Statistical Approximation Properties of a Generalization of Positive Linear Operators

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1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5 No ISSN SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 9 JUNE - 0 JULY 0 ISTANBUL TURKEY Statistical Approximatio Properties of a Geeralizatio of Positive Liear Operators Reyha Caata Ogü Doğru Departmet of Mathematics Graduate School of Natural ad Applied Sciece Akara Uiversity Gölbaşi Akara Turkey Gazi Uiversity Faculty of Sciece ad Arts Departmet of Mathematics Tekik Okullar Akara Turkey Abstract. I the preset paper we itroduce a geeralizatio of positive liear operators ad obtai its Korovki type statistical approximatio properties. The rates of statistical covergece of this geeralizatio is also obtaied by meas of modulus of cotiuity ad Lipschitz type maximal fuctios. Secodly we costruct a bivariate geeralizatio of these operators ad ivestigate the statistical approximatio properties. We also get a partial differetial equatio such that the secod momet of our bivariate operators is a particular solutio of it. Fially we obtai a Voroovskaja type formulae via statistical limit. 000 Mathematics Subject Classificatios: 4A0 4A5 4A36 Key Words ad Phrases: Sequece of positive liear operators Korovki theorem for statistical approximatio modulus of cotiuity Lipschitz type maximal fuctios. Itroductio There are a lot of approximatig operators that their Korovki type error estimates approximatio properties ad rates of covergece are ivestigated (see[] for details). I the preset paper Korovki type statistical approximatio properties of a geeralizatio of positive liear operators icludig may well-kow operators which was defied by Doğru i[4] are ivestigated. These operators are itroduced as Correspodig author. L (f ; x)= ϕ (x) f( )ϕ () a =0 () addresses: ÖÝÒºÒØÒÑкÓÑ (R. Caata) ÓÙÒºÓÖÙÞºÙºØÖ (O. Doğru) 75 c 0 EJPAM All rights reserved.

2 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) where a = ϕ() lim =µ 0 x< µ ad f C[0 µ ). Hereϕ (x) C satisfies the followig coditios: ϕ ( ) a (i) Every elemet of the sequece ϕ is aalytic o a domai D cotaiig the disk B= z : z < µ (ii) ϕ () d = ϕ d x (x) x=0 > 0 for=... (iii) ϕ (x)>0 for each x [0 µ ) (iv) There exists a sequece of c such that + a + a c ad st lim c = 0. Firstly let us recall some otatios ad defiitios o the cocept of statistical covergece. A sequece x= x k is said to be statistically coverget to a umber of L if for every ǫ> 0 δ k : xk L ǫ = 0 whereδ(k) := lim { the umber k:k K} wheever the limit exist[see e.g. istace δ()=δ{k : k }= adδ k : k = 0. 8]. For Notice that ay coverget sequece is statistically coverget but ot coversely. For example the sequece L = m x k = (m=3...) L m is statistically coverget to L but ot coverget i ordiary sese whe L L. I this paper we also defie the bivariate operators for these operators ad examie their statistical covergece ad fially a applicatio to partial differetial equatios is give.. Korovki Type Statistical Approximatio Properties I[5] Gadjiev ad Orha proved the followig Korovki-type statistical approximatio theorem for ay sequece of positive liear operators. Theorem ([5]). If the sequece of positive liear operators A : C M [a b] C[a b]

3 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) satisfies the coditios st lim A (e ) e C[ab] = 0with e (t)=t for= 0 the for ay fuctio f C M [a b] we have st lim A (f) f C[ab] = 0. The space of all fuctios f which are cotiuous i[a b] ad bouded all positive axis is deoted by C M [a b]. To obtai mai results of this part let us recall some lemmas give i[4] Lemma ([4]). For all N x [0 a](0< a< ) we have µ Lemma ([4]). For all N x [0 a](0< a< ) we have µ L (e 0 x)=. () L (e x)= x. (3) Lemma 3 ([4]). For all N x [0 a](0< a< ) we have µ L (e x) x c x. (4) Now we ca obtai the followig mai result for the operators give by (). Theorem. For all f C M [0 a](0< a< ) we have µ L (f ;.) f C[0a] = 0. Proof. By Lemma ad Lemma it is clear that L (e 0 ;.) e 0 C[0a] = 0 (5) ad L (e ;.) e C[0a] = 0. (6) From Lemma 3 we have L (e ;.) e C[0a] c a. (7) Now for a giveǫ> 0 let us defie the followig sets: T := k : Lk (e ;.) e C[0a] ǫ ad T := k : c k aǫ.

4 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) We ca see that T T by (7) so we get δ k: Lk (e ;.) e C[0a] ǫ δ k:c k aǫ. Usig the c = 0 we have Cosequetly we ca write So the proof is completed from Theorem. L (e ;.) e C[0a] = 0. (8) L (e ;.) e C[0a] = 0 for= 0. (9) 3. Rates of Statistical Covergece Let f C[0 a] the modulus of cotiuity of f deoted by ω( f δ) is defied as ω(f δ) := sup f(t) f(x). (0) xt [0a] t x δ At this poit let us recall followig theorems which were proved i[4]. Theorem 3 ([4]). Let f C[0 a]. If L is defied by () the we have L (f ;.) f (+ a)ω(f c ) () whereω(f c ) is modulus of cotiuity defied i (0) ad lim c = 0. The Lipschitz type maximal fuctios of orderαitroduced by Leze[7] as follows ω α (f x) := sup t x; t [0a] f(t) f(x) t x α x [0 a]α (0]. Notice that the boudedess ofω α (f x) is equivalet to f Lip M (α). Now let us compute the rate of covergece for the differece L (f ; x) f(x) with the help of Lipschitz type maximal fuctios. Theorem 4 ([4]). If L is defied by () the we have L (f ; x) f(x) c x α ω α (f x). () Remark. Achievig a fast order of statistical covergece is importat i approximatio by positive liear operators. If we replace lim c = 0 by c = 0 i Theorem 3 ad Theorem 4 it is obvious that ω(f c )=0. So Theorems 3 ad 4 give us the rates of statistical covergece of the operators L (f ;.) to f.

5 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) ad where ad 4. Costructio of the Bivariate Operators Let I =[0 a] [0 a](0< a< µ ) ad f C([0 a] ) L x L y m a = ϕ() f ; x y = ϕ (x) f ; x y = ϕ m y ϕ ( ) lim f =0 y a f x =0 b m ϕ () ϕ () y m! a =µ 0 x< µ ad f C([0 µ ) [0 µ )) b m = ϕ() m b m lim ϕ ( ) m =µ 0 y< µ ad f C([0 µ ) [0 µ )). Hereϕ (x) C adϕ m y C satisfy the followig coditios: (a) Every elemet of the sequece ϕ ad ϕm are aalytic o a domai D cotaiig the disk B= z : z < µ (b) ϕ () d = ϕ d x (x) x=0 > 0 for=... adϕ () m = d ϕ d y m (y) y=0 > 0 for =... (c) ϕ (x)>0 for each x [0 µ ) adϕ m y > 0 for each y [0 µ ) (d) There exists a sequece of c such that + a + a c ad c = 0 ad a sequece of d m such that + b m+ b m dm ad m d m= 0. Now we ca defie the followig bivariate geeralizatio of liear ad positive operators L m f ; x y = f ϕ () y ϕ (x) ϕ m y a ϕ() m!. (3) =0=0 Lemma 4. For the operators (3) we have L m f ; x y = L x Proof. Followig calculatios reveal that L x L y m f ; x y = ϕ m y b m L y m f ; x y = L y m =0 ϕ (x) f =0 a L x f ; x y. b m ϕ () ϕ() m y!

6 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) = f ϕ (x) ϕ m y = L m f ; x y. Similarly we ca easily show that L y m =0=0 a b m ϕ () L x f ; x y = Lm f ; x y. ϕ() m y! 5. Statistical Approximatio Properties of the Bivariate Operators If we have m fm f C([ab] [cd]) = 0 the we say that the sequece of fuctios f m statistically coverget to f uiformly. Where fc([ab] [cd]) = max f x y. (x y) [ab] [cd] Volkov[9] gave the first Korovki type theorem for bivariate fuctios. Subsequetly H.H. Goska C. Badea ad I. Badea established a simpler form of Volkov s theorem as follows: Theorem 5 ([6]). Let a b c d be real umbers satisfyig the iequalities a < b c < d ad let L m : C([a b] [c d]) C([a b] [c d]) be a positive liear operators havig the properties for ay x y [a b] [c d] () L m e00 ; x y = +u m x y () L m e0 ; x y = x+ v m x y (3) L m e0 ; x y = y+ w m x y (4) L m e0 + e 0 ; x y = x + y + h m x y. If the sequeces u m x y vm x y wm x y hm x y coverge to zero uiformly o[a b] [c d] the(l m f) coverges to f uiformly o[a b] [c d] for ay f C([a b] [c d]) where e ij = x i y j are two dimesioal test fuctios. Lemma 5. The bivariate operators i (3) satisfy the followig items: (i) L m e00 ; x y = (ii) L m e0 ; x y = x (iii) L m e0 ; x y = y (iv) Lm e0 + e 0 ; x y x y c x+ d m y where c ad d satisfy the properties i (d).

7 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) Proof. (i) It is obvious that L m e00 ; x y = L m ; x y = ϕ (x) ϕ m y By usig Lemma we have L m e00 ; x y =. ϕ () =0=0 ϕ() m y!. (ii) L m e0 ; x y = ϕ (x) ϕ m y =0=0 ϕ () a by usig Lemma we ca easily see that L m e0 ; x y = x. ϕ() m y! (iii) It is prove by similarly way like(ii). (iv) Sice L m e0 + e 0 ; x y = ϕ (x) ϕ m y ϕ () =0=0 y ϕ() m! a + b m by usig Lemma 3 the proof is completed. Theorem 6. The sequece L m f defied by (3) coverges statistically to f C([0 a] [0 a]) uiformly i[0 a] [0 a]. Proof. ad from the property(d) we ca easily obtai m Lm e00 ;.. e 00 =0 (4) m Lm e0 ;.. e 0 =0 (5) m Lm e0 ;.. e 0 =0 (6) m Lm e0 + e 0 ;.. e 0 e 0 =0. (7) Usig (4) (5) (6) (7) i the light of Theorem 3 we have m Lm f ;.. f =0. (8)

8 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) Estimatio of the Rate of Statistical Covergece of the Bivariate Operators Defiitio ([]). I =[0 a] [0 a] f C I for ayδ > 0δ > 0 ω f ;δ δ = sup (ts) I (xy) I t x δ s y δ Theorem 7. If L m f is defied by (3) the we have f(ts) f x y. (9) Lm f ;.. fc(i ) f ω ; c d m a+. (0) Proof. Usig the properties for modulus (9) we have f(ts) f x yω t x s y f ;δ δ + +. () δ δ O the other had for ay x y I we have Lm f ; x y f x y Lm f(ts) f x y ; x y. () If we use () i () the we get Lm f ; x y f x y ω f ;δ δ t x s y L m + + ; x y δ δ = ω f ;δ δ δ δ ϕ (x) ϕ m y x a y b m =0=0 y ϕ()! +ω f ;δ δ δ ϕ (x) ϕ m y x a ϕ() =0=0 +ω f ;δ δ δ ϕ (x) ϕ () ϕ() m y ϕ() m ϕ m y ϕ() m y! y b m =0=0! +ω f ;δ δ.

9 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) By usig Cauchy-Schwarz iequality ad Lemmas ad 3 the we obtai Lm f ; x y f x y ω f ;δ δ δ δ x ϕ () ϕ (x) a =0 y ϕ () y ϕ m y m b =0 m! + ω f ;δ δ δ x ϕ () ϕ (x) a =0 + ω f ;δ δ δ y ϕ () y ϕ m y m b =0 m! +ω f ;δ δ = ω f ;δ δ c x dm y + ω f ;δ δ c x δ δ δ + ω f ;δ δ dm y +ω f ;δ δ. δ If we chooseδ = c δ = d m i the last iequality the we have Lm f ;.. fc(i ) ω f ; a c d m a+ω f ; a c d m +ω f ; a+ω c d m f ; c d m = ω f ; c d m a+ a+ = ω f ; c d m a+. Remark. Sice c ad d m satisfy c = 0 ad d m= 0 we ca easily say that m Lm f is statistically coverget to f o I. 7. Applicatio to Partial Differetial Equatios Let L m f be as i (3) the we ca give the followig theorem.

10 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) Theorem 8. Let ad The we have x s x L y m f ; x y + t m Proof. Usig the equalities x L m d d x ϕ (x)=h (x)ϕ (x) (3) d d y ϕ m y = hm y ϕm y (4) g y L m a b m ϕ f ; x y = (x) ϕ (x) ϕ m y f a = s + t m. (5) f ; x y = x h (x) y h m y s t m =0=0 + ϕ (x)ϕ m y f a =0=0 L m f ; x y + Lm f g; x y. (6) b m b m ϕ () ϕ () ϕ() m y! y ϕ() m! ad y L m ϕ f ; x y = m y y =0=0 ϕm ϕ (x) f a + ϕ (x)ϕ m y f a =0=0 b m b m ϕ () ϕ () ϕ() m ϕ() m y! y! we get the proof immediately.

11 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) Voroovskaja Type Approximatio Properties It ca be give the followig theorem for Voroovskaja type operators via statistical limit. Lemma 6. It ca be easily showed that L t 3 ; x x 3 + 3x c + xc (7) ad L t 4 ; x x 4 + 6c x 3 + 4c x + c 3 x (8) Theorem 9. Let L f ; x as i() L (t; x) f(x) = f c (x) x (9) Proof. Proof. Necessity. It will be used the same techique i[3] for this proof. It is kow from the Taylor expasio f(t)= f(x)+ f (x)(t x)+ f (x) (t x) (t x) (30) where(t x)= f (x) (t x)+... ad it is a cotiuous fuctio ad teds to zero for 3! t x. Let s choose t= i(30) the a f = f(x)+ f (x) x + f (x) x + x ( x). a a a a a (3) Siceis a cotiuous fuctio it is bouded ad there exists a positive costat H so for all h we ca write (h) H. If (3) is multiplied with ad take sum from= 0 to ifiity from both side of it we have ϕ (x) ϕ() so f( )ϕ () ϕ (x) a =0 = f(x) L (; x)+ f (x) L (t x; x) + f (x) L (t x) ; x ( x) a + ϕ (x) =0 x a ϕ () L f ; x = f(x)+ f (x) L (t; x) x + f (x) L t ; x x L (t; x)+ x + I. (3)

12 R. Caata O. Doğru/Eur. J. Pure Appl. Math 5 (0) where I= ϕ (x) = ϕ (x) ( x) x a a =0 + ϕ (x) =0 a x δ =0 a x >δ ( a x) ϕ () a x ϕ () ( x) x a a ϕ () Becauseis a cotiuous fuctio for everyǫ> 0 there exists aδ(ǫ) x ǫ a ad is bouded for x>δ we have a x < H. If these expressios are used a i (33) we have Iǫ ( x) ϕ () + HJ (34) ϕ (x) a where Due to x>δ ( a By usig (34) ad (36) J= ϕ (x) a x) δ ad usig (8) i (37) we obtai so >. So =0 J δ ϕ (x) =0 a x >δ ( x) ϕ () a (33) (35) ( x) 4 ϕ () a. (36) =0 IǫL (t x) ; x + H δ L (t x) 4 ; x (37) L (t; x) f(x) c L (t; x) f(x)= o(c ) f (x) x+ǫx+ H δ xc f (x) x+ǫx+ H δ xc Because c = 0 adǫ is a arbitrary positive costat we have the proof.

13 REFERENCES 87 Remark 3. It is obvious that sice c teds to zero statistically we have a better order of approximatio i Theorem 9 tha Theorem 3. Refereces [] F. Altomore ad M. Campiti. Korovki Type Approximatio Theory ad its Applicatios. Walter de Gruyter Berli 994. [] G. Aastassiou ad S. Gal. Approximatio Theory: Moduli of Cotiuity ad Global Smoothess Preservatio. Birkhäuser Bosto 000. [3] O. Doğru. Necessary coditios to obtai Voroovskaja type asymptotic formulae via statistical limit. Proc. of the th WSEAS It. Cof. o Applied Mathematics.pages 8-3 Cairo Egypt 007. [4] O. Doğru. Approximatio properties of a geeralizatio of positive liear operators. J. Math. Aal. Appl. 34: [5] A. Gadjiev ad C. Orha. Some Approximatio Theorems via statistical covergece. Rocky Moutai J. Math. 3: [6] H. Goska C. Badea ad I. Badea. A Test Fuctio Theorem ad Approximatio by Pseudopolyomials. Bull. Austral. Math. Soc. 34: [7] B. Leze. Berstei-Baskakov-Katorovich Operators ad Lipschitz-Type maximal fuctios. i: Approx. Th. Kecskemét Hugary Colloq. Math. Soc. Jáos Bolyai. 58: [8] I. Nive H. Zuckerma ad H. Motgomery. A Itroductio to the Theory of Numbers. 5th Editio Wiley New York 99. [9] V. Volkov. O the Covergece Sequeces of Liear Positive Operators i the Space of Cotiuous Fuctios of Two Variables. (Russia) Dokl. Akad. Nauk. SSSR (N. S.). 5: