# 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 $ℓ ∞$ denote the spaces of all convergent and bounded sequences, respectively, and note that $c⊂ ℓ ∞$. 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 $ℓ ∞$ and this extended functional is known as the Banach limit. In 1948, Lorentz  used this notion of a generalized limit to define a new type of convergence, known as almost convergence. Later on, Raimi  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 $ℓ ∞$ of all bounded sequences is called an invariant mean (or a σ-mean; cf. ) 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. ) 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 $x∈c$ if and only if σ has no finite orbits (cf. ) and $c⊂ V σ ⊂ ℓ ∞$.

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 .

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

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

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

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 : 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  by using λ-statistical convergence, while in  lacunary statistical convergence has been used. Boyanov and Veselinov  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 , 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 ) | ≤2M,−∞
(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 )

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

□

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 .

Example 2.1 Consider the sequence of classical Baskakov operators 

$V n (f;x):= ∑ k = 0 ∞ f ( k n ) ( n − 1 + k k ) x k ( 1 + x ) − n − k ,$

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 .

## 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. □

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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 