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Invariant mean and a Korovkin-type approximation theorem

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.

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)=limx for all xc 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)-limx.

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)|, fC[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 fC[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 fC(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 fC(I). Then there exists a constant M>0 such that |f(x)|M for xI. Therefore,

| f ( s ) f ( x ) | 2M,<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 xI.

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 ),|sx|<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 xI. 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 0x,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 fC(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. □

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Correspondence to Saleh Abdullah Al-Mezel.

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Al-Mezel, S.A. Invariant mean and a Korovkin-type approximation theorem. J Inequal Appl 2013, 103 (2013). https://doi.org/10.1186/1029-242X-2013-103

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Keywords

  • invariant mean
  • σ-convergence
  • Korovkin-type approximation theorem