Miskolc Mathematical Notes
Vol. 14 (2013), No. 1, pp. 219–231
HU e-ISSN 1787-2413
THE VORONOVSKAJA TYPE THEOREM FOR A GENERAL
CLASS OF SZÁSZ-MIRAKJAN OPERATORS
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
Received June 8, 2011
Abstract. The paper is devoted to defining a new general class of linear and positive operators
depending on a certain function '. This new class of linear and positive operators generalizes the
Szász-Mirakjan operators. For this new class of operators we establish a convergence theorem
and the evaluation of the rate of convergence, in terms of the modulus of continuity.
2000 Mathematics Subject Classification: 41A10; 41A25; 41A36
Keywords: Voronovskaja’s type theorem, Szász-Mirakjan operators, positivity, linearity, modulus of continuity
1. I NTRODUCTION
Let N be the set of positive integers and N0 D N [ f0g: In this section we recall
some results from [12], which we shall use in the present paper.
Let I; J be real intervals and I \ J ¤ ¿: For any n; k 2 N0 , n ¤ 0 consider the
functions 'n;k W J ! R, with the property that 'n;k .x/ 0, for any x 2 J and the
linear positive functionals An;k W E.I / ! R:
For any n 2 N define the operator Ln W E.I / ! F .J /, by
.Ln f /.x/ D
1
X
'n;k .x/An;k .f /;
(1.1)
kD0
where E.I / is a linear space of real-valued functions defined on I , for which the
operators (1.1) are convergent and F .J / is a subset of the set of real-valued functions
defined on J .
Remark 1. [12] The operators .Ln /n2N are linear and positive on E.I \ J /.
For n; i 2 N0 , n ¤ 0, let the function
Œ0; C1ŒŒ0; C1Œ and define Tn;i
by
.Tn;i
Ln /.x/ D ni .Ln
i
x /.x/
D ni
x
1
X
be defined by
'n;k .x/An;k .
x .t /
i
x /;
Dt
x, .x; t / 2
x 2 I \ J:
(1.2)
kD0
c 2013 Miskolc University Press
220
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
In what follows s 2 N0 is even and we suppose that the following two conditions hold:
there exists the smallest ˛s ; ˛sC2 2 Œ0; C1Œ, so that
Ln /.x/
.Tn;j
lim
D Bj .x/ 2 R;
n!1
n˛j
for any x 2 I \ J and j 2 fs; s C 2g,
˛sC2 < ˛s C 2
(1.3)
(1.4)
I \ J is an interval.
Theorem 1 ([12]). Let f 2 E.I / be a function. If x 2 I \ J and f is s times
differentiable in a neighborhood of x, f .s/ is continuous in x, then
!
s
.i / .x/
X
f
lim ns ˛s .Ln f /.x/
.Tn;i
Ln /.x/ D 0:
(1.5)
n!1
ni i Š
i D0
Assume that f is s times differentiable on I and there exists an interval K I \ J
such that, there exists n.s/ 2 N and the constants kj 2 R depending on K, so that for
n n.s/ and x 2 K, the following
.Tn;j
Ln /.x/
n˛j
kj
(1.6)
holds, for j 2 fs; s C 2g.
Then, the convergence expressed by (1.5) is uniform on K and
ˇ
ˇ
s
ˇ
ˇ
X
f .i / .x/ ˇ
s ˛s ˇ
n
.T
L
/.x/
(1.7)
.L
f
/.x/
ˇ
ˇ n
n;i n
ˇ
ˇ
ni i Š
i D0
1
1
.s/
.ks C ksC2 /!1 f I p
;
sŠ
n2C˛s ˛sC2
for any x 2 K, n n.s/, where !1 .f I ı/ denotes the modulus of continuity [1], (the
first order modulus of smoothness) of the function f .
In [11], C. Mortici defined the sequence of operators
'Sn W C 2 .Œ0; C1Œ/ ! C 1 .Œ0; C1Œ/;
given by
1
1 X ' .k/ .0/
.'Sn f /.x/ D
.nx/k f
'.nx/
kŠ
kD0
k
;
n
(1.8)
for any x 2 Œ0; C1Œ and n 2 N, where ' W R !0; C1Œ is an analytic function. These
are called the '-Szász-Mirakjan operators, because in the case when '.y/ D e y , they
reduce to the classical Mirakjan-Favard-Szász operators [5, 10] and [15].
THE VORONOVSKAJA TYPE THEOREM
221
Remark 2. Similar generalization of this type are the operators defined and studied
by Jakimovski and Leviatan [7] or the operators defined by Baskakov in 1957 (see,
e.g., the book [2], subsection 5.3.11, p. 344, where they are attributed to Mastroianni). Another recently similar generalization of this type are the operators defined
and studied by authors in [3, 4, 8, 9] and [13].
Remark 3. The classical Mirakjan-Favard-Szász operators Sn W C2 .Œ0; C1Œ/ !
C.Œ0; C1Œ/ are defined by
1
X
k
.nx/k
f
;
(1.9)
.Sn f /.x/ D e nx
kŠ
n
kD0
where
f .x/
exists and is finite :
C2 .Œ0; C1Œ/ D f 2 C.Œ0; C1Œ/ W lim
x!1 1 C x 2
In the following, we shall use the classical definition of Mirakjan-Favard-Szász operators, i.e., f 2 C2 .Œ0; C1Œ/.
The purpose of this paper is to introduce a new linear positive operators of SzászMirakjan type, in order to use it in the theory of uniform approximation of functions,
which generalize some new, respectively older results, cited at the adequate moment.
We shall prove uniform convergence, Voronovskaja type formulas and the order of
approximation, for these new linear positive operators.
2. AUXILIARY RESULTS
We consider an analytic function ' W Œ0; C1Œ!0; C1Œ and it follows
'.x/ D
1
X
' .k/ .0/
kD0
kŠ
xk :
(2.1)
By differentiation of the relation (2.1), we get
'
.1/
1
X
' .k/ .0/ k
.x/ D
x
.k 1/Š
1
(2.2)
1
X
' .k/ .0/ k
x
.k 2/Š
2
(2.3)
kD1
and
' .2/ .x/ D
:
kD2
We consider the function ˛ W Œ0; C1Œ! R defined by
˛.x/ D
for any x 2 Œ0; C1Œ.
' .1/ .x/
'.x/
1;
(2.4)
222
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
Remark 4. From (2.4) it follows that ˛ is also an analytic function.
Let .an /n2N be a sequence, so that for any n 2 N, an > 0 and we assume that the
function ' has the properties
an ' .1/ .an x/
D1
x!1 n '.an x/
(2.5)
a 2 ' .2/ .a x/
n
n
D 1;
x!1 n
'.an x/
(2.6)
lim
and
lim
where x 2 Œ0; C1Œ.
Lemma 1. The following identity
Rx
xC ˛.t /dt
'.x/ D c e
(2.7)
0
holds, for any x 2 Œ0; C1Œ, where c > 0.
Proof. From (2.4), we get
' .1/ .x/
.1 C ˛.x//'.x/ D 0;
for any x 2 Œ0; C1Œ. This relation is equivalent to
0
1.1/
Rx
x
˛.t /dt
@'.x/ e
0
A
D 0;
for any x 2 Œ0; C1Œ. Because ' W Œ0; C1Œ!0; C1Œ exists c > 0, so that
x
'.x/ e
Rx
˛.t /dt
0
D c;
for any x 2 Œ0; C1Œ.
The relation (2.7) follows immediately from the above equality.
3. T HE '-S Z ÁSZ -M IRAKJAN TYPE OPERATORS
We define the sequence of operators 'Sn W C2 .Œ0; C1Œ/ ! C.Œ0; C1Œ/, given by
1
X
' .k/ .0/
k
1
k
.'Sn f /.x/ D
.an x/ f
;
(3.1)
'.an x/
kŠ
n
kD0
for any x 2 Œ0; C1Œ and n 2 N. The operators (3.1) are called the '-Szász-Mirakjan
type operators, because in the case when an D n, for any n 2 N, the '-Szász-Mirakjan
operators [11] are obtained. In the case when an D n, for any n 2 N and '.x/ D e x ,
for any x 2 Œ0; C1Œ, the operators (3.1) become the classical Mirakjan-Favard-Szász
operators.
THE VORONOVSKAJA TYPE THEOREM
223
Lemma 2. Let ej .x/ D x j , j 2 f0; 1; 2g be the test functions. The '-SzászMirakjan type operators satisfy
i) .'Sn e0 /.x/ D 1;
an ' .1/ .an x/
ii) .'Sn e1 /.x/ D
x;
n '.an x/
a 2 ' .2/ .a x/
1 an ' .1/ .an x/
n
n
iii) .'Sn e2 /.x/ D
x2 C
x;
n
'.an x/
n n '.an x/
for any x 2 Œ0; C1Œ and n 2 N.
Proof. We get
1
.'Sn e0 /.x/ D
X ' .k/ .0/
1
.an x/k D 1;
'.an x/
kŠ
kD0
taking (2.1) into account.
Next, we get
1
.'Sn e1 /.x/ D
X ' .k/ .0/
k
1
.an x/k
'.an x/
kŠ
n
kD0
1
X
an x
' .k/ .0/
D
.an x/k
n '.an x/
.k 1/Š
kD1
1
D
an ' .1/ .an x/
x;
n '.an x/
taking (2.2) into account. In a similar way, one obtains
2
1
X
' .k/ .0/
1
k k
.an x/
.'Sn e2 /.x/ D
'.an x/
kŠ
n
kD0
D
D
a 2
n
n
1
1
2
n '.an x/
1
X
' .k/ .0/
kD0
kŠ
.an x/k .k.k
1
x 2 X ' .k/ .0/
.an x/k
'.an x/
.k 2/Š
kD2
1/ C k/
1
2
C
X ' .k/ .0/
an x
.an x/k
n2 '.an x/
.k 1/Š
1
kD1
a 2 ' .2/ .a x/
1 an ' .1/ .an x/
n
n
x2 C
x;
D
n
'.an x/
n n '.an x/
taking (2.2) and (2.3) into account.
224
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
Lemma 3. Let ej .x/ D x j , j 2 f3; 4g be the test functions. The '-Szász-Mirakjan
type operators satisfy
a 3 ' .3/ .a x/
3 an 2 ' .2/ .an x/ 2
n
n
.'Sn e3 /.x/ D
x3 C
x
(3.2)
n
'.an x/
n n
'.an x/
C
1 an ' .1/ .an x/
x
n2 n '.an x/
and
a 4 ' .4/ .a x/
6 an 3 ' .3/ .an x/ 3
n
n
x4 C
x
n
'.an x/
n n
'.an x/
7 an 2 ' .2/ .an x/ 2
1 an ' .1/ .an x/
C 2
x C 3
x;
n
n
'.an x/
n n '.an x/
for any x 2 Œ0; C1Œ and n 2 N:
.'Sn e4 /.x/ D
(3.3)
Proof. Because the function ' is analytic by differentiation, it follows
' .3/ .x/ D
1
X
' .k/ .0/ k
x
.k 3/Š
3
; ' .4/ .x/ D
kD3
1
X
' .k/ .0/ k
x
.k 4/Š
4
:
kD4
Using the above relations and (2.1), (2.2), then from (3.1), it follows
3
1
X
1
k
' .k/ .0/
.'Sn e3 /.x/ D
.an x/k
'.an x/
kŠ
n
kD0
D
D
C
1
1
3
n '.an x/
a 3
n
n
kŠ
kD0
.an x/k u1 .k/
1
x 3 X ' .k/ .0/
.an x/k
'.an x/
.k 3/Š
3 a 2
n
n
1
X
' .k/ .0/
n
x2
kD3
1
X
'.an x/
kD2
3
' .k/ .0/
.an x/k
.k 2/Š
1
2
C
X ' .k/ .0/
1 an x
.an x/k
n2 n '.an x/
.k 1/Š
kD1
a 3 ' .3/ .a x/
3 an 2 ' .2/ .an x/ 2
1 an ' .1/ .an x/
n
n
D
x3 C
x C 2
x;
n
'.an x/
n n
'.an x/
n n '.an x/
where u1 .k/ D k.k 1/.k 2/ C 3k.k 1/ C k.
4
1
X
' .k/ .0/
1
k k
.'Sn e4 /.x/ D
.an x/
'.an x/
kŠ
n
kD0
D
1
1
4
n '.an x/
1
X
' .k/ .0/
kD0
kŠ
.an x/k u2 .k/
1
THE VORONOVSKAJA TYPE THEOREM
1
1
4
6 an 3 x 3 X ' .k/ .0/
C
.an x/k
n n
'.an x/
.k 3/Š
3
1
7 an 2 x 2 X ' .k/ .0/
.an x/k
C 2
n
n
'.an x/
.k 2/Š
1
X
1 an x
' .k/ .0/
2
C 3
.an x/k
n n '.an x/
.k 1/Š
1
D
a 4
n
n
x 4 X ' .k/ .0/
.an x/k
'.an x/
.k 4/Š
225
kD4
kD2
kD3
kD1
a 4 ' .4/ .a x/
6 an 3 ' .3/ .an x/ 3
7 an 2 ' .2/ .an x/ 2
n
n
x4 C
x C 2
x
D
n
'.an x/
n n
'.an x/
n
n
'.an x/
1 an ' .1/ .an x/
x;
n3 n '.an x/
where u2 .k/ D k.k 1/.k
C
2/.k
3/ C 6k.k
2/ C 7k.k
1/.k
1/ C k.
Lemma 4. For any x 2 Œ0; C1Œ and n 2 N, the following
.Tn;0
'Sn /.x/ D 1;
.Tn;1
'Sn /.x/ D n
.Tn;2
'Sn /.x/
2
Dn
an ' .1/ .an x/
n '.an x/
a 2 ' .2/ .a x/
n
n
n
'.an x/
C
(3.4)
!
1 x;
!
an ' .1/ .an x/
2
C 1 x2
n '.an x/
!
(3.5)
(3.6)
1 an ' .1/ .an x/
x ;
n n '.an x/
a 3 ' .3/ .a x/
n
n
(3.7)
n
'.an x/
!
a 2 ' .2/ .a x/
an ' .1/ .an x/
n
n
C6
4
C 1 x4
n
'.an x/
n '.an x/
!
a 2 ' .2/ .a x/ a ' .1/ .a x/
6 an 3 ' .3/ .an x/
n
n
n
n
C
2
C
x3
n
n
'.an x/
n
'.an x/
n '.an x/
!
!
a 2 ' .2/ .a x/
1 an ' .1/ .an x/
an ' .1/ .an x/
1
n
n
2
C 2 7
4
x C 3
x
n
n
'.an x/
n '.an x/
n n '.an x/
.Tn;4
'Sn /.x/ D n4
a 4 ' .4/ .a x/
n
n
n
'.an x/
4
hold.
Proof. Taking into account relations (1.2), (3.1), Lemma 2 and Lemma 3, we get
.Tn;0
'Sn /.x/ D .'Sn e0 /.x/ D 1;
.Tn;1
'Sn /.x/ D n.'Sn
x /.x/
D n ..'Sn e1 /.x/
x.'Sn e0 /.x//
226
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
an ' .1/ .an x/
Dn
n '.an x/
!
1 x;
.Tn;2
'Sn /.x/ D n2 .'Sn x2 /.x/ D n2 .'Sn e2 /.x/ 2x.'Sn e1 /.x/ C x 2 .'Sn e0 /.x/
!
!
a 2 ' .2/ .a x/
.1/ .a x/
.1/ .a x/
a
'
1
a
'
n
n
n
n
n
n
2
C 1 x2 C
x ;
D n2
n
'.an x/
n '.an x/
n n '.an x/
.Tn;4
'Sn /.x/ D n4 .'Sn
4
x /.x/
3
D n4 ..'Sn e4 /.x/
4x.'Sn e3 /.x/
C6x .'Sn e2 /.x/ 4x .'Sn e1 /.x/ C x .'Sn e0 /.x/
a 3 ' .3/ .a x/
a 4 ' .4/ .a x/
n
n
n
n
4
4
Dn
n
'.an x/
n
'.an x/
!
a 2 ' .2/ .a x/
an ' .1/ .an x/
n
n
C6
4
C 1 x4
n
'.an x/
n '.an x/
!
a 2 ' .2/ .a x/ a ' .1/ .a x/
6 an 3 ' .3/ .an x/
n
n
n
n
x3
C
2
C
n
n
'.an x/
n
'.an x/
n '.an x/
!
!
a 2 ' .2/ .a x/
1
an ' .1/ .an x/
1 an ' .1/ .an x/
n
n
2
C 2 7
4
x C 3
x :
n
n
'.an x/
n '.an x/
n n '.an x/
2
4
Remark 5. Taking (2.5) and (2.6) into account, for x 2 Œ0; C1Œ we have
!
a 2 ' .2/ .a x/
an ' .1/ .an x/
n
n
2
C 1 D 0:
lim
n!1
n
'.an x/
n '.an x/
In the following, we assume that there exist ; ı, so that 0 < 1, ı 2, < ı,
and the function ' verifies the conditions
!
a 2 ' .2/ .a x/
an ' .1/ .an x/
n
n
lim n
2
C 1 D ˇ2 .x/;
(3.8)
n!1
n
'.an x/
n '.an x/
!
a 3 ' .3/ .a x/
a 2 ' .2/ .a x/ a ' .1/ .a x/
n
n
n
n
n
n
lim nı 1
2
C
D ˇ3 .x/
n!1
n
'.an x/
n
'.an x/
n '.an x/
(3.9)
and
a 4 ' .4/ .a x/
a 3 ' .3/ .a x/
a 2 ' .2/ .a x/
n
n
n
n
n
n
lim nı
4
C6
(3.10)
n!1
n
'.an x/
n
'.an x/
n
'.an x/
THE VORONOVSKAJA TYPE THEOREM
227
!
an ' .1/ .an x/
C 1 D ˇ4 .x/;
4
n '.an x/
for any x 2 Œ0; C1Œ, where ˇ2 ; ˇ3 ; ˇ4 are functions, ˇ2 ; ˇ3 ; ˇ4 W Œ0; C1Œ! R.
Lemma 5. For any x 2 Œ0; C1Œ, the following identities
lim .Tn;0
'Sn /.x/ D 1;
n!1
'Sn /.x/
.Tn;2
D ˇ2 .x/ x 2 C . / x;
n!1
n2 'Sn /.x/
.Tn;4
lim
D ˇ4 .x/ x 4 C 6ˇ3 .x/ x 3 C 3.ı/ x 2
n!1
n4 ı
lim
(3.11)
(3.12)
(3.13)
hold and there exists n0 2 N, so that
.Tn;0
'Sn /.x/ D 1 D k0 ;
'Sn /.x/
.Tn;2
.Tn;4
'Sn /.x/
n2
m2 .K/ b 2 C b C 1 D k2 ;
(3.14)
(3.15)
m4 .K/ b 4 C 6m3 .K/ b 3 C 3b 2 C 1 D k4 ;
(3.16)
n4 ı
for any x 2 K D Œ0; b, b > 0, any n 2 N; n n0 and m2 .K/ WD sup jˇ2 .x/j;
( x2K
1; D 1
m3 .K/ WD sup jˇ3 .x/j, m4 .K/ WD sup jˇ4 .x/j, where . / D
0; 0 < < 1
x2K
x2K
(
1; ı D 2
and .ı/ D
0; ı < 2:
Proof. The identities (3.11)-(3.13) follow from Lemma 4, while (3.14)-(3.16) follow from (3.11)-(3.13) by taking the definition of the limit into account.
Let us assume that I D J D Œ0; C1Œ; E.I / D C2 .Œ0; C1Œ/, F .J / D C.Œ0; C1Œ/,
1 ' .k/ .0/
the functions 'n;k W Œ0; C1Œ! R let be defined by 'n;k .x/ D
.an x/k ,
'.an x/ kŠ
for any x 2 Œ0; C1Œ, any n; k 2 N0, n¤ 0 and the functionals An;k W C2 .Œ0; C1Œ/ !
R let be defined by An;k .f / D f kn , for any n; k 2 N0 , n ¤ 0. In this way we get
the '-Szász-Mirakjan type operators.
228
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
Theorem 2. Let the function f 2 E.I / be given. If x 2 Œ0; C1Œ, f is s times
differentiable in a neighborhood of x, then:
lim .'Sn f /.x/ D f .x/;
(3.17)
n!1
if s D 0 and
lim n
n!1
!
!
an ' .1/ .an x/
.'Sn f /.x/ f .x/
1 xf .1/ .x/
n '.an x/
1
D ˇ2 .x/ x 2 C . / x f .2/ .x/;
2
(3.18)
if s D 2.
If f is s times differentiable on Œ0; C1Œ, then the convergence from (3.17) and (3.18)
is uniform on any compact interval K D Œ0; b Œ0; C1Œ.
Moreover, we get
1
j.'Sn f /.x/ f .x/j .1 C k2 /!1 f I p
;
(3.19)
n
for any f 2 C2 .Œ0; C1Œ/, any x 2 K and n 2 N, n n0 .
Proof. It follows from Theorem 1, with ˛0 D 0; ˛2 D 2 and ˛4 D 4 ı, Lemma
4 and Lemma 5.
Remark 6. If an D n, for any n 2 N and '.x/ D e x , for any x 2 Œ0; C1Œ we get
the well-known results for the classical Szász-Mirakjan operators. In this case D 1,
ı D 2, ˇ2 .x/ D ˇ3 .x/ D ˇ4 .x/ D 0, for any x 2 Œ0; C1Œ and k2 D b, k4 D 3b 2 C b,
(see [12]).
4. OTHER FORM FOR THE S Z ÁSZ -M IRAKJAN OPERATORS
In this section, we assume that c D 1 in the relation (2.7). In [6] or [14], we can
find some known results about differentiation of composed functions.
Theorem 3 ([14]). Let I; J be two intervals of real numbers and let f W I !
R, g W J ! R be two real-valued functions, such that f .I / J . If f is k times
differentiable on I and g is k times differentiable on J , k 2 N, then the function
g ı f W I ! R is also k times differentiable on I and the following identity
.g ı f /.k/ .x/ D
k X
g .i / ı f .x/
i D1
f .1/ .x/
1Š
holds, for any x 2 I .
X
a1 Ca2 C:::Cak Di
a1 C2a2 C:::Ckak Dk
!a1
f .2/ .x/
2Š
!a2
kŠ
a1 Ša2 Š : : : ak Š
f .k/ .x/
:::
kŠ
!ak
(4.1)
THE VORONOVSKAJA TYPE THEOREM
229
Theorem 4. If k 2 N, then the following identity
'
.k/
.0/ D
k
X
i D1
X
a1 Ca2 C:::Cak Di
a1 C2a2 C:::Ckak Dk
˛ .1/ .0/
2Š
kŠ
a1 Ša2 Š : : : ak Š
!a2
:::
˛ .k
1/ .0/
1 C ˛.0/
1Š
a1
(4.2)
!ak
kŠ
holds.
Rx
Proof. One applies Theorem 3, with g.x/ D e x , for x 2 R and f .x/ D x C ˛.t /dt ,
0
for x 2 Œ0; C1Œ.
Remark 7. The relation (3.1) represents the form of '-Szász-Mirakjan type operators, by using the function '. By replacing the form of ' .k/ .0/, from the relation
(4.2) in (3.1), we get the form of the '-Szász-Mirakjan type operators, where appears
the function ˛.
Now we can give some applications.
Example 1. If an D n, n 2 N and ˛.x/ D 0, for any x 2 Œ0; C1Œ, from (4.2)
follows that ' .k/ .0/ D 1, for any k 2 N. In this case, we get the classical SzászMirakjan operators.
Example 2. Suppose that ˛.x/ D
. 1/p .p C 1/Š
, '.x/ D e xC1
.x C 1/pC2
' .k/ .0/ D
k
X
i D1
1
xC1
1
, for any x 2 Œ0; C1Œ. Then ˛ .p/ .x/ D
.x C 1/2
, for any x 2 Œ0; C1Œ, any p 2 N0 and we get
X
a1 Ca2 C:::Cak Di
a1 C2a2 C:::Ckak Dk
kŠ
2a1 . 1/1a2 C2a3 C:::C.k
a1 Ša2 Š : : : ak Š
1/ak
:
Taking into account a1 C 2a2 C : : : C kak D k and a1 C a2 C : : : C ak D i , we get
'
.k/
.0/ D
k
X
i D1
X
a1 Ca2 C:::Cak Di
a1 C2a2 C:::Ckak Dk
kŠ
. 1/k i 2a1 ;
a1 Ša2 Š : : : ak Š
from where, it follows
'
.k/
.0/ D . 1/
k
k
X
i D1
X
a1 Ca2 C:::Cak Di
a1 C2a2 C:::Ckak Dk
. 1/i
kŠ
2 a1 :
a1 Ša2 Š : : : ak Š
(4.3)
230
OVIDIU T. POP, DAN MICLĂUŞ, AND DAN BĂRBOSU
If we consider an D n C n1 , for any n 2 N, the operators defined by (3.1) become
. 1/k
1
1
.'Sn f /.x/ D e .nC n /xC1
.nC n1 /x
1
1
X
k
P
i D1
P
a1 Ca2 C:::Cak Di
a1 C2a2 C:::Ckak Dk
kD0
. 1/i a1 Ša2kŠŠ:::ak Š 2a1
kŠ
k k
1
nC
x f
;
n
n
for any f 2 C2 .Œ0; C1Œ/, x 2 Œ0; C1Œ and n 2 N.
ACKNOWLEDGEMENT
We thank the reviewer for his suggestions leading to the improvement of the first
version of this paper.
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THE VORONOVSKAJA TYPE THEOREM
231
Authors’ addresses
Ovidiu T. Pop
National College Mihai Eminescu, Mihai Eminescu 5, 440014 Satu Mare, Romania
E-mail address: ovidiutiberiu@yahoo.com
Dan Miclăuş
North University of Baia Mare, Department of Mathematics and Computer Science, Victoriei 76,
430122 Baia Mare, Romania
E-mail address: danmiclausrz@yahoo.com
Dan Bărbosu
North University of Baia Mare, Department of Mathematics and Computer Science, Victoriei 76,
430122 Baia Mare, Romania
E-mail address: barbosudan@yahoo.com