Local Approximation Properties for certain King type Operators

Transcription

1 Filomat 27:1 (2013, DOI /FIL O Published by Faculty of Scieces ad athematics, Uiversity of Niš, Serbia Available at: Local Approimatio Properties for certai Kig type Operators ehmet Ali Özarsla a, Hüseyi Aktuğlu b a Departmet of athematics, Easter editerraea Uiversity,Gazimagusa, TRNC, ersi 10, Turkey b Departmet of athematics, Easter editerraea Uiversity,Gazimagusa, TRNC, ersi 10, Turkey Abstract I this paper, we cosider a certai Kig type operators which icludes geeral families of Szász-irakja, Baskakov, Post-Widder ad Stacu operators By itroducig two parameter family of Lipschitz type space, which provides global approimatio for the above metioed operators, we obtai the rate of covergece of this class Furthermore, we give local approimatio results by usig the first ad the secod modulus of cotiuity 1 Itroductio Kig was the first, who costructed a o-trivial Berstei operators preservig the test fuctios 1 ad 2 ad provide a better error estimatio for 0 < 1 3 [8] Research i this directio followed by several authors i [4],[5],[6],[7],[9],[11] ad [13] The geeralizatio of Kig-type operators was give i [1] Now, let p N 0 := 0} N = 0, 1, 2, }, I [0,, ω p ( := (1 + p 1, e k ( := k (k = 0, 1, 2 for I Throughout the paper, we cosider the followig fuctio spaces: C(I : The space of all real valued cotiuous fuctios o I C B (I : The space of all real valued cotiuous bouded fuctios o I edowed with the orm f = sup I f ( B p (I : The space of all fuctios f : I R, for which f ω p is bouded o I, edowed with the orm f p = sup I ω p ( f ( 2010 athematics Subject Classificatio 41A25; 41A36 Keywords Positive liear operators, Lipschitz type space, Szász-irakja operators, Baskakov operators, Post-Widder operators, Stacu operators, modulus of cotiuity Received: 30 December 2011; Accepted: 23 ay 2012 Commuicated by Gradimir ilovaović addresses: mehmetaliozarsla@emuedutr (ehmet Ali Özarsla, huseyiaktuglu@emuedutr (Hüseyi Aktuğlu

2 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, C p (I : f B p (I for which f ω p is uiformly cotiuous o I } f ( E[0, := f C [0, : lim is coverget I this paper we cosider a certai class of positive liear operators L } N satisfyig the followig coditios: (i L : C p (I B p (I for every p N (ii Let L (e 0, = 1, L (e 1, = for I (iii There eist umbers a, b 0, a 2 +b 2 > 0 ad a positive icreasig ubouded sequece ( such that L (e 2 ; = 2 + a2 + b for every I Note that the requiremets (ii ad (iii are the same as i [13, see Eqs (24-(25] Furthermore, i [1], the family of liear positive operators L satisfyig L (e 0, = 1, L (e 1, =, L (e 2, = a 2 + b + c was cosidered by O Agratii Hece choosig a := 1 + a, b := b, c := 0 oe ca obtai the operators cosidered i this paper Takig ito accout the above family, for all f C p, we cosider the operators T ( f ; = L ( f, u (, I, 0 (1 where (u is a sequece of cotiuous fuctios o [0, such that u (0 = 0, 0 u ( for every I [0, ad N (2 Note that by takig u ( = u ( := b + b (a + 2, 2(a + oe ca obtai the Kig-type operators (a class of operators presevig e 2 ( cosidered earlier by Rempulska ad Tomczak [13], where they obtaied some global results for the sub-family of these operators The family T ( f,, satisfyig the above coditios, icludes may well kow operators such as Szász- irakja (a = 0, b = 1, =, Baskakov (a = 1, b = 1, =, Post-Widder (a = 1, b = 0, = ad Stacu operators (a = 1, b = 1, = 1 It is obvious by (ii ad (iii that T (e 0 ; = 1, T (e 1, = u (, T (e 2 ; = u 2 ( + au2 ( + bu ( for every I Furthermore, T (φ (t, = u (, T (φ 2 (t, = ( u ( 2 + au2 ( + bu (, (3

3 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, for every, t I, where φ (t := (t Now we start with givig the followig Korovki-type approimatio theorem for the operators T give by (1 Theorem 11 Let (u be a sequece of fuctios o I satisfyig (2 If lim u ( = (4 uiformly with respect to [0, b] with b > 0, the, for all f E[0,, we have lim T ( f ; = f ( uiformly with respect to [0, b] Proof For a fied b > 0, cosider the lattice homomorphism H b : C[0, + C[0, b] defied by H b ( f := f [0,b] for every f C[0, + I this case, from (4, we see that, for each i = 0, 1, 2, lim H b (T (e i = H b (e i uiformly o [0, b] Hece, with the uiversal Korovki-type property with respect to mootoe operators (see Theorem 414 (vi of [2, p 199], we obtai that, for all f E[0,, lim T ( f ; = f ( uiformly with respect to [0, b] I the preset paper, by defiig the two parameter family of Lipschitz-type space, we study the approimatio properties of the operators T i this space Local approimatio behavior of the operators T with the help of the first ad the secod modulus of smoothess is also studied Fially i the last sectio we give applicatios of the mai results 2 ai Results I this sectio, we first defie two parameter family of Lipschitz-type space Let a, b > 0 be fied We cosider the followig Lipschitz-type space Lip (a,b (α := f C[0, : y α f (y f ( ( y + a2 + b ;, y (0,, α/2 where is ay positive costat ad 0 < α 1 We should ote that the space Lip (0,1 (α coicides with the space Lip (α cosidered by Otto Szász [14] We have the followig local approimatio result Theorem 21 For ay f Lip (a,b (α, α (0, 1], ad for every (0,, N, we have T ( f ; f ( } α/2 ( u (a 2 + b α/2 ( 2 + au2 ( + bu (

4 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, Proof Assume that α = 1 The, for f Lip (a,b (1 ad (0,, we have T ( f ; f ( T ( f (y f ( ; y T ( y + a2 + b ; 1/2 (a 2 + b 1/2 T ( y ; Applyig Cauchy-Schwarz iequality, we get by (3 that T ( f ; f ( (a 2 + b 1/2 = (a 2 + b 1/2 T ( ( y 2 ; ( u ( 2 + au2 ( + bu ( } 1/2 Now assume that α (0, 1 The, for f Lip (a,b (α ad (0,, we have T ( f ; f ( T ( f (y f ( ; y α T ( y + a2 + b ; α/2 (a 2 + b α/2 T ( y α ; Takig p = 1 α ad q = 1 1 α, for ay f Lip(a,b (α, ad applyig the Hölder iequality, we have T ( f ; f ( [ T ( y ] α ; (a 2 + b α/2 Fially, by Cauchy-Schwarz iequality, we get from (3 T ( f ; f ( Whece the result (a 2 + b α/2 (a 2 + b α/2 [ T ( ( ] α y 2 ; ( u ( 2 + au2 ( + bu ( } α/2 Takig u ( := i the above theorem we ca state the followig corollary which gives global result: Corollary 22 For ay f Lip (a,b (α, α (0, 1], we have L ( f ; f ( λ α/2 uiformly for I (0, as

5 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, By C 2 B [0, we deote the space of all fuctios f C B[0, such that f, f C B [0, Let f deote the usual supremum orm of a bouded fuctio f The, the classical Peetre s K-fuctioal ad the secod modulus of smoothess of a fuctio f C B [0, are defied respectively by K( f, δ := if f + δ } C 2 B [0, ad ω 2 ( f, δ := sup 0<h δ, [0, f ( + 2h 2 f ( + h + f (, where δ > 0 The, by Theorem 24 of [3, p 177], there eists a costat C > 0 such that K( f, δ Cω 2 ( f, δ (5 We ow get the followig local approimatio result Theorem 23 For ay f C B [0, ad for every [0,, N, we have T ( f ; f ( Cω2 f, ( u ( 2 + au2 ( + bu ( + ω( f, u ( for some costat C := C ( > 0, where ω deotes the usual modulus of cotiuity of f Proof Usig the operator T give by (1, defie a ew operator T : C B [0, C B [0, as T ( f ; = T ( f ; f (u ( + f (, (6 where (u satisfies (2 The T (e 1 e 0 ; = u ( u ( + = 0 (7 Let C 2 [0, ad [0, Usig Taylor s formula, we have B t (t ( = ( (t + (t u (udu, t [0, Thus, from (6 ad (7, we obtai ( e1 (t T( ; ( = (T (e 1 e 0 ; + T (e 1 (t u (udu; ( e1 (t = T (e 1 (t u (udu; ( e1 (t u ( = T (e 1 (t u (udu; (u ( u (udu ( e1 (t T (e 1 (t u (udu ; + (u ( u (udu u ( 2 T ( (e1 e 0 2 ; + ( u ( 2 2

6 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, By (3, we have T( ; ( ( 2 ( u ( 2 + au2 ( + bu ( (8 2 The, for ay f C B [0,, it follows from (8 that T ( f ; f ( T ( f ; ( f ( + T ( ; ( + f (u ( f ( 4 ( f + 2 ( u ( 2 + au2 ( + bu ( + f (u ( f ( Fially, by (5, we have T ( f ; f ( ( 4 f + ( u ( 2 + au2 ( + bu ( } + ω( f, u ( ( 4K f, ( u ( 2 + au2 ( + bu ( + ω( f, u ( Cω 2 f, ( u ( 2 + au2 ( + bu ( + ω( f, u ( 3 Applicatios of the mai results As it is metioed i itroductio sectio, the operators L ( f ; icludes may well kow operators such as Szász- irakja (a = 0, b = 1, =, Baskakov (a = 1, b = 1, =, Post-Widder (a = 1, b = 0, = ad Stacu operators (a = 1, b = 1, = 1 Now, we list the applicatios of the mai results for the above metioed operators: 31 Szász- irakja operators For every (0,, N, the Szász- irakja operators S ( f ; are defied by ( k ( S ( f ; = e k f k! k=0 Now cosiderig the family of Szász- irakja operators defied by S ( f ; = S ( f, u (, I, 0 where (u satisfies (2 As a cosequece of Theorem 21, Corollary 22 ad Theorem 23, we have: Corollary 31 For ay f Lip (0,1 (α, α (0, 1], ad for every (0,, N, we have S ( f ; f ( α/2 ( u ( 2 + u } α/2 (

7 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, The followig global result holds true: Corollary 32 For ay f Lip (0,1 (α, α (0, 1], we have S ( f ; f ( α/2 uiformly for (0, as Note that this result was obtaied by Otto Szász [14] Corollary 33 For ay f C B [0, ad for every (0,, N, we have S ( f ; f ( Cω2 f, ( u ( 2 + u ( + ω( f, u ( for some costat C := C ( > 0, where ω deotes the usual modulus of cotiuity of f Note that Theorems 11 ad 12 of [10] are special cases of Corollaries 5 ad 7, respectively 32 Baskakov operators The Baskakov operators V ( f ; are defied by V ( f ; = (1 + f k=0 ( k ( + k 1 k ( k, + 1 where (0,, N Cosider the family of Baskakov operators defied by V ( f ; = V ( f, u (, I, 0 where (u satisfies (2 We have: Corollary 34 For ay f Lip (1,1 (α, α (0, 1], ad for every (0,, N, we have V ( f ; f ( The followig global result holds true: } α/2 ( u ( 2 + α/2 ( 2 + u2 ( + u ( Corollary 35 For ay f Lip (1,1 (α, α (0, 1], we have V ( f ; f (, α/2 uiformly for (0, as Corollary 36 [12, Theorem 41] For ay f C B [0, ad for every [0,, N, we have V( f ; f ( Cω2 f, ( u ( 2 + u2 ( + u ( + ω( f, u ( for some costat C := C ( > 0, where ω deotes the usual modulus of cotiuity of f

8 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, Post-Widder operators The Post-Widder operators are defied, for every (0,, N, by P ( f ; = 0 f (t (/ t 1 ( 1! ( t ep dt Cosider the family of Post-Widder operators defied by P ( f ; = P ( f, u (, I, 0 where (u satisfies (2 Corollary 37 For ay f Lip (1,0 (α, α (0, 1], ad for every (0,, N, we have P ( f ; f ( } α/2 ( α u ( 2 + u2 ( The followig global result holds true: Corollary 38 For ay f Lip (1,0 (α, α (0, 1], we have P ( f ; f (, α/2 uiformly for (0, as Corollary 39 For ay f C B [0, ad for every [0,, N, we have P ( f ; f ( Cω2 f, ( u ( 2 + u2 ( + ω( f, u ( for some costat C := C ( > 0, where ω deotes the usual modulus of cotiuity of f 34 Stacu operators For every (0,, 2, 3, 4, }, Stacu operators are defied by L ( f ; = 0 t 1 f (t B(, + 1 (1 + t + 1 dt, with the Euler beta fuctio B Itroduce the family of Stacu operators by L ( f ; = L ( f, u (, I, 0 where (u satisfies (2 Corollary 310 For ay f Lip (1,1 (α, α (0, 1], ad for every (0,, 2, 3, 4, }, we have L ( f ; f ( } α/2 ( u ( 2 + α/2 ( 2 + u2 ( + u ( 1

9 ehmet Ali Özarsla, Hüseyi Aktuğlu / Filomat 27:1 (2013, The followig global result holds true: Corollary 311 For ay f Lip (1,1 (α, α (0, 1], we have L ( f ; f ( ( 1, α/2 uiformly for (0, as Corollary 312 For ay f C B [0, ad for every [0,, N, we have L ( f ; f ( Cω2 f, ( u ( 2 + u2 ( + u ( 1 + ω( f, u ( for some costat C := C ( > 0, where ω deotes the usual modulus of cotiuity of f Refereces [1] O Agratii, Liear operators that preserve some test fuctios, Iteratioal Joural of athematics ad athematical Scieces, Vol 2006, Article ID94136, Pages 1-11 [2] F Altomare ad Campiti, Korovki Type Approimatio Theory ad its Applicatios, Walter de Gruyter Publ Berli, 1994 [3] RA DeVore, GG Loretz, Costructive Approimatio Spriger-Verlag, Berli (1993 [4] O Doğru, Örkçü, Statistical approimatio by a modificatio of q-eyer- Koig ad Zeller operators, Appl ath Lett 23 (3 ( [5] O Duma, A Özarsla, Szász irakja type operators providig a better error estimatio, Appl ath Lett 20 (12 ( [6] O Duma, A Özarsla, H Aktuğlu, Better error estimates for Szász irakja Beta operators, J Comput Aal ad App 10 (1 ( [7] H Goska, P Piţul, I Raşa, Geeral kig-type operators, Results ath 53 ( [8] JP Kig, Positive liear operators which preserve 2, Acta ath Hugar, 99(2003, o 3, [9] N ahmudov, Korovki-type theorems ad applicatios, Cet Eur J ath 7 (2 ( [10] A Özarsla ad O Duma, Local approimatio results for Szász-irakja type operators Arch ath (Basel 90 (2008, o 2, [11] A Özarsla, O Duma, KZ type operators providig a better estimatio o [1/2, 1, Caad ath Bull 50 ( [12] A Özarsla ad O Duma, NI ahmudov, Local approimatio properties of modified Baskakov operators Results ath 59 (2011, o 1-2, 1 11 [13] L Rempulska ad K Tomczak, Approimatio by certai liear operators preservig 2, Turk J ath, 33, (2009, [14] O, Szasz, Geeralizatio of S Berstei s polyomials to the ifiite iterval, J Research Nat Bur Stadards 45, (1950,