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Some approximation results for generalized Kantorovich-type operators

  • 1,
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  • 2Email author
Journal of Inequalities and Applications20132013:585

https://doi.org/10.1186/1029-242X-2013-585

  • Received: 25 September 2013
  • Accepted: 18 November 2013
  • Published:

Abstract

In this paper, we construct a new family of operators, prove some approximation results in A-statistical sense and establish some direct theorems for Kantorovich-type integral operators.

MSC: Primary 41A10, 41A25, 41A36.

Keywords

  • Lagrange polynomial
  • Korovkin approximation theorems
  • A-statistical convergence
  • Kantorovich-type operators
  • positive linear operators
  • modulus of continuity
  • Peetre’s K-functional
  • Lipschitz class

1 Introduction and preliminaries

Statistical convergence [1] and its variants, extensions and generalizations have been proved to be an active area of recent research in summability theory, e.g., lacunary statistical convergence [2], λ-statistical convergence [3], A-statistical convergence [4], statistical A-summability [5], statistical summability ( C , 1 ) [6], statistical summability ( H , 1 ) [7], statistical summability ( N ¯ , p ) [8] and statistical σ-summability [9]etc. Following the work of Gadjiv and Orhan [10], these statistical summability methods have been used in establishing many approximation theorems (e.g., [5, 1120] and [21]). Recently, the statistical approximation properties have also been investigated for several operators. For instance, in [22] Butzer and Hahn operators; in [23] and [24]q-analogue of Stancu-Beta operators; in [25] Bleimann, Butzer and Hahn operators; in [26] Baskakov-Kantorovich operators; in [27] Szász-Mirakjan operators; in [28] analogues of Bernstein-Kantorovich operators; and in [29]q-Lagrange polynomials were defined and their statistical approximation properties were investigated. Most recently, the statistical summability of Walsh-Fourier series has been discussed in [30]. In this paper, we construct a new family of operators with the help of Erkuş-Srivastava polynomials, establish some A-statistical approximation properties and direct theorems.

Let us recall the following definitions.

Let denote the set of all natural numbers. Let K N and K n = { k n : k K } . Then the natural density of K is defined by δ ( K ) = lim n n 1 | K n | if the limit exists, where | K n | denotes the cardinality of the set K n . A sequence x = ( x k ) of real numbers is said to be statistically convergent to L (cf. Fast [1]) provided that for every ϵ > 0 the set { k N : | x k L | ϵ } has natural density zero, i.e., for each ϵ > 0 ,
lim n 1 n | { k n : | x k L | ϵ } | = 0 .

In this case, we write st - lim k x k = L . Note that every convergent sequence is statistically convergent but not conversely.

Let A = ( a n k ) , n , k = 1 , 2 , 3 ,  , be an infinite matrix. For a given sequence x = ( x k ) , the A-transform of x is defined by A x = ( ( A x ) n ) , where ( A x ) n = k = 1 a n k x k , provided the series converges for each n. We say that A is regular if lim n ( A x ) n = L = lim x . Let A be a regular matrix.

We say that a sequence x = ( x k ) is A-statistically convergent to a number L (cf. Kolk [4]) if for every ϵ > 0 ,
lim n k : | x k L | ϵ a n k = 0 .

In this case, we denote this limit by st A - lim n x n = L .

Note that for A = C 1 : = ( c j n ) , the Cesàro matrix of order 1, A-statistical convergence reduces to the statistical convergence.

2 Construction of a new operator and its properties

The well-known (two-variable) polynomials g n ( α , β ) ( x , y ) , which are generated by
( 1 x z ) α ( 1 y z ) β = n = 0 g n ( α , β ) ( x , y ) z n ( | z | < min { | x | 1 , | y | 1 } ) ,
(2.1)
are the Lagrange polynomials which occur in certain problems in statistics [31]. Recently, Chan [32] introduced and systematically investigated the multivariable extension of the classical Lagrange polynomials g n ( α , β ) ( x , y ) . These multivariable Lagrange polynomials, which are popularly known in the literature as the Chan-Chyan-Srivastava polynomials, are generated by (see [32] and [33])
j = 1 r { ( 1 x j z ) α j } = n = 0 g n ( α 1 , , α r ) ( x 1 , , x r ) z n , α j C ( j = 1 , 2 , , r ) ; | z | < min { | x 1 | 1 , , | x r | 1 } .
(2.2)
Clearly, the defined generating function (2.2) yields the explicit representation given by [[34], p.140, Eq. (6)]
g n ( α 1 , , α r ) ( x 1 , , x r ) = k 1 + + k r = n ( α 1 ) k 1 ( α r ) k r x 1 k 1 ! k 1 x r k r ! k r
(2.3)
or, equivalently, by [[14], p.522, Eq. (17)]
g n ( α 1 , , α r ) ( x 1 , , x r ) = n r 1 = 0 n n r 2 = 0 n r 1 n 1 = 0 n 2 ( α 1 ) n 1 ( α 2 ) n 2 n 1 ( α r ) n n r 1 n 1 ! ( n 2 n 1 ) ! ( n n r 1 ) ! x 1 n 1 x 2 n 2 n 1 x r n n r 1 .
(2.4)
On the other hand, Altin and Erkuş [34] presented a multivariable extension of the so-called Lagrange-Hermite polynomials generated by
j = 1 r { ( 1 x j z j ) α j } = n = 0 h n ( α 1 , , α r ) ( x 1 , , x r ) z n , α j C ( j = 1 , 2 , , r ) ; | z | < min { | x 1 | 1 , , | x r | 1 / r } .
(2.5)

The case r = 2 of the polynomials given by (2.5) corresponds to the familiar (two-variable) Lagrange-Hermite polynomials considered by Dattoli et al. [23].

The multivariable polynomials
U n , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) ,
which are defined by the following generating function [[32], p.268, Eq. (3)]:
j = 1 r { ( 1 x j z j ) α j } = n = 0 U n , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) z n , α j C ( j = 1 , 2 , , r ) ; j N ( j = 1 , 2 , , r ) ; | z | < min { | x 1 | 1 / 1 , , | x r | 1 / r } ,
(2.6)
are a unification (and generalization) of several known families of multivariable polynomials including (for example) Chan-Chyan-Srivastava polynomials
g n ( α 1 , , α r ) ( x 1 , , x r )
defined by (2.2) (see [35] for details). Obviously, the Chan-Chyan-Srivastava polynomials
g n ( α 1 , , α r ) ( x 1 , , x r )
follow as a special case of the polynomials due to Erkuş and Srivastava [35]
U n , 1 , , r α 1 , , α r ( x 1 , , x r ) ,
when
j = 1 ( j = 1 , , r ) ,
where (as well as in what follows)
N = { 1 , 2 , 3 , } and N 0 = { 0 , 1 , 2 , } = N { 0 } .
Moreover, the Lagrange-Hermite polynomials
h n ( α 1 , , α r ) ( x 1 , , x r )
follow as a special case of the polynomials [35]
U n , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) ,
when
j = 1 ( j = 1 , , r ) .
The generating function (2.6) yields the following explicit representation ([[35], p.268, Eq. (4)]):
U n , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) = 1 k 1 + + r k r = n ( α 1 ) k 1 ( α r ) k r x 1 k 1 k 1 ! x r k r k r ! ,
(2.7)
which, in the special case when
j = 1 ( j = 1 , , r ) ,

corresponds to (2.3).

The following relationship is established between the polynomials due to Erkuş and Srivastava [35] and the Chan-Chyan-Srivastava polynomials by applying the generating functions (2.2) and (2.6) in [36].
n = 0 U n , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) z n = i = 1 r { ( 1 x i z i ) α i } = i = 1 r j = 1 i { ( 1 ω ( i , j ) z ) α i } = n = 0 g n ( α 1 , , α 1 , , α r , , α r ) ( ω 11 , , ω 1 1 , , ω r 1 , , ω r r ) z n ,
(2.8)
where it is tacitly assumed that the following set:
ω ( i , j ) : 1 i r and 1 j i ( i N ; i = 1 , , r ) ,
which depends upon the i distinct values of the factor x i 1 i occurring in the expression
1 ( x i 1 i z ) i ( i = 1 , , r ) ,
exists such that
( 1 x i z i ) α i = j = 1 i { ( 1 ω ( i , j ) z ) j } ( i = 1 , , r ) .
Thus, by assertion (2.8), we obtain the desired relationship as follows:
U n , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) = g n ( α 1 , , α 1 , , α r , , α r ) ( ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ) .
Now by using the Erkuş-Srivastava multivariable polynomials given by (2.2), we introduce the following family of positive linear operators on C [ 0 , 1 ] :
T n ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ( f ; x ) = i = 1 r { ( 1 x i z i ) } n m = 0 U m , 1 , , r ( α 1 , , α r ) ( x 1 , , x r ) z m k r n + k r 1 k r + 1 n + k r 1 f ( t ) d t = i = 1 r j = 1 i { ( 1 ω ( i , j ) z ) } n m = 0 g m ( ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ) z n × ( n + k r 1 ) k r n + k r 1 k r + 1 n + k r 1 f ( t ) d t ,
(2.9)
where
α j C ( j = 1 , 2 , , r ) ; j N ( j = 1 , 2 , , r ) ; | z | < min { | x 1 | 1 / 1 , , | x r | 1 / r } .
Throughout this paper, we assume that
ω ( i , j ) = { ω ( n ) ( i , j ) } n N , 1 i r  and  1 j i ( i N ; i = 1 , , r ) ,
are sequences of real numbers such that
0 < ω ( i , j ) < 1 .
For convenience, taking r = 1 , i = 2 , α 1 = α 2 = n in (2.9), we have
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f ; x ) = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 g m ( n , n ) ( ω ( 1 , 1 ) , ω ( 1 , 2 ) ) x m k n + k 1 k + 1 n + k 1 f ( t ) d t = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 { k 1 = m ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 k n + k 1 k + 1 n + k 1 f ( t ) d t } x m .
(2.10)
Lemma 2.1 For each x [ 0 , 1 ] and n N ,
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 0 ; x ) = 1 ( f 0 ( x ) = 1 ) .
Lemma 2.2 For each x [ 0 , 1 ] and n N ,
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ; x ) x ω n ( 1 , 1 ) + 1 2 n ( f 1 ( x ) = x ) .
Proof Let each x [ 0 , 1 ] be fixed. Then from (2.10) we get
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ; x ) = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 ( n + k 1 1 ) { k 1 = m ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 k n + k 1 k + 1 n + k 1 t d t } x m = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 ( n + k 1 1 ) { k 1 = m ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 [ t 2 2 ] k n + k 1 k + 1 n + k 1 } x m = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 k 1 = m ( 2 k 1 + 1 ) 2 ( n + k 1 ) ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 x m = x ω n ( 1 , 1 ) x ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 1 k 1 = 1 m ( ω n ( 1 , 1 ) ) k 1 1 k 1 1 ! ( n ) k 1 1 x m 1 = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 k 1 = m 1 2 ( n + k 1 ) ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 x m ω n ( 1 , 1 ) x + 1 2 n , 0 < ω n ( 1 , 1 ) < 1 , ω n ( 1 , 1 ) 1 .

 □

Lemma 2.3 For each x [ 0 , 1 ] and n N ,
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) x 2 ( ω n ( 1 , 1 ) ) 2 + 2 x ( ω n ( 1 , 1 ) ) n + 1 3 n 2 ( f 2 ( x ) = x 2 ) .
Proof Let each x [ 0 , 1 ] be fixed. Then from (2.10) we get
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ; x ) = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 ( n + k 1 1 ) { k 1 = m ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 k n + k 1 k + 1 n + k 1 t 2 d t } x m = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 ( n + k 1 1 ) { k 1 = m ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 [ t 3 3 ] k n + k 1 k + 1 n + k 1 } x m = ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 0 k = m ( n + k 1 ) ( ω n ( 1 , 1 ) ) k 1 ( k 1 ) ! ( n ) k 1 × { k 1 2 ( n + k 1 1 ) 3 + k 1 ( n + k 1 1 ) 3 + 1 3 ( n + k 1 1 ) 3 } x m = x ω n ( 1 , 1 ) ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 1 k = 1 m ( k n + k 1 ) { ( ω n ( 1 , 1 ) ) k 1 1 ( k 1 1 ) ! ( n ) k 1 1 } x m 1 + x ω n ( 1 , 1 ) ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 1 k = 1 m ( 1 n + k 1 ) × { ( ω n ( 1 , 1 ) ) k 1 1 ( k 1 1 ) ! ( n ) k 1 1 } x m 1 + x ω n ( 1 , 1 ) ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 1 k = 1 m ( 1 3 ( n + k 1 ) 2 ) { ( ω n ( 1 , 1 ) ) k 1 k 1 ! ( n ) k 1 } x m x 2 ( ω n ( 1 , 1 ) ) 2 ( 1 ω ( 1 , 1 ) x ) n ( 1 ω ( 1 , 2 ) x ) n m = 2 k = 2 m ( ( n + k 2 ) n + k 1 ) × { ( ω n ( 1 , 1 ) ) k 1 2 ( k 1 2 ) ! ( n ) k 1 2 } x m 2 + 2 x ω n ( 1 , 1 ) n + 1 3 n 2 x 2 ( ω n ( 1 , 1 ) ) 2 + 2 x ω n ( 1 , 1 ) n + 1 3 n 2 , T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) f 2 ( x ) x 2 ( ( ω n ( 1 , 1 ) ) 2 1 ) + 2 x ω n ( 1 , 1 ) n + 1 3 n 2 .
(2.11)
On the other hand, since
0 T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( y x ) 2 ; x ) = T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) 2 x T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ; x ) + x 2 ,
it follows from Lemma 2.1 and Lemma 2.2 that
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) f 2 ( x ) 2 x 2 ( ( ω n ( 1 , 1 ) ) 2 1 ) .
(2.12)
Combining (2.11) and (2.12), we have
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) f 2 ( x ) | x 2 ( 1 ( ω n ( 1 , 1 ) ) 2 ) + 2 x ω n ( 1 , 1 ) n + 1 3 n 2 .
Then, taking supremum over x [ 0 , 1 ] , we have
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) f 2 ( x ) x 2 ( 1 ( ω n ( 1 , 1 ) ) 2 ) + 2 x ω n ( 1 , 1 ) n + 1 3 n 2 .
(2.13)

 □

Remark 2.1
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( t x ; x ) = x ω n ( 1 , 1 ) x + 1 2 n = x ( ω n ( 1 , 1 ) 1 ) 1 2 n .
Remark 2.2 Let x [ 0 , 1 ] , since T n ω ( 1 , 1 ) , ω ( 1 , 2 ) is linear, we get
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( y x ) 2 ; x ) = T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ; x ) 2 x T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ; x ) + x 2 T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 0 ; x ) x 2 ( ω ( 1 , 1 ) ) 2 + 2 x ω ( 1 , 1 ) n 2 x 2 ω ( 1 , 1 ) + x 2 2 x [ x ω n ( 1 , 1 ) + 1 2 n ] + x 2 x 2 ( ω ( 1 , 1 ) ) 2 + 2 x ω ( 1 , 1 ) n 2 x 2 ω ( 1 , 1 ) + x 2 + 1 3 n 2 x n .

3 A-statistical approximation

Let C [ a , b ] be a linear space of all real-valued continuous functions f on [ a , b ] , and let T be a linear operator which maps C [ a , b ] into itself. We say that T is positive if for every non-negative f C [ a , b ] , we have T ( f , x ) 0 for all x [ a , b ] . We know that C [ a , b ] is a Banach space with the norm
f C [ a , b ] : = sup x [ a , b ] | f ( x ) | , f C [ a , b ] .

For typographical convenience, we will write in place of C [ a , b ] if no confusion arises.

Theorem 3.1 Let A = ( a j n ) be a non-negative regular summability matrix. Then
st A - lim n ω n ( 1 , 1 ) = 1
(3.1)
if and only if for all f C [ 0 , 1 ] ,
st A - lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f ) f = 0 .
(3.2)
Proof Suppose that (3.2) holds for all f C [ 0 , 1 ] . Then we have
st A - lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ) f 1 = 0
(3.3)
since f 1 C [ 0 , 1 ] . By Lemma 2.2, we have
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ) f 1 = 1 ω n ( 1 , 1 ) .
(3.4)
By (3.3) and (3.4), we immediately get
st A - lim n ω n ( 1 , 1 ) = 1 .
Conversely, suppose that (3.1) holds. Then from Lemma 2.1 we have lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 0 ) f 0 = 0 . Hence
st A - lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 0 ) f 0 = 0 ( f 0 ( x ) = 1 ) .
(3.5)
Also from Lemma 2.2 it follows that
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ) f 1 = 1 ω n ( 1 , 1 ) .
Therefore, by using (3.1), we get
st A - lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 1 ) f 1 = 0 ( f 1 ( x ) : = x ) .
(3.6)
Now we claim that
st A - lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ) f 2 = 0 ( f 2 ( x ) : = x 2 ) .
(3.7)
By Lemma 2.3, we have
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ) f 2 2 ( 1 ω n ( 1 , 1 ) ) + 2 ω n ( 1 , 1 ) n + 1 3 n 2 .
(3.8)
Now, for a given ϵ > 0 , we define the following sets:
D : = { n : T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f 2 ) f 2 ϵ } , D 1 : = { n : 1 ω n ( 1 , 1 ) ϵ 4 } , D 2 : = { n : ω ( 1 , 1 ) n n ϵ 2 } , D 3 : = { n : 1 n 2 ϵ } .
From (3.8), it is easy to see that D D 1 D 2 D 3 . Then, for each j N , we get
n D a j n n D 1 a j n + n D 2 a j n + n D 3 a j n .
(3.9)
Using (3.3), we get
st A - lim n ( 1 ω n ( 1 , 1 ) ) = 0
and
st A - lim n ω n ( 1 , 1 ) n = 0 .
Now, using the above facts and taking the limit as j in (3.9), we conclude that
lim j n D a j n = 0 ,

which gives (3.7). Now, combining (3.5)-(3.7), and using the statistical version of the Korovkin approximation theorem (see Gadjiv and Orhan [10], Theorem 1), we get the desired result.

This completes the proof of the theorem. □

In a similar manner, we can extend Theorem 3.1 to the ( i , j ) -dimensional case for the operators T n ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ( f ; x ) given by (2.9) as follows.

Theorem 3.2 Let A = ( a j n ) be a non-negative regular summability matrix. Then
st A - lim n ω n ( i , j ) = 1
if and only if for all f C [ 0 , 1 ] ,
st A - lim n T n ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ( f ) f = 0 .

Remark 3.1 If in Theorem 3.2 we replace A = ( a j n ) by the identity matrix, we immediately get the following theorem which is a classical case of Theorem 3.2.

Theorem 3.3 lim n ω n ( i , j ) = 1 if and only if for all f C [ 0 , 1 ] , the sequence
T n ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ( f )

is uniformly convergent to f on [ 0 , 1 ] .

Finally, we display an example which satisfies all the hypotheses of Theorem 3.2, but not of Theorem 3.3. Therefore, this indicates that our A-statistical approximation in Theorem 3.2 is stronger than its classical case.

Take A = C 1 : = ( c j n ) , the Cesàro matrix of order 1 and
ω ( i , j ) : = ( ω n ( i , j ) ) n N ( j = 1 , , r 1 )
are sequences of real numbers defined by
ω n ( i , j ) : = { 1 2 if  n = m 2 ( m N ) ; 1 1 n + i j otherwise .
(3.10)
We then observe that
0 < ω n ( i , j ) < 1 ( n N )
and also that
st A - lim n ω n ( i , j ) = 1 .
Therefore, by Theorem 3.2, we have that for all f C [ 0 , 1 ] ,
st A - lim n T n ω ( 1 , 1 ) , , ω ( 1 , 1 ) , , ω ( r , 1 ) , , ω ( r , r ) ( f ) f = 0 .

However, since the sequence ω n ( i , j ) defined by (3.10) is non-convergent, Theorem 3.3 does not hold in this case.

4 Direct theorems

By C B [ 0 , 1 ] , we denote the space of all real-valued continuous bounded functions f on the interval [ 0 , 1 ] , the norm on the space C B [ 0 , 1 ] is given by
f = sup 0 x 1 | f ( x ) | .
Peetre’s K-functional is defined by
K 2 ( f , δ ) = inf [ { f g + δ g : g W 2 } ] ,
where
W 2 = { g C B [ 0 , 1 ] : g , g C B [ 0 , 1 ] } .
By [14] there exists a positive constant c > 0 s.t.
K 2 ( f , δ ) c w 2 ( f , δ 1 / 2 ) , δ > 0 ,
where the second-order modulus of smoothness is
w 2 ( f , δ ) = sup 0 h δ sup 0 x 1 | f ( x + 2 h ) 2 f ( x + h ) + f ( x ) | .
Also, for f C B [ 0 , 1 ] , the usual modulus of continuity is given by
w ( f , δ ) = sup 0 h δ sup 0 x 1 | f ( x + h ) f ( x ) | .
Theorem 4.1 Let f C B [ 0 , 1 ] and 0 ω n ( i , j ) < 1 . Then, for all x [ 0 , 1 ] and n N , there exists an absolute constant C > 0 s.t.
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f ; x ) f ( x ) | C w 2 ( f , δ n ( x ) ) ,
where
δ n 2 ( x ) = x 2 [ ( ω n ( 1 , 1 ) ) 2 2 ω n ( 1 , 1 ) + 1 ] + x ω n ( 1 , 1 ) n .
Proof Let g W 2 . From Taylor’s expansion
g ( t ) = g ( x ) + g ( x ) ( t x ) + x t ( t x ) g u d u , t [ 0 , 1 ] ,
and from Lemmas (2.1), (2.2) and (2.3), we get
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( g , x ) = g ( x ) + T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( x t ( t x ) g ( u ) d u , x ) ,
hence
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( g , x ) g ( x ) | | T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( x t ( t x ) g ( u ) d u , x ) | | L n u ( 1 ) , u ( 2 ) ( ( t x ) 2 , x ) | g .
Using Remark 2.2, we obtain
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( g , x ) g ( x ) | [ x 2 ( ω ( 1 , 1 ) ) 2 + 2 x ω ( 1 , 1 ) n 2 x 2 ω ( 1 , 1 ) + x 2 + 1 3 n 2 x n ] g .
On the other hand, by the definition of T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) , we have
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f ; x ) | f .
Next
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) f ( x ) | | T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( f g ) ; x ) ( f g ) ( x ) | + | T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( g , x ) g ( x ) | f g + [ x 2 ( ω ( 1 , 1 ) ) 2 + 2 x ω ( 1 , 1 ) n 2 x 2 ω ( 1 , 1 ) + x 2 + 1 3 n 2 x n ] g .
Hence, taking infimum on the right-hand side over all g W 2 , we get
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) f ( x ) | C K 2 ( f , δ n 2 ( x ) ) .
In view of the property of K-functional, for every 0 < ω n ( i , j ) < 1 , we get
| T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) f ( x ) | C w 2 ( f , δ n ( x ) ) .

This completes the proof of the theorem. □

Theorem 4.2 Let f C B [ 0 , 1 ] be such that f , f C B [ 0 , 1 ] and 0 < ω n ( i , j ) < 1 , j = 1 , 2 , 3 , , n , such that ω n ( i , j ) 1 as n . Then the following equality holds:
lim n n ( T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) f ( x ) ) = x 2 f ( x )

uniformly on [ 0 , 1 ] .

Proof By the Taylor’s formula, we may write
f ( t ) = f ( x ) + f ( x ) ( t x ) + 1 2 f ( x ) ( t x ) 2 + r ( t , x ) ( t x ) 2 ,
(4.1)
where r ( t , x ) is the remaining term and lim t x r ( t , x ) = 0 . Applying T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f ; x ) to (4.1), we obtain
n ( T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) f ( x ) ) = n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( t x ; x ) f ( x ) + n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( t x ) 2 ; x ) f ( x ) 2 + n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( r ( t , x ) ( t x ) 2 ; x ) .
By the Cauchy-Schwartz inequality, we have
T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( r ( t , x ) ( t x ) 2 ; x ) T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( r 2 ( t , x ) 2 ; x ) T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( t x ) 4 ; x ) .
(4.2)
Observe that r 2 ( x , x ) = 0 and r 2 ( , x ) C [ 0 , 1 ] . Then it follows from Theorem 4.1 that
lim n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( r 2 ( t , x ) ; x ) = r 2 ( x , x ) = 0
(4.3)

uniformly with respect to x [ 0 , 1 ] .

Now, from (4.2), (4.3) and Remark 2.2, we get
lim n n T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( r ( t , x ) ( t x ) 2 ; x ) = 0 .
Finally, using Remark 2.1, we get the following:
lim n n ( T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( f , x ) f ( x ) ) = lim n n ( f ( x ) T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( t x ) ; x ) ) + 1 2 f ( x ) T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( t x ) 2 ; x ) + 1 2 f ( x ) T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( ( t x ) 2 ; x ) + T n ω ( 1 , 1 ) , ω ( 1 , 2 ) ( r ( t , x ) ( t x ) 2 ; x ) = x 2 f ( x ) .

 □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India
(2)
Department of Mathematics, Faculty of Science, University Putra Malaysia (UPM), Serdang, Selangor Darul Ehsan, 4300, Malaysia

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© Mursaleen et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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