Statistical Approximation Operators Sahib K. Jassim Zainab I. AbdulNaby Department of Mathematics / College of Science/Al–Mustansiriya University Received in : 20 January 2013 , Accepted in : 12 May 2013 Abstract In this paper we obtain some statistical approximation results for a general class of max- product operators including the paused linear positive operators. Keywords: Statistical approximation, max-product operators, pseudo linear operators. 350 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Introduction Let (xn) be sequences of numbers. Then, (xn) is called statistically convergent to a number L iff or everyε > 0,limj ⋕{𝑛≤𝑗:|𝑥𝑛−𝐿|≥𝜖} 𝑗 = 0 .Where ⋕ B denoted the cardinality of the subset of B [1]. We denote this statistical limit by st − limjxn = L. Now, let A = (aj,n) be a finite summability matrix. Then, the A-transform of 𝑥, denoted by Ax = (Ax)j = ∑ aj,n∞n=1 xn, and provided the series converges for each j. We say that A is a regular if limj((Ax)j) = L, where ever limjxj = L [2].Assume now that A is nonnegative regular summability matrix. Then, a sequence (xn) is said to be A-statistically convergent to L if, for every > 0 , limj ∑ aj,nn:|xn−L|≥ ε = 0 …(1.1) It is denoted by st − limjxn = L , [2]. Properties of A Statistical We recall some basic properties of A-statistical convergence as follows: 1) Let𝐾 be a subset of 𝑁, the set of all natural numbers. The A- density of 𝐾, denoted byδA(K) , is defined byδA(K) = limj ∑ aj,n𝜒𝐾(𝑛) nϵK , provided that the limit exists, where χKthe characteristic function of isK .By (1.1), we easily see that in [3], st − limjxn = L if and only ifδA(K) = ({n: |xn − L| ≥ ε}) = 0, for everyε > 0. 2) Every convergent sequence is A- statistically convergent to the same value for any non- negative regular matrixA , but its converse is not true. For example in [3], for the sequence 𝑥 = (𝑥𝑛), is defined as 𝑥𝑘 = � 1, 𝑖𝑓 𝑘 𝑖𝑠 𝑠𝑞𝑢𝑎𝑟 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. It is easy to see that st − lim xk = 0. Let 𝑋 = [0, ∞) and 𝐶(𝑋) is the space of all real-valued continuous functions on 𝑋 and 𝐶𝐵(𝑋) = {𝑓 ∈ 𝐶(𝑋): 𝑓 is bounede on 𝑋} with the norm on 𝐶𝐵(𝑋) is given by: ‖f − g‖ = ⋁ |f(x) − g(x)|x∈X …(2.1) Let f: C(X) → 𝐶𝐵(𝑋) be bounded, let also xk ∈ 𝑋,𝑘 ∈ {0, … , 𝑛}, 𝑛 ≥ 1 be fixed sampled data. Then the general discrete form of a max-product approximation is given by [3]: Pn(f, x) = � Kn(x, xk). f(xk) n k=0 Where Kn(x, xk): C(X) → 𝐶𝐵(𝑋), k = 0, … , n. Are continuous functions onXsuch that, δA({n ∈ N: ⋁ Kn(x, xk) = 1 n k=0 }) = 1 …(2.2) Let 𝑋 be normed space and f: X → [0, ∞)be a target. Let xk ∈ 𝑋 ,𝑦𝑘 = 𝑓(𝑥𝑘) be sampled data. Then the Shepard-type max-product operator associated to 𝑓 is define by [3]: Snλ(f, x) = ⋁ Kn(x, xk). f(xk) n k=0 …(2.3) Where the Shepard Kernel for x ≠ xk is: Kn(x, xk) = 1 �x−xk� λ ⋁ 1 �x−xk� λ n k=0 = 1 …(2.4) so the max-product Shepard operator is: Snλ(f, x) = ⋁ 1 �x−xk� λ ⋁ 1 �x−xk� λ n k=0 . f(xk) = ⋁ f(xk) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 n k=0 …(2.5) We put 𝑍 = {𝑛 ∈ 𝑁: ⋁ 𝐾𝑛(𝑥, 𝑥𝑘) = 1 𝑛𝑘=0 } …(2.6) 351 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 So by (2.2), we may write that δA(Z) = 1 and δA(𝑁 𝑍� ) = 0 …(2.7) Then we define theTn(f, x) max-product operators from C(X) into CB(X) as [1]: Tn(f, x) = (1 + un)Snλ(f, x), xϵX …(2.8) Where un is a convergent sequence. In order to study approximation properties of the operator defined above in (2.8) we need the following definition and lemmas. Definition 1: [4] Let 𝑓(𝑥) be defined on an interval 𝐼 and suppose that we can find two positive constants 𝑀 and 𝛼 , such that|𝑓(𝑥1) − 𝑓(𝑥2)| ≤ 𝑀|𝑥1 − 𝑥2|𝛼, for all 𝑥1, 𝑥2 ∈ 𝐼 then 𝑓 is said to satisfy Lipschitz condition of order 𝛼 and we say that 𝑓 ∈ 𝑙𝑖𝑝(𝛼). Lemma 2.1: [5] For any functions ak, bk ∈ [0, ∞), k ∈ {0, … , n}, we have: �� ak n k=0 − � bk n k=0 � ≤ �|ak − bk| n k=0 Now we prove lemma (2.2), since using in our work. Lemma 2.2: Let x ∈ X and f ∈ C(X, [0, ∞))be fixed. Then the inequality: |f(y) − f(x)| ≤ ε + 2Mf δ2 φx(y) is valid for sufficiently Largent, where Mf = ⋁{|f(y)|, y ∈ X} andφx(y) = 𝑑2(𝑥, 𝑦) . Proof: Let 𝑥, 𝑦 ∈ 𝑋 = [0, ∞), and by uniformly continuity of 𝑓on [0, ∞), for each ∈> 0 there exists a number 𝛿𝛿 > 0 such that|𝑓(𝑥) − 𝑓(𝑦)| < 𝜖𝜖, where �𝑥2 − 𝑦2 < 𝛿𝛿. Now let 𝑥, 𝑦 ∈ [0, ∞) and let 𝑧 ∈ [0, ∞) such that0 ≤ 𝑧 < ∞ , since 𝑓 is continuous on the boundary point also for each 𝜖𝜖 > 0 there exists a 𝛿𝛿 > 0 such that, |𝑓(𝑥) − 𝑓(𝑦)| ≤ |𝑓(𝑥) − 𝑓(𝑧)| + |𝑓(𝑧) − 𝑓(𝑦)| < 𝜖𝜖 Where �𝑥2 − 𝑦2 < 𝛿𝛿, and𝛿𝛿 = 𝑖𝑛𝑓𝑦∈𝑋 𝑑(𝑥, 𝑦). Finally let 𝑥, 𝑦 ∈ [0, ∞) and let �𝑥2 − 𝑦2 > 𝛿𝛿, then easy calculations show that and by lemma (2.1)we have |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝑀|𝑥 − 𝑦| By using the notation Euclidean n-space 𝑅𝑛 in [6],𝑛 ≥ 1is equipped with the distance |𝑥 − 𝑦| = (∑ (𝑥𝑖 − 𝑦𝑖)2 𝑛 𝑖=1 ) 1 2. Then we have that, |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝑀|𝑥 − 𝑦| ≤ 𝑀�(𝑥 − 𝑦)2 ≤ 2 𝑀𝑓 �𝑥2−𝑦2� 𝛿2 ≤ 𝑀𝑓 2 𝛿2 𝑑2(𝑥, 𝑦) ≤ 𝜖𝜖 + 2𝑀𝑓 𝛿2 𝑑2(𝑥, 𝑦) ≤ 𝜖𝜖 + 2𝑀𝑓 𝛿2 𝜑𝑥(𝑦) 352 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Theorem 2.1: Let Tn, n ∈ 𝑁 be the linear positive operator in (2.8), which are mappings from C(X) into 𝐶𝐵(𝑋), is continuous in the uniform distance and it is pseudo linear in the sense that Tn(α. f ∨ β. g) = α. Tn(f, x) ∨ β. Tn(g, x) Proof: By (2.8), (2.5), we have |Tn(f, x) − Tn(g, x)| = �(1 + un)Snλ(f, x) − (1 + un)Snλ(g, x)� =|(1 + un) ⋁ Kn(x, xk). f(xk) n k=0 − (1 + un) ⋁ Kn(x, xk). g(xk) n k=0 | By lemma (2.1) |Tn(f, x) − Tn(g, x)| ≤ (1 + un) � Kn(x, xk) n k=0 | f(xk) − g(xk)| By (2.2) |Tn(f, x) − Tn(g, x)| ≤ (1 + un) � Kn(x, xk) n k=0 ‖f − g‖ SinceKn(x, xk) are continuous and bounded then we have |Tn(f, x) − Tn(g, x)| ≤ M‖f − g‖ Where M = (1 + un) ⋁ Kn(x, xk) n k=0 the pseud linearity of Tn is obvious. Theorem 2.2: Let Tn, n ∈ 𝑁, be a positive linear operators from C(X) into𝐶𝐵(𝑋)be an arbitrary normed space and letA = (aj,n) be non-negative regular summability matrix. If 𝑠𝑡𝐴 − limnxn{⋁{|Tn(φx, x)|: xϵX}} = 0, withφx(y)=d2(x, y) …(2.9) Then, for allf: C(X) → 𝐶𝐵(𝑋) , we have 𝑠𝑡𝐴 − lim𝑛 xn ��{|Tn(f, x) − f(x)|: xϵX}� = 0 Proof: |Tn(f, x) − f(x)| = �(1 + un)Snλ(f, x) − f(x)� = |(1 + un) ⋁ Kn(x, xk) n k=0 . f(xk) − f(x)| By Kn(x, xk) = 1 �x−xk� λ ⋁ 1 �x−xk� λ n k=0 = 1 in (2.4) we get |Tn(f, x) − f(x)| = �(1 + un) � Kn(x, xk) n k=0 . f(xk) − � Kn(x, xk) n k=0 . f(x)� |Tn(f, x) − f(x)| = �(1 + un)(� 1 ‖x − xk‖λ ⋁ 1 ‖x − xk‖λ n k=0 n k=0 f(xk)) − � 1 ‖x − xk‖λ ⋁ 1 ‖x − xk‖λ n k=0 n k=0 . f(x)� = �(1 + un) ⋁ f�xk� �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 − ⋁ f(x) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 � = � ⋁ f(xk) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 + un (⋁ f�xk� �x−xk� λ ⋁ 1 �x−xk� λ n k=0 n k=0 ) − ⋁ f(x) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 � = �un ⋁ f(xk) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 + ⋁ f(xk) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 − ⋁ f(x) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 � 353 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ≤ �un ⋁ 𝑓(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � + � ⋁ 𝑓(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 − ⋁ 𝑓(𝑥) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � ≤ �un ⋁ 𝑓(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � + � ⋁ 𝑓(𝑥𝑘)−𝑓(𝑥) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � By lemma (2.3) ≤ �un ⋁ 𝑓(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � + �� ⋁ 𝜖+ 2𝑀𝑓 𝛿2 𝜑𝑥(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 �� ≤ �un ⋁ 𝑓(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � + �(𝜖𝜖 + 2𝑀𝑓 𝛿2 ). ⋁ 𝜑𝑥(𝑥𝑘) �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 ⋁ 1 �𝑥−𝑥𝑘� 𝜆 𝑛 𝑘=0 � Since |𝑓(𝑥𝑘)| < |𝜑𝑥(𝑥𝑘)| for all𝑥 ∈ 𝑋, 𝑘 = 1, … , 𝑛 we have < �un ⋁ φx(xk �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 � + �(ϵ + 2Mf δ2 ). ⋁ φx(xk) �x−xk� λ n k=0 ⋁ 1 �x−xk� λ n k=0 � By (2.4), (2.5) and lemma (2.2) we have ≤ |un ⋁ 𝐾𝑛(𝑥, 𝑥𝑘). 𝜑𝑥(𝑥𝑘)𝑛𝑘=0 | + �((𝜖𝜖 − 1) + 1 + 2𝑀𝑓 𝛿2 ). ⋁ 𝐾𝑛(𝑥, 𝑥𝑘). 𝜑𝑥(𝑥𝑘) 𝑛𝑘=0 � ≤ (|1 + un| + (𝜖𝜖 − 1) + 2𝑀𝑓 𝛿2 ) ⋁ 𝐾𝑛(𝑥, 𝑥𝑘). 𝜑𝑥(𝑥𝑘)𝑛𝑘=0 ≤ (𝜖𝜖 − 1 + 2𝑀𝑓 𝛿2 )(1 + un) ⋁ 𝑆𝑛𝜆(𝑓, 𝑥). 𝜑𝑥(𝑥𝑘)𝑛𝑘=0 ≤ (𝜖𝜖 − 1) + 2𝑀𝑓 𝛿2 𝑇𝑛(𝜑𝑥, 𝑥) Now, taking maximum over 𝑥 ∈ 𝑋 , in the last inequality gives, for all𝑛 ∈ 𝐾 , that ⋁{|𝑇𝑛(𝑓, 𝑥) − 𝑓(𝑥)|: 𝑥 ∈ 𝑋} ≤ (𝜖𝜖 − 1) + 2𝑀𝑓 𝛿2 ⋁{|𝑇𝑛(𝜑𝑥, 𝑥): 𝑥 ∈ 𝑋|} …(2.10) For a given 𝑟 > 0 , choose an ∈ −1 > 0 such that∈ −1 < 𝑟 , and then define the sets 𝐷 ∶ {𝑛 ∈ 𝑁 ∶ (⋁{|𝑇𝑛(𝑓, 𝑥) − 𝑓(𝑥)|: 𝑥 ∈ 𝑋}) ≥ 𝑟}, 𝐷/ ∶ �𝑛 ∈ 𝑁 ∶ (�{|𝑇𝑛(𝜑𝑥, 𝑥) − 𝑓(𝑥)|: 𝑥 ∈ 𝑋}) ≥ (𝑟 − 𝜖𝜖 + 1)𝛿𝛿2 2𝑀𝑓 � The inequality (2.7), implies𝐷⋂𝑍 ⊆ 𝐷/⋂𝑍which yields for every𝑗 ∈ 𝑁 , that ∑ 𝑎𝑗𝑛𝑛∈𝐷⋂𝑍 ≤ ∑ 𝑎𝑗𝑛𝑛∈𝐷/⋂𝑍 ≤ ∑ 𝑎𝑗𝑛𝑛∈𝐷/ …(2.11) Taking limit as 𝑗 → ∞ on the both-sides of the inequality (2.11), we get lim𝑗 ∑ 𝑎𝑗𝑛𝑛∈𝐷⋂𝑍 = 0 …(2.12) On the other hand, since � 𝑎𝑗𝑛 𝑛∈𝐷 = � 𝑎𝑗𝑛 𝑛∈𝐷⋂𝑍 + � 𝑎𝑗𝑛 𝑛∈𝐷⋂(𝑁 𝑍� ) ≤ � 𝑎𝑗𝑛 𝑛∈𝐷⋂𝑍 + � 𝑎𝑗𝑛 𝑛∈(𝑛 𝑍� ) Holds for every 𝑗 ∈ 𝑁, letting 𝑗 → ∞ in the last inequality and using (2.12) and also the fact thatδA�𝑁 𝑍� � = 0 , we have lim𝑗 ∑ 𝑎𝑗𝑛𝑛∈𝐷 = 0 This means that st − lim n ��{|𝑇𝑛(𝑓, 𝑥) − 𝑓(𝑥)|: xϵX}� = 0 354 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Reference 1- Dumen O. (2005) k-positive linear operators, TOBB University of economics and Technology faculty of art and sciences, pp. 569-576. 2- Agratini O. (2006) On approximation of functions by positive linear operators, stud.Cercet.Stiint, Ser.Mat, proceeding sof ICMI 45, Bacau, pp.17-28. 3- Dirik F. (2007) Statistical convergence and rate of convergence of a sequence of positive Linear operators, Faculty of sciences and arts, Sinop University, 57000 Sinop, Turkey, pp. 147-153. 4- Heinonen J. (2004) Lecturec on Lipchitz analysis, lecturec at the 14th Jyvaskyla summer school, NSF grant DMS 0353544 and DMS0244421, pp.1-77. 5- Bede1 B.; Nobuhara2 H.; Dankova3 M. and Nola4 A. di. (2007) Approximation by pseudo- linear operator Preprint submitted to Elsevier Science, pp.1-26. 6- CLuckers1 R.; Comte2 G. and Loesoer3 F. (2008), Lipchitz continuity properties, Modnet Barcelona conference, pp.1-26. 355 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 مؤثر التقریب االحصائي صاحب كحیط الساعدي زینب عیسى عبد النبي الجامعة المستنصریة /كلیة العلوم /قسم الریاضیات 2013آیار 12، قبل في 2013كانون الثاني 20أستلم البحث في : الخالصھ فضاء جداء المؤثرات العام متضمنة المؤثرات في ھذا البحث حصلنا على بعض النتائج في التقریب االحصائي في .شبة الخطیة الموجبة فضاء جداء المؤثرات ، مؤثرات شبھ خطیة.التقریب االحصائي ، الكلمات المفتاحیة: 356 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013