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On the Korovkin approximation theorem and Volkov-type theorems

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.

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:EC(X) is called positive if A(f)0 in C(X) whenever f0 in E. Here f0 means that f(x)0 in for all xX.

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 xX. 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 |AB|.

Definition 1.1 Let X and Y be compact Hausdorff spaces, Z be the product space of X and Y, and let hC(Z×Z) and fC(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 hC(Z×Z) be given. We define H w , h as the set of all continuous functions fC(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 hC(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

0A( 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 ) ) |<δimplies|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)|2f2f [ ( φ u , v + Φ u , v ) ] ( x , y ) δ 2 .

Hence, for all (u,v)Z, we have

|f(u,v)f|ϵ+2f [ ( 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=2K+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 hC(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 =uand 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.

References

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  3. Altomare F, Campiti M: Korovkin-Type Approximation Theory and Its Applications. de Gruyter, Berlin; 1994.

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  5. Volkov VI: On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR 1957, 115: 17-19. (in Russian)

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Correspondence to Nihan Uygun.

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Uygun, N. On the Korovkin approximation theorem and Volkov-type theorems. J Inequal Appl 2014, 89 (2014). https://doi.org/10.1186/1029-242X-2014-89

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Keywords

  • positive linear operator
  • Korovkin theorem
  • Volkov-type theorem
  • modules of continuity
  • compact Hausdorff spaces