A Korovkin Type Approximation Theorem and Its Applications

Abstract

We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions via A-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz's representation theory and Lebesgue-Stieltjes integral-i, for Riesz's functional supremum formula via statistical limit.

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Academic Editor:Ivanka Stamova

Department of Mathematics and Computer Application, College of Sciences, University of Al-Muthanna, Samawa, Iraq

Received 28 January 2014; Accepted 20 March 2014; 17 April 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Main Results

Some will accept the notes and definitions used in this paper. The concept of A -statistical approximation for regular summability matrix (see [1, 2]). Let A=(_ank ) , n,k=1,2,... , be an infinite summability matrix. For a given sequence x=(_xk ) , the A -transform of x , denoted by Ax=_(Ax)n , is given by _(Ax)n =^∑k=1∞_ankxk , provided that the series converges for each n . A is said to be regular if _{lim...n[arrow right]∞(Ax)n} =L , whenever lim......x=L . Then _{lim...n[arrow right]∞ank} =0 , for all k∈N . In [3], Dzyubenko and Gilewicz have given the notion.

A is nonnegative regular summability matrix. Then x is A -statistically convergent to L , if, for every ∈>0 , _{lim...n[arrow right]∞∑k:|xk -L|...5;∈ank} =0 .

We denote by _C2π (...) the space of all 2π -periodic and continuous functions on ... . Endowed with the norm _||·||2π , this space is a Banach space, where _||f||2π =sup...{|f(t)|:f∈_C2π (...),t∈...} . Now, recall that, in [4], the m th order Ditzian-Totik modulus of smoothness in the uniform metric is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the symmetric m th difference. We have to recall the Korovkin type theorem.

Theorem 1 (see [2]).

Let A=_{(^An )n...5;1} be a sequence of infinite nonnegative real matrices such that _sup...n,k^{∑j=1∞akjn} <∞ and let {_Lj } be a sequence of positive linear operators mapping _C2π (...) into _C2π (...) . Then, for all _f∈C2π (...) , we have _{lim...k[arrow right]∞}^{∑j=1∞akjn}_{||Lj f-f||2π} =0 uniformly in n , if and only if _{lim...k[arrow right]∞}^{∑j=1∞akjn}_{||Ljfi -fi ||2π} =0 (i=1,2,3 ), uniformly in n , where _f1 (t)=1 , _f2 (t)=cos......t , and _f3 (t)=sin...t , for all t∈... .

It is worth noting that the statistical analog of Theorem 1 has been studied by Radu [2], as follows.

Theorem 2.

Let A=_{(^An )n∈N} be a sequence of nonnegative regular summability matrices and let {_Lj } be a sequence of positive linear operators mapping _C2π (...) into _C2π (...) . Then, for all _f∈C2π (...) , we have _stA -_{lim...j[arrow right]∞||Lj f-f||2π} =0 , uniformly in n , if and only if _stA -_{lim...j[arrow right]∞||Ljfi -fi ||2π} =0 (i=1,2,3 ), uniformly in n , where _f1 (t)=1 , _f2 (t)=cos......t , and _f3 (t)=sin...t , for all t∈... .

The following notations are used this paper (see [5, 6]).

Let n be fixed and sufficiently large. If _yi ∈_Ij(i) and 1...4;i...4;k , then it is convenient to denote [figure omitted; refer to PDF] and therefore |^{Ii[variant prime]} |~|_ρi |~_hj(i) , for x∈^{Ii[variant prime]} . Recall that [figure omitted; refer to PDF] is the sign of f on [a,b] .

Now, let us introduce our theorems as follows.

Theorem 3.

Let A=_{(^An )n...5;1} be a sequence of infinite nonnegative real matrices such that _sup...n,k^{∑j=1∞akjn} <∞ and let {_Lj } be a sequence of positive linear operators mapping _C2π (...) into _C2π (...) . Then, for all _f∈C2π (...) , we have [figure omitted; refer to PDF] uniformly in n , if and only if [figure omitted; refer to PDF] uniformly in n , where _f1 (t)=1 , _f2 (t)=(t-^{yi[variant prime]} )/(^{yi[variant prime][variant prime]} -_yi ) , and _f3 (t)=(t-^{yi[variant prime][variant prime]} )/(_yi -^{yi[variant prime]} ) , for all t∈... . And c the constant does not depend on j .

Theorem 4.

Let A=_{(^An )n∈N} be a sequence of nonnegative regular summability matrices and let {_Lj } be a sequence of positive linear operators mapping _C2π (...) into _C2π (...) . Then, if there exists f∈_C2π (...) , we have [figure omitted; refer to PDF] uniformly in n , if and only if [figure omitted; refer to PDF] uniformly in n , where _f1 (t)=1 , _f2 (t)=(t-^{yi[variant prime]} )/(^{yi[variant prime][variant prime]} -_yi ) , and _f3 (t)=(t-^{yi[variant prime][variant prime]} )/(_yi -^{yi[variant prime]} ) , for all t∈... .

2. Proofs of Theorems 3 and 4

Proof of Theorem 3.

Since _fζ (ζ=1,2,3 ) belong to _C2π (...) , implications (5) [implies] (6) are obvious. Now, assume that (6) holds. Let _f∈C2π (...) , and, I be a closed subinterval of length 2π of ... . And let _Lj be defined by [figure omitted; refer to PDF] and also where _Lj (^{yi[variant prime]} ) and _Lj (^{yi[variant prime][variant prime]} ) are chosen so that [figure omitted; refer to PDF]

In [5] Kopotun, we have |f(x)-L(f;x)|...4;c^ω3[varphi] (f,^n-1 ) and x∈I , where [figure omitted; refer to PDF] is the Lagrange polynomial of degree ...4;2 , which interpolates f at _yi , ^{yi[variant prime]} , and ^{yi[variant prime][variant prime]} . Inequality (11) is an analog of Whitney's inequality for Ditzian-Totik moduli. Using (11) and the above presentations of _Lj and L(f;x) , we write, for x∈I , [figure omitted; refer to PDF] Taking supremum over x and =1/(^ω3[varphi] (f,^n-1 )sgn...(f)) , we obtain [figure omitted; refer to PDF] Suppose B>0 , let us write sets as follows: [figure omitted; refer to PDF] Consequently, we get [vartheta]⊂_[vartheta]1 ∪_[vartheta]2 ∪_[vartheta]3 and ^{∑j∈[vartheta]akjn} ...5;^{∑j∈_[vartheta]1akjn} ...5;^{∑_{j∈[vartheta]2}akjn} ...5;^{∑_{j∈[vartheta]3}akjn} implies [figure omitted; refer to PDF]

Proof of Theorem 4.

Since _fζ (ζ=1,2,3 ) belong to _C2π (...) , implications (8) [implies] (7) are obvious. Assume that the condition (7) is satisfied. Let _f∈C2π (...) and I be a closed subinterval of length 2π of ... ; we have [figure omitted; refer to PDF]

Now, given K(j)>0 , choose B>0 , where B=sup...{|f(x)|:x∈I} implied K<B , and define the following set: [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] where _p3 (x)=(x-^{yi[variant prime][variant prime]} )/(_yi -^{yi[variant prime]} )_∈C2π (...) polynomial and x∈... . Since x is A -statistically convergent, we can easily show that [vartheta]⊃_[vartheta]1 ⊃_[vartheta]2 ⊃_[vartheta]3 implies ^{∑j∈[vartheta]akjn} ...5;^{∑j∈_[vartheta]1akjn} ...5;^{∑_{j∈[vartheta]2}akjn} ...5;^{∑_{j∈[vartheta]3}akjn} .

Now, let B`=^ω3[varphi] (_fζ ,π/n,[-π,π]) , and using (7) implies [figure omitted; refer to PDF] This is a complete proof.

3. Application to Functional Approximation

In this section we give some applications which satisfy our theorems, but it's not the classical Korovkin theorem. It has been treated with the Weierstrass second approximation theorem via A -statistical convergence (see [6-8]). If _f∈C2π (...) , then there is a sequence of polynomials and A -statistically uniformly convergent to f on [-π,π] (not uniformly convergent). Observe that Fejer operators may be written in the form of [figure omitted; refer to PDF] We now consider the linear operator _Tn defined by [figure omitted; refer to PDF] where {^ξk(n) } (n=1,2,...;k=1,2,...,n) is a matrix of real numbers and also _ak and _bk are Fourier coefficients. Now, let A=(_ank ) be a nonnegative regular summability matrice. Assume that the following statements are satisfied:

(i) _stA -_lim...n^ξ1(n) =1 ;

(ii) (1/2)+^{∑k=1nξk(n)} (t-^{yi[variant prime]} )/(^{yi[variant prime][variant prime]} -_yi )...5;c(n)^ω3[varphi] (f,π/n,[-π,π]) . We get [figure omitted; refer to PDF] where {_Tn } is the sequence of linear operators given by (21).

In [9], Sakaoglu and Ünver proved the following theorem by using _LP [a,b;c,d] and denoted the space of all functions f defined on [a,b]×[c,d] , for which ^{∫cd∫ab|f(x,y)|P} dx dy<∞ , 1...4;P<∞ . In this case, the _LP norm of a function f in _LP [a,b;c,d] , denoted by _||f||P , is given by _||f||P =^{(∫cd∫ab|f(x,y)|P dx dy)1/P} .

Theorem 5 (see [9]).

Let A=(_ajn ) be a nonnegative regular summability matrix and let {_Tn } be an A -statistically uniformly bounded sequence of positive linear operators from _LP [a,b;c,d] into _LP [a,b;c,d] and 1...4;P<∞ . Then, for any function_f∈LP [a,b;c,d] , _stA -_lim...n ||_Tn (f;x,y)_-f(x,y)||P =0 if and only if _stA -_{lim...n||Tn (fi ;x,y)-fi (x,y)||P} =0 , i=1,2,3,4 where _f1 (t,v)=1 , _f2 (t,v)=t , _f3 (t,v)=v , and _f4 (t,v)=^t2 +^v2 .

The theory of the Lebesgue integral can be developed in several distinct ways (see [10, 11]). Only one of these methods will be discussed here.

Now, let us introduce our definition as follows.

Definition 6 (Lebesgue-Stieltjes integral-i ).

Let S be measurable set, f:S[arrow right]R be a bounded function, and _gi :S[arrow right]R be nondecreasing function for i∈I . For ...AB; Lebesgue partition of S , put LS_(f,...AB;,g_)=^{∑j=1n∏i∈I}_mjgi (μ(_Sj )) and LS¯(f,...AB;,g_)=^{∑j=1n∏i∈I}_Mjgi (μ(_Sj )) such that μ measurable function of S ; _mj =inf...{f(x):x∈_Sj } , _Mj =sup...{f(x):x∈_Sj } , and g_=_g1 ,_g2 ,... . Also, _gi (_xj )-_gi (_xj-1 )>0 , LS_(f,...AB;,g_)...4;LS¯(f,...AB;,g_) , ^∏i∈I_∫i _fdg_=sup...{LS_(f,g_)} , and ^∏i∈I∫i ¯fdg_=inf...{LS¯(f,g_)} , where LS_(f,g_)={LS_(f,...AB;,g_):...AB; part of set S} and LS¯(f,g_)={LS¯(f,...AB;,g_):...AB; part of set S} . If ^∏i∈I_∫i _fdg_=^∏i∈I∫i ¯fdg_ , where dg_=_dg1 ×_dg2 ×...×_dgn ... . Then f is integral _∫i according to _gi for i∈I .

Now, we can provide our theorem as follows as a case which is an illustrative application of approximation theory in functional analysis using functional supremum to limit convergence that acts as support and reinforcement of the concept of Riesz's representation.

Theorem 7.

If a sequence _Gn (f) is positive linear functional and bounded on C(S) , f is bounded measurable function to S . Then, there exists nondecreasing function to S such that _stA -_{lim...μ(S)[arrow right]0} ...(_sup...nGn (f)-f)=0 .

Proof.

Assume that functional supremum _Gn is as follows: [figure omitted; refer to PDF] where _{[straight phi]t,n} (x)=(1-n(x-t))/(^{yi[variant prime][variant prime]} -^{yi[variant prime]} ) converges to r∈R ; that is, let ...AB;=^{{_{S[cursive l]} }[cursive l]=0m} be Lebesgue partition such that [figure omitted; refer to PDF] where _πt,n (x)=_{[straight phi]t,n} (x)(^{yi[variant prime][variant prime]} -^{yi[variant prime]} ) .

Since G positive linear functional and bounded on C(S) , then [figure omitted; refer to PDF] also, respect between sum LS(f,...AB;,_{[straight phi]t,n} _) and Lebesgue-Stieltjes integral-i are 2[straight epsilon] , we have [figure omitted; refer to PDF] as m[arrow right]∞ ; hence G satisfies Lebesgue-Stieltjes integral-i of f .

Now, since _Gn (f) is functional supremum and satisfies Lebesgue-Stieltjes integral-i , and us Definition 6, we have [figure omitted; refer to PDF] Note that effect sum on measurable function μ(_Sj ) by using Lebesgue partition [figure omitted; refer to PDF] let n∈N , choose _...A6;n >0 , and define the following sets: [figure omitted; refer to PDF] Then _L⊂L1∪L2 ∪_L3 , which gives [figure omitted; refer to PDF] we obtain that _stA -_{lim...μ(S)[arrow right]0}^{∑μ(S)⊂Lakjn} =0 implies _stA -_{lim...μ(S)[arrow right]0} (_{sup...n Gn} (f)-f)=0 .

Now, in this paper we have proved Riesz's representation theory with Lebesgue-Stieltjes integral-i , by using Korovkin type approximation which is one of the threads in the development of Riesz's theorem to support the definition of Lebesgue integral, Rudin [10]. This integration toxicity ratio for the world on behalf of the French Lebesgue, who came in his thesis for a doctorate in 1902.

Acknowledgments

The author is grateful for hospitality at the University of Kufa. He thanks his fellows for the fruitful discussions while preparing this paper. He was partially supported by University of Al-Muthanna.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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