On general Gamma-Taylor operators on weighted spaces

Transcription

1 It. J. Adv. Appl. Math. ad Mech ISSN: Joural homepage: IJAAMM Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics O geeral Gamma-Taylor operators o weighted spaces Research Article Alok Kumar a Artee a D. K. Vishwakarma a Rajat Kaushik b a Departmet of Computer Sciece Dev Saskriti Vishwavidyalaya Haridwar Uttarakhad Idia b Departmet of Mathematics Idia Istitute of Techology Roorkee Roorkee Uttarakhad Idia Received 1 March 16; accepted i revised versio 1 April 16 Abstract: MSC: I the preset paper we cosider ew operators by combiig geeral Gamma type operators ad Taylors polyomials. We establish covergece properties of these operators i weighted spaces. 41A5 4A35 41A36 Keywords: Gamma type operators Taylor polyomials Modulus of cotiuity Weighted space 16 The Authors. This is a ope access article uder the CC BY-NC-ND licese 1. Itroductio I 7 Mao [17] defied the followig Gamma type liear ad positive operators M k f ; x = k 1!x1 =! k! g xug k u tf tdud t We ca rewrite the operators M k f ; x as where M k f ; x = t k f td t x >. x t k K k x tf td t 1 k 1!x1 K k x t = x t.! k! x t k t k The rate of covergece of these operators for fuctios with derivatives of bouded variatio was studied i [1]. Some approximatio results for these operators based o q itegers were obtaied i [15]. The Voroovskaja type theorem ad the local rate of covergece for the operators M k were give i [9]. I [1] global approximatio theorems for these operators were obtaied. I this paper we cosider ew operators by combiig geeral Gamma type operators ad Taylor polyomials of r times differetiable fuctio f i weighted space o a ] which expads to whe. We study the covergece of these ew operators. Correspodig author. addresses: alokkpma@gmail.com Alok Kumar artee.varma@dsvv.ac.i Artee dkvishwa7@gmail.com D. K. Vishwakarma bittoo96983@gmail.com Rajat Kaushik

2 1 O geeral Gamma-Taylor operators o weighted spaces By C r we deote the set of all real valued fuctios f such that r th r = 1... order derivatives are cotiuous. For ay f C r ad t we cosider Taylor polyomials of order r T r f ; x = f j t x t j. j = Combie 1 ad we obtai M kr f ; x = K k x t f j t x t j d t. 3 j = It is clear that M k f ; x = M k f ; x. Let νx = 1 x < x < ad B ν be the set of all fuctios f defied o the real axis satisfyig the coditio f x C f νx where C f is a costat depedig oly o f. B ν is a ormed space with the orm f ν = x f x νx f B ν. C ν deotes the subspace of all cotiuous fuctios i B ν ad C k ν deotes the subspace of all fuctios f C ν for which f x lim x νx <. B νa ] C νa ] ad C k νa ] are defied as B ν C ν ad C k ν respectively oly with the domai a ] istead of real axis R ad the orm is take as f νa ]= x a ] f x νx. I the sequel it will be assumed that lim a =.. Auxiliary results I this sectio we give some prelimiary results which will be used i the mai part of this paper. Let us cosider e m t = t m ϕ xm t = t x m m N x t. Lemma.1 [1]. For ay m N set of o-egative itegers m k M k t m ; x = [ k m] m x m [] m 4 where k N ad [x] m = xx 1...x m 1[x] = 1 x R. I particular for m = 1... i 4 we get i M k 1; x = 1 ii M k t; x = k 1 x iii M k t k k 1 ; x = x. 1 Lemma. [1]. Let m N ad fixed x the m M k ϕ xm ; x = j m m k m x m. j = j! k!

3 Lemma.3. For m = 134 oe has i M k ϕ x ; x = 1 ii M k ϕ x1 ; x = 1 k x iii M k ϕ x ; x = k 5k 4 x 1 Alok Kumar et al. / It. J. Adv. Appl. Math. ad Mech iv M k ϕ x3 ; x = k3 1k 17k 18 1k 4 x 3 1 v M k ϕ x4 ; x = k4 k 3 k k x vi M k ϕ xm ; x = O [m1/]. Usig Lemma. we get Lemma.3. Remark.1. Usig Lemma.3 we get M k t x m ; x λ m x m m where λ m is a costat depedig oly o m. Lemma.4. m! k m! Let a km =. The for all we have! k! a km e. k m k m 1... k m m 1 a km = m m mk < e. Lemma.5. For sufficietly large the followig iequalities holds: x m i M k t x m ; x λ m m/ ii M k t x m t l ; x λ m e xml m/ iii M k t x m t x j x ; x λ mj m λ j mj / iv M k t x m t l t x j ; x λ m λ j e 1/4 where lm j N. x mlj mj /

4 1 O geeral Gamma-Taylor operators o weighted spaces i ad i i i follow by the usig Hölder s iequality ad Remark.1. Also by Hölder s iequality Remark.1 ad Lemma.4 M k t x m t l ; x M k t x m ; x M k t l ; x ad x λ ml m a kl m/ λ m e xml m/ M k t x m t l t x j ; x M k t x m ; x M k t l t x j ; x x m λ m M k t x 4j ; x M k t 4l ; x m/ λ m λ j e 1/4 x mlj. mj / 1/ Let {b } be a sequece with positive terms b 1 > b lim b b = lim =. Theorem.1. For every f C k νb ] we have lim M kf f νb ]=. From [3] we kow that it is sufficiet to verify the followig three coditios lim M kt m ; x x m νb ]= m = 1. 5 Sice M k 1; x = 1 the coditio i 5 holds for m =. By usig Lemma.1 we have M k t; x x νb ] = x b ] 1 k M k t; x x 1 x x 1 x 1 k x b ] which implies that the coditio i 5 holds for m = 1. Similarly we ca write for > 1 M k t ; x x M k t ; x x νb ] = x b ] 1 x k 3k 4 k 1 which implies that lim M k t ; x x νb ]= the equatio 5 holds for m =. This completes the proof of theorem. 3. Rate of covergece of M k f ; x ad M kr f ; x i weighted spaces Now we wat to fid the rate of covergece of the operators {M k } ad {M kr }. It is well kow that the first order modulus of cotiuity ωf ;δ = t x δ xt [ab] f t f x does ot ted to zero as δ o ay ifiite iterval.

5 Alok Kumar et al. / It. J. Adv. Appl. Math. ad Mech A weighted modulus of cotiuity Ω f ;δ was defied i [5] which teds to zero as δ o a ifiite iterval. A similar defiitio of the modulus of cotiuity ca be foud i [1]. For each f C k νb ] it is give by { } f x h f x Ω f ;δ = 1 x 1 h : h δ x b ]. 6 For every f C k νb ] followig properties of Ω f ;δ were show i [5] lim Ω f ;δ = 7 δ f t f x 1 δ 1 x Ω f ;δ S t; x 8 where S t; x = 1 t x t x 1. It is easy to see that 1 δ t x δ S t; x 1 δ t x4 δ 4 t x δ. Theorem 3.1. Let f C k νb ]. The for all sufficietly large M k f f νb ] C Ω f ; b where C is a positive costat. δ 9 Usig Lemma.1 we get M k f ; x f x M k f t f x ; x Usig 9 we get S t; x 1 δ 1 1 δ 1 x Ω f ;δ Mk S t; x; x. t x4 δ 4 for all x b ] ad t. Thus for > 3 x b ] usig Lemma.3 we get M k f ; x f x 4 1 δ 1 x 1 1 c k Ω f ;δ δ δ 1 x 1 36 δ x4 Ω f ;δ. where c k = k 4 k 3 k k ad δ = that δ 1 for sufficietly large ad the statemet of the theorem follows. b 4 b b. Sice lim = we have Now we eed the followig modified Taylor formula. By Taylor s theorem [19] p we have f x = j = = j = f j t x t j f j t Let s = t ux t the f x = j = = j = f j t f j t x x t j x tr r 1! x t j x tr r 1! x t j x tr r 1! t f r s r 1! x sr d s x s x t r f r s x t d s. 1 u r f r t ux tdu 1 u r f r t ux t f r t du.

6 14 O geeral Gamma-Taylor operators o weighted spaces Theorem 3.. Let f f r C k νb ]. The for all sufficietly large b r / M kr f f νb ] C r Ω f r ; where C r is a positive costat depeds oly o r r = 1... b Usig modified Taylor s formula Lemma.1 ad 3 we get M kr f ; x f x where Θr t = x t K k x t Θr td t r 1! 1 u r f r t ux t f r t du. Usig 8 we get f r t ux t f r t 1 δ 1 t S xu tω f r δ where x b ] t u [1] ad S xu t = 1 u t x 1 u t x. δ It is easy to see that 1 u1 u δ t x δ S xu t 1 u1 u δ t x4 δ 4 t x δ. So for all x b ] t ad u [1] Thus S xu t 1 u 1 u δ 1 t x4 M kr f ; x f x C rδ Ω f r δ Mk 1 t t x r 1 δ 4. t x4 1 where C rδ = 1 δ r! 1 r 1! δ r! δ3. r 3! Usig Lemma.4 we get M kr f ; x f x C rδ Ω f r λ r x r λr ex r λr λ 4 δ r / r / Thus we have C rδ Ω f r δ 1 x λ r x r r / M kr f ; x f x x b ] 1 x C rδ Ω f r δ where A r = λ r λ r eb r = λ r λ 4 λ r λ 8e 1/4. b Choosig δ = ad takig ito accout that b 3 1 C rδ 4 := ξ r r j = b r / A r B r δ 4 δ 4 δ 4 λr ex r r / b ; x x r 4 r 4/ λr λ 4 δ 4 λ r λ 8 e 1/4 x r 4 δ 4 r 4/ x r 6 r 4/ λ r λ 8 e 1/4 b 1 for sufficietly large sice lim = we obtai δ 4 x r 6 r 4/. ad b r / M kr f f νb ] ξ r A r B r Ω f r b ;. Hece the statemet of the theorem follows with C r = ξ r A r B r.

7 4. Ackowledgemet Alok Kumar et al. / It. J. Adv. Appl. Math. ad Mech The authors are very thakful to Head of Departmet Computer Sciece Dev Saskriti Vishwavidyalaya Haridwar Uttarakhad Idia for providig ecessary facilities ad iformatios. Authors would also wish to express his gratitude to his parets for their moral port. Refereces [1] N. I. Akhieser Lectures o the theory of approximatio OGIZ Moscow-Leigrad 1947i Russia Theory of approximatio i Eglish Traslated by Hyma Frederick Ugar Publishig Co. New York [] R. A. DeVore G. G. Loretz Costructive Approximatio. Spriger Berli [3] A. D. Gadjiev Theorems of the type of P. P. korovki s theorems Matematicheskie Zametki [4] A. D. Gadjiev R. O. Efediyev E. Ibikli O Korovki type theorem i the space of locally itegrable fuctios Czechoslovak Math. J [5] N. Ispir O modified Baskakov operators o weighted spaces Turk. J. Math [6] A. İ zgi Voroovskaya type asymptotic approximatio by modified gamma operators Appl. Math. Comput [7] A. Izgi Rate of approximatio by modified Gamma-Taylor operators Eurasia Math. J [8] A. İ zgi I. Büyükyazici Approximatio ad rate of approximatio o ubouded itervals Kastamou Edu. J. Okt i Turkish. [9] A. Kumar Voroovskaja type asymptotic approximatio by geeral Gamma type operators It. J. of Mathematics ad its Applicatios 3 4-B [1] A. Kumar D. K. Vishwakarma Global approximatio theorems for geeral Gamma type operators. It. J. Adv. Appl. Math. ad Mech [11] H. Karsli Rate of covergece of a ew Gamma type operators for the fuctios with derivatives of bouded variatio Math. Comput. Modell [1] H. Karsli O covergece of geeral Gamma type operators Aal. Theory Appl [13] H. Karsli M. A. Özarsla Direct local ad global approximatio results for operators of gamma type Hacet. J. Math. Stat [14] H. Karsli V. Gupta A. Izgi Rate of poitwise covergece of a ew kid of gamma operators for fuctios of bouded variatio Appl. Math. Letters [15] H. Karsli P. N. Agrawal M. Goyal Geeral Gamma type operators based o q-itegers Appl. Math. Comput [16] A. Lupas M. Müller Approximatioseigeschafte der GammaoperatÃűre Mathematische Zeitschrift [17] L. C. Mao Rate of covergece of Gamma type operator J. Shagqiu Teachers Coll [18] S. M. Mazhar Approximatio by positive operators o ifiite itervals Math. Balkaica [19] M. Spivak Calculus Secod Editio Publish or Perish Ic Submit your mauscript to IJAAMM ad beefit from: Regorous peer review Immediate publicatio o acceptace Ope access: Articles freely available olie High visibility withi the field Retaiig the copyright to your article Submit your ext mauscript at editor.ijaamm@gmail.com