Open Access

On the Korovkin approximation theorem and Volkov-type theorems

Journal of Inequalities and Applications20142014:89

https://doi.org/10.1186/1029-242X-2014-89

Received: 3 September 2013

Accepted: 5 February 2014

Published: 20 February 2014

Abstract

In this short paper, we give a generalization of the classical Korovkin approximation theorem (Korovkin in Linear Operators and Approximation Theory, 1960), Volkov-type theorems (Volkov in Dokl. Akad. Nauk SSSR 115:17-19, 1957), and a recent result of (Taşdelen and Erençin in J. Math. Anal. Appl. 331(1):727-735, 2007).

MSC:41A36, 41A25.

Keywords

positive linear operatorKorovkin theoremVolkov-type theoremmodules of continuitycompact Hausdorff spaces

1 Introduction

In this paper, the classical Korovkin theorem (see [1]) and one of the key results (Theorem 1) of [2] will be generalized to arbitrary compact Hausdorff spaces. For a topological space X, the space of real-valued continuous functions on X, as usual, will be denoted by C ( X ) . We note that if X is a compact Hausdorff space, then C ( X ) is a Banach space under pointwise algebraic operations and under the norm
f = sup x X | f ( x ) | .

Let X be a compact Hausdorff space and E be a subspace of C ( X ) . Then a linear map A : E C ( X ) is called positive if A ( f ) 0 in C ( X ) whenever f 0 in E. Here f 0 means that f ( x ) 0 in for all x X .

For more details on abstract Korovkin approximations theory, we refer to [3] and [4].

Constant-one function on a topological space X will be denoted by f 0 , that is, f 0 ( x ) = 1 for all x X . If A = ( a , b ) and B = ( c , d ) are elements of R 2 , then the Euclidean distance between A and B, given by
| ( a , b ) ( c , d ) | = ( a c ) 2 + ( b d ) 2 ,

is denoted by | A B | .

Definition 1.1 Let X and Y be compact Hausdorff spaces, Z be the product space of X and Y, and let h C ( Z × Z ) and f C ( Z ) be given. The module of continuity of f with respect to h is a function w h ( f ) : [ 0 , ) R defined by w ( f ) ( 0 ) = 0 , and
w h ( f ) ( δ ) = sup { | f ( u , v ) f ( x , y ) | : ( u , v ) , ( x , y ) Z  and  | h ( ( u , v ) , ( x , y ) ) | < δ }
whenever δ > 0 , with the following additional properties:
  1. (i)

    w ( f ) is increasing;

     
  2. (ii)

    lim δ 0 = 0 .

     

We note that the above definition is motivated from [[2], p.729] and generalizes the definition which is given there.

Definition 1.2 Let X, Y, and Z be as in Definition 1.1. Let h C ( Z × Z ) be given. We define H w , h as the set of all continuous functions f C ( X × Y ) such that for all ( u , v ) , ( x , y ) X × Y , one has
| f ( u , v ) f ( x , y ) | w h ( f ) ( | h ( ( u , v ) , ( x , y ) ) | ) .
When H w , h is mentioned, we always suppose that h satisfies the property for H w , h being a vector subspace of C ( X × X ) . We note that H w , h has been considered in [2] by taking X = [ 0 , A ] , Y = [ 0 , B ] ( A , B > 0 ),
h ( ( u , v ) , ( x , y ) ) = ( f 1 ( u , v ) , f 2 ( u , v ) ) ( f 1 ( x , y ) , f 2 ( x , y ) ) ,
where
f 1 ( u , v ) = u 1 u and f 2 ( u , v ) = v 1 v .

The main result of this paper will be obtained via the following lemma.

2 Main result

Lemma 2.1 Let X and Y be compact Hausdorff spaces and Z be a product space of X and Y. Let f 1 , f 2 C ( Z ) and h C ( Z × Z ) be defined by
h ( ( u , v ) , ( x , y ) ) = | ( f 1 ( u , v ) , f 2 ( u , v ) ) ( f 1 ( x , y ) , f 2 ( x , y ) ) |
so that H w , h is a subspace C ( X × Y ) and f 1 , f 2 H w , h ( Z ) . Let A : H w , h C ( Z ) be a positive linear map. Let ( u , v ) Z be given, and define φ u , v , Φ u , v C ( Z ) by
φ u , v = ( f 1 ( u , v ) f 0 f 1 ) 2 and Φ u , v = ( f 2 ( u , v ) f 0 f 2 ) 2 .
Then, for all ( u , v ) Z , one has
0 A ( φ u , v + Φ u , v ) C 1 [ A ( f 0 ) f 0 ] ( u , v ) C 2 [ A ( f 1 + f 2 ) ( f 1 + f 2 ) ] + [ A ( f 1 2 + f 2 2 ) ( f 1 2 + f 2 2 ) ] ,
where
C 1 = ( f 1 ( u , v ) 2 + f 2 ( u , v ) 2 ) and C 2 = 2 ( f 1 ( u , v ) + f 2 ( u , v ) ) .
Proof Note that
0 φ u , v = f 1 ( u , v ) 2 f 0 2 f 1 ( u , v ) f 1 + f 1 2 .
Applying the linearity and positivity of A, we have
0 A ( φ u , v ) = f 1 ( u , v ) 2 A ( f 0 ) 2 f 1 ( u , v ) A ( f 1 ) + A ( f 1 2 ) .
Then one can have
0 A ( φ u , v ) ( u , v ) = f 1 ( u , v ) 2 A ( f 0 ) ( u , v ) 2 f 1 ( u , v ) A ( f 1 ) ( u , v ) + A ( f 1 2 ) ( u , v ) = f 1 2 ( u , v ) [ A ( f 0 ) ( u , v ) f 0 ( u , v ) + f 0 ( u , v ) ] 2 f 1 ( u , v ) [ A ( f 1 ) ( u , v ) f 1 ( u , v ) + f 1 ( u , v ) ] + [ A ( f 1 2 ) ( u , v ) f 1 ( u , v ) 2 + f 1 ( u , v ) 2 ] = f 1 2 ( u , v ) [ A ( f 0 ) f 0 ] ( u , v ) 2 f 1 ( u , v ) [ A ( f 1 ) f 1 ] ( u , v ) + [ A ( f 1 2 ) f 1 2 ] ( u , v ) .
Similarly, we have
A ( Φ u , v ) ( u , v ) = f 2 2 ( u , v ) [ A ( f 0 ) f 0 ] ( u , v ) 2 f 2 ( u , v ) [ A ( f 2 ) f 2 ] ( u , v ) + [ A ( f 2 2 ) f 2 2 ] ( u , v ) .

Now applying A, which is linear, to φ u , v + Φ u , v completes the proof. □

Lemma 2.2 Let X and Y be compact Hausdorff spaces and f 1 , f 2 , and h be defined as in Lemma  2.1. Let f H w , h be given. For each ϵ > 0 , there exists δ > 0 such that
| f ( u , v ) f ( x , y ) | < ϵ + 2 f δ 2 h 2 ( ( u , v ) , ( x , y ) ) .
Proof Let ϵ > 0 be given. Since w ( f ) : [ 0 , ) R is continuous, there exists δ > 0 such that w ( f , δ ) = w ( f ) ( δ ) < ϵ for all 0 δ < δ . This implies, since
| f ( u , v ) f ( x , y ) | w ( f , | h ( ( u , v ) , ( x , y ) ) | ) for all  ( u , v ) , ( x , y ) Z ,
that
[ ( φ u , v + Φ u , v ) ] 1 2 ( x , y ) = | h ( ( u , v ) h ( x , y ) ) | < δ implies | f ( u , v ) f ( x , y ) | < ϵ ,
where φ u , v and Φ u , v are defined as in Lemma 2.1. If [ ( φ u , v + Φ u , v ) ] 1 2 ( x , y ) δ , then
| f ( u , v ) f ( x , y ) | 2 f 2 f [ ( φ u , v + Φ u , v ) ] ( x , y ) δ 2 .
Hence, for all ( u , v ) Z , we have
| f ( u , v ) f | ϵ + 2 f [ ( φ u , v + Φ u , v ) ] δ 2 .

This completes the proof. □

Lemma 2.3 Suppose that the hypotheses of Lemma  2.2 are satisfied. Let f H w , h and ϵ > 0 be given. Then there exists C > 0 such that
A ( f ) f < ϵ + C ( A ( f 0 ) f 0 + A ( f 1 + f 2 ) ( f 1 + f 2 ) + A ( f 1 2 + f 2 2 ) ( f 1 2 + f 2 2 ) ) .
Proof Set K : = 2 f δ 2 . From Lemma 2.2, there exists δ > 0 such that for each ( u , v ) Z we have
| f ( u , v ) f 0 f | ϵ + 2 f δ 2 [ φ u , v + Φ u , v ] ϵ + 2 f δ 2 [ f 1 2 ( u , v ) f 0 + f 2 2 ( u , v ) f 0 2 f 1 ( u , v ) f 1 2 f 2 ( u , v ) f 2 + ( f 1 2 + f 2 2 ) ] ,
whence
| [ A ( f ) f ( u , v ) A ( f 0 ) ] ( u , v ) | ϵ A ( f 0 ) ( u , v ) + K ( A ( φ u , v ) + A ( Φ u , v ) ) = ϵ + ϵ [ A ( f 0 ) f 0 ] ( u , v ) + K A ( φ u , v + Φ u , v ) .
In particular, we have
| A ( f ) f | ( u , v ) | [ A ( f ) f ( u , v ) A ( f 0 ) ] ( u , v ) | + | f ( u , v ) | | ( A ( f 0 ) f 0 ) ( u , v ) | ϵ + K A ( φ u , v + Φ u , v ) ( u , v ) + ( f + ϵ ) A ( f 0 ) f 0 .
Now, applying Lemma 2.1 and taking
C = 2 K + f ,

we have what is to be shown. □

We note that in the above theorem C depends only on f and ϵ, and is independent of the positive linear operator A.

We are now in a position to state the main result of the paper.

Theorem 2.4 Let X and Y be compact Hausdorff spaces and Z be the product space of X and Y. Let f 1 , f 2 C ( Z ) , and h C ( Z × Z ) be defined by
h ( ( u , v ) , ( x , y ) ) = ( f 1 ( u , v ) , f 2 ( u , v ) ) ( f 1 ( x , y ) , f 2 ( x , y ) )
so that H w , h is a subspace C ( X × Y ) and f 1 , f 2 H w , h ( Z ) . Let ( A n ) n N be a sequence of positive operators from H w , h into C ( X × Y ) satisfying:
  1. (i)

    A n ( f 0 ) f 0 0 ;

     
  2. (ii)

    A n ( f 1 ) f 1 0 ;

     
  3. (iii)

    A n ( f 2 ) f 2 0 ;

     
  4. (iv)

    A n ( f 1 2 + f 2 2 ) ( f 1 2 + f 2 2 ) 0 .

     
Then, for all f H w , h , we have
A n ( f ) f 0 .
Proof Let f H w , h and ϵ > 0 be given. By Lemma 2.3, there exists C > 0 (depending only on f and ϵ > 0 ) such that for each n,
A n ( f ) f ϵ + C ( A n ( f 0 ) f 0 + A n ( f 1 + f 2 ) ( f 1 + f 2 ) + A n ( f 1 2 + f 2 2 ) ( f 1 2 + f 2 2 ) ) .
Since ϵ > 0 is arbitrary and the last three terms of the inequality converge to zero by the assumption, we have
A n ( f ) f .

This completes the proof. □

Note also that in Theorem 1 of [2] it is not necessary to take a double sequence of positive operators: as the above result reveals, one can take ( A n ) instead of ( A n , m ) .

Remarks
  1. (1)
    If X = [ 0 , 1 ] , and Y = { y } and f 1 , f 2 C ( X × Y ) are defined by
    f u , v = u and f 2 = 0 ,
     
then Theorem 2.4 becomes the classical Korovkin theorem.
  1. (2)
    If one takes X = [ 0 , A ] , Y = [ 0 , B ] ( 0 < A , B < 1 ), and f 1 and f 2 are defined by
    f 1 ( u , v ) = u 1 u and f 2 ( u , v ) = v 1 v ,
     
then the above theorem becomes Theorem 1 of [2].
  1. (3)

    For linear positive operators of two variables, Theorem 2.4 generalizes the result of Volkov in [5].

     
  2. (4)

    We believe that the above theorem can be generalized to n-fold copies by taking Z = X 1 × X 2 × × X n instead of Z = X × Y , where X 1 , X 2 , , X n are compact Hausdorff spaces.

     
  3. (5)

    The above theorem is also true if one replaces C ( X ) by C b ( X ) , the space of bounded continuous functions, in the case of an arbitrary topological space X.

     

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Abant İzzet Baysal University, Gölköy Kampüsü

References

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Copyright

© Uygun; licensee Springer. 2014

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