Open Access

Invariant mean and a Korovkin-type approximation theorem

Journal of Inequalities and Applications20132013:103

https://doi.org/10.1186/1029-242X-2013-103

Received: 29 May 2012

Accepted: 24 February 2013

Published: 15 March 2013

Abstract

In this paper we apply this form of convergence to prove some Korovkin-type approximation theorem by using the test functions 1, e x , e 2 x , which generalizes the results of Boyanov and Veselinov (Bull. Math. Soc. Sci. Math. Roum. 14(62):9-13, 1970).

MSC:41A65, 46A03, 47H10, 54H25.

Keywords

invariant meanσ-convergenceKorovkin-type approximation theorem

1 Introduction and preliminaries

Let c and denote the spaces of all convergent and bounded sequences, respectively, and note that c . In the theory of sequence spaces, an application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on c can be extended to the whole of and this extended functional is known as the Banach limit. In 1948, Lorentz [1] used this notion of a generalized limit to define a new type of convergence, known as almost convergence. Later on, Raimi [2] gave a slight generalization of almost convergence and named it σ-convergence. Before proceeding further, we recall some notations and basic definitions used in this paper.

Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ defined on the space of all bounded sequences is called an invariant mean (or a σ-mean; cf. [2]) if it is non-negative, normal and φ ( x ) = φ ( ( x σ ( n ) ) ) .

A sequence x = ( x k ) is said to be σ-convergent to the number L if and only if all of its σ-means coincide with L, i.e., φ ( x ) = L for all φ. A bounded sequence x = ( x k ) is σ-convergent (cf. [3]) to the number L if and only if lim p t p m = L uniformly in m, where
t p m = x m + x σ ( m ) + x σ 2 ( m ) + + x σ p ( m ) p + 1 .

We denote the set of all σ-convergent sequences by V σ and in this case we write x k L ( V σ ) and L is called the σ-limit of x. Note that a σ-mean extends the limit functional on c in the sense that φ ( x ) = lim x for all x c if and only if σ has no finite orbits (cf. [4]) and c V σ .

If σ is a translation then the σ-mean is called a Banach limit and σ-convergence is reduced to the concept of almost convergence introduced by Lorentz [1].

In [5], the idea of statistical σ-convergence is defined which is further applied to prove some approximation theorems in [6] and [7].

If m = 1 , then we get ( C , 1 ) convergence, and in this case we write x k ( C , 1 ) , where = ( C , 1 ) - lim x .

Remark 1.1 Note that
  1. (a)

    a convergent sequence is also σ-convergent;

     
  2. (b)

    a σ-convergent sequence implies ( C , 1 ) convergence.

     
Example 1.1 Let σ ( n ) = n + 1 . Define the sequence z = ( z n ) by
z n = { 1 if  n  is odd , 0 if  n  is even .

Then x is σ-convergent to 1/2 but not convergent.

Let C [ a , b ] be the space of all functions f continuous on [ a , b ] . We know that C [ a , b ] is a Banach space with the norm f : = sup a x b | f ( x ) | , f C [ a , b ] . Suppose that T n : C [ a , b ] C [ a , b ] . We write T n ( f , x ) for T n ( f ( t ) , x ) and we say that T is a positive operator if T ( f , x ) 0 for all f ( x ) 0 .

The classical Korovkin approximation theorem states the following [7]: Let ( T n ) be a sequence of positive linear operators from C [ a , b ] into C [ a , b ] . Then lim n T n ( f , x ) f ( x ) = 0 , for all f C [ a , b ] if and only if lim n T n ( f i , x ) f i ( x ) = 0 , for i = 0 , 1 , 2 , where f 0 ( x ) = 1 , f 1 ( x ) = x and f 2 ( x ) = x 2 .

Quite recently, such type of approximation theorem has been studied in [8, 9] and [10] by using λ-statistical convergence, while in [11] lacunary statistical convergence has been used. Boyanov and Veselinov [12] have proved the Korovkin theorem on C [ 0 , ) by using the test functions 1, e x , e 2 x . In this paper, we generalize the result of Boyanov and Veselinov by using the notion of σ-convergence. Our results also generalize the results of Mohiuddine [13], in which the author has used almost convergence and the test functions 1, x, x 2 .

2 Korovkin-type approximation theorem

We prove the following σ-version of the classical Korovkin approximation theorem.

Theorem 2.1 Let ( T k ) be a sequence of positive linear operators from C ( I ) into C ( I ) . Then, for all f C ( I ) ,
σ - lim k T k ( f ; x ) f ( x ) = 0
(2.1)
if and only if
σ - lim k T k ( 1 ; x ) 1 = 0 ,
(2.2)
σ - lim k T k ( e s ; x ) e x = 0 ,
(2.3)
σ - lim k T k ( e 2 s ; x ) e 2 x = 0 .
(2.4)
Proof Since each 1, e x , e 2 x belongs to C ( I ) , conditions (2.2)-(2.4) follow immediately from (2.1). Let f C ( I ) . Then there exists a constant M > 0 such that | f ( x ) | M for x I . Therefore,
| f ( s ) f ( x ) | 2 M , < s , x < .
(2.5)
It is easy to prove that for a given ε > 0 there is a δ > 0 such that
| f ( s ) f ( x ) | < ε ,
(2.6)

whenever | e s e x | < δ for all x I .

Using (2.5), (2.6), putting ψ 1 = ψ 1 ( s , x ) = ( e s e x ) 2 , we get
| f ( s ) f ( x ) | < ε + 2 M δ 2 ( ψ 1 ) , | s x | < δ .
This is,
ε 2 M δ 2 ( ψ 1 ) < f ( s ) f ( x ) < ε + 2 M δ 2 ( ψ 1 ) .
Now, we operate T σ k ( n ) ( 1 , x ) for all n to this inequality since T σ k ( n ) ( f , x ) is monotone and linear. We obtain
T σ k ( n ) ( 1 ; x ) ( ε 2 M δ 2 ( ψ 1 ) ) < T σ k ( n ) ( 1 ; x ) ( f ( s ) f ( x ) ) < T σ k ( n ) ( 1 ; x ) ( ε + 2 M δ 2 ( ψ 1 ) ) .
Note that x is fixed and so f ( x ) is a constant number. Therefore
ε T σ k ( n ) ( 1 ; x ) 2 M δ 2 T σ k ( n ) ( ψ 1 ; x ) < T σ k ( n ) ( f ; x ) f ( x ) T σ k ( n ) ( 1 ; x ) < ε T σ k ( n ) ( 1 ; x ) + 2 M δ 2 T σ k ( n ) ( ψ 1 ; x ) .
(2.7)
But
T σ k ( n ) ( f ; x ) f ( x ) = T σ k ( n ) ( f ; x ) f ( x ) T σ k ( n ) ( 1 ; x ) + f ( x ) T σ k ( n ) ( 1 ; x ) f ( x ) = [ T σ k ( n ) ( f ; x ) f ( x ) T σ k ( n ) ( 1 ; x ) ] + f ( x ) [ T σ k ( n ) ( 1 ; x ) 1 ] .
(2.8)
Using (2.7) and (2.8), we have
T σ k ( n ) ( f ; x ) f ( x ) < ε T σ k ( n ) ( 1 ; x ) + 2 M δ 2 T σ k ( n ) ( ψ 1 ; x ) + f ( x ) ( T σ k ( n ) ( 1 ; x ) 1 ) .
(2.9)
Now
T σ k ( n ) ( ψ 1 ; x ) = T σ k ( n ) ( ( e s e x ) 2 ; x ) = T σ k ( n ) ( e 2 s 2 e s e x + e 2 x ; x ) = T σ k ( n ) ( e 2 s ; x ) 2 e x T σ k ( n ) ( e s ; x ) + ( e 2 x ) T σ k ( n ) ( 1 ; x ) = [ T σ k ( n ) ( e 2 s ; x ) e 2 x ] 2 e x [ T σ k ( n ) ( e s ; x ) e x ] + e 2 x [ T σ k ( n ) ( 1 ; x ) 1 ] .
Using (2.9), we obtain
T σ k ( n ) ( f ; x ) f ( x ) < ε T σ k ( n ) ( 1 ; x ) + 2 M δ 2 { [ T σ k ( n ) ( ( e 2 s ) ; x ) e 2 x ] 2 e x [ T σ k ( n ) ( e s ; x ) e x ] + e 2 x [ T σ k ( n ) ( 1 ; x ) 1 ] } + f ( x ) ( T σ k ( n ) ( 1 ; x ) 1 ) = ε [ T σ k ( n ) ( 1 ; x ) 1 ] + ε + 2 M δ 2 { [ T σ k ( n ) ( ( e 2 s ) ; x ) e 2 x ] 2 e x [ T σ k ( n ) ( e s ; x ) e x ] + e 2 x [ T σ k ( n ) ( 1 ; x ) 1 ] } + f ( x ) ( T σ k ( n ) ( 1 ; x ) 1 ) .
Since ε is arbitrary, we can write
T σ k ( n ) ( f ; x ) f ( x ) ε [ T σ k ( n ) ( 1 ; x ) 1 ] + 2 M δ 2 { [ T σ k ( n ) ( ( e 2 s ) ; x ) e 2 x ] 2 e x [ T σ k ( n ) ( e s ; x ) e x ] + e 2 x [ T σ k ( n ) ( 1 ; x ) 1 ] } + f ( x ) [ T σ k ( n ) ( 1 ; x ) 1 ] .
Therefore
| T σ k ( n ) ( f ; x ) f ( x ) | ε + ( ε + M ) | T σ k ( n ) ( 1 ; x ) 1 | + 2 M δ 2 | e 2 x | | T σ k ( n ) ( 1 ; x , y ) 1 | + 2 M δ 2 | T σ k ( n ) ( e 2 s ; x ) | | e 2 x | + 4 M δ 2 | e x | | T σ k ( n ) ( e s ; x ) e x | ε + ( ε + M + 4 M δ 2 ) | T σ k ( n ) ( 1 ; x ) 1 | + 2 M δ 2 | e 2 x | | T σ k ( n ) ( 1 ; x ) 1 | + 2 M δ 2 | T σ k ( n ) ( e 2 s ; x ) e 2 x | + 4 M δ 2 | T σ k ( n ) ( e s ; x ) e x |
since | e x | 1 for all x I . Now, taking sup x I
T σ k ( n ) ( f ; x ) f ( x ) ε + K ( T σ k ( n ) ( 1 ; x ) 1 + T σ k ( n ) ( e s ; x ) e x + T σ k ( n ) ( e 2 s ; x ) e 2 x ) ,
where K = max { ε + M + 4 M δ 2 , 2 M δ 2 } . Now writing
D n , p ( f , x ) = 1 p k = 0 p 1 T σ k ( n ) ( f , x ) ,
we get
D n , p ( f , x ) f ( x ) ( ϵ + 2 M b 2 δ 2 + M ) D n , p ( 1 , x ) 1 + 4 M b δ 2 D n , p ( t , x ) e x + 2 M δ 2 D n , p ( t 2 , x ) e 2 x .
Letting p and using (2.2), (2.3), (2.4), we get
lim p D n , p ( f , x ) f ( x ) = 0 , uniformly in  n .

 □

In the following example we construct a sequence of positive linear operators satisfying the conditions of Theorem 2.1 but not satisfying the conditions of the Korovkin theorem of Boyanov and Veselinov [12].

Example 2.1 Consider the sequence of classical Baskakov operators [14]
V n ( f ; x ) : = k = 0 f ( k n ) ( n 1 + k k ) x k ( 1 + x ) n k ,

where 0 x , y < .

Let the sequence ( L n ) be defined by L n : C ( I ) C ( I ) with L n ( f ; x ) = ( 1 + z n ) V n ( f ; x ) , where z n is defined as above. Since
L n ( 1 ; x ) = 1 , L n ( e s ; x ) = ( 1 + x x e 1 n ) n , L n ( e 2 s ; x ) = ( 1 + x 2 x 2 e 1 n ) n ,
and the sequence ( P n ) satisfies the conditions (2.1), (2.2) and (2.3). Hence we have
σ - lim L n ( f , x ) f ( x ) = 0 .
On the other hand, we get L n ( f , 0 ) = ( 1 + z n ) f ( 0 ) since L n ( f , 0 ) = f ( 0 ) , and hence
L n ( f , x ) f ( x ) | L n ( f , 0 ) f ( 0 ) | = z n | f ( 0 ) | .

We see that ( L n ) does not satisfy the classical Korovkin theorem since lim sup n z n does not exist. Hence our Theorem 2.1 is stronger than that of Boyanov and Veselinov [12].

3 A consequence

Now we present a slight general result.

Theorem 3.1 Let ( T n ) be a sequence of positive linear operators on C ( I ) such that
lim n sup m 1 n k = 0 n 1 T n T σ k ( m ) = 0 .
If
σ - lim n T n ( e ν s , x ) e ν x = 0 ( ν = 0 , 1 , 2 ) ,
(3.1)
then, for any function f C ( I ) bounded on the real line, we have
lim n T n ( f , x ) f ( x ) = 0 .
(3.2)
Proof From Theorem 2.1, we have that if (3.1) holds, then
σ - lim n T n ( f , x ) f ( x ) = 0 ,
which is equivalent to
lim n sup m D m , n ( f , x ) f ( x ) = 0 .
Now
T n D m , n = T n 1 n k = 0 n 1 T σ k ( m ) = 1 n k = 0 n 1 ( T n T σ k ( m ) ) .
Therefore
T n sup m D m , n = sup m 1 n k = 0 n 1 ( T n T σ k ( m ) ) .
Hence, using the hypothesis, we get
lim n T n ( f , x ) f ( x ) = lim n sup m D m , n ( f , x ) f ( x ) = 0 ,

that is, (3.2) holds. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University

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© Al-Mezel; licensee Springer. 2013

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