes the proof of saturation theorem.
1. Gupta V., Srivastava G. S. Simultaneous approximation by Baskakov – Szasz type operators //
Bull. Math. Soc. Sci. (N. S.) – 1993. – 37. – P. 73 – 85.
2. Deo N. Simultaneous approximation by Lupas operators with weighted function of Szasz operators
// J. Inequal Pure and Appl. Math. – 2004. – 5, # 4. – Art. 113.
3. Gupta V., Srivastava G. S. On simultaneous approximation by combinations of Baskakov – Szasz
type operators // Fasciculi Mat. – 1997. – 27. – P. 29 – 42.
4. May C. P. Saturation and inverse theorems for combination of a class of exponential type operators
// Can. J. Math. – 1976. – 28. – P. 1224 – 1250.
5. Gupta P., Gupta V. Rate of convergence on Baskakov – Szasz type operators // Fasciculi Mat. –
2001. – 31. – P. 37 – 44.
6. Gupta V., Maheshwari P. On Baskakov – Szasz type operators // Kyungpook Math. J. – 2003. –
43, # 3. – P. 315 – 325.
Received 21.07.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
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