Degree of Approximationof Functionsby Newly Defined Polynomials onan unbounded interval

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1 IOSR Joural of Mathematics (IOSRJM) ISSN: Volume, Issue (May-Jue 0), PP Degree of Approximatioof Fuctiosy Newly Defied Polyomials oa uouded iterval Awar Hai Departmetof Geeral Studies JuailIdustrial College, KSA ABSTRACT : Chlodovsy( 937) has proved the theorem. &. for Berstei Polyomials f( B (x)=b f (x; )= )( x )( ) ( x ) o a uouded iterval. The oject of this paper is to exted the aove theorems for ewly defied polyomials + A (x) = A f (x, α; ) = ( + ) { f(t)dt} p x, ; α o a uouded iterval. Keywords:Berstei Polyomials, Lesgue Itegrale fuctio, L orm, geeratig fuctio, Modified Polyomials I. Itroductio & results If f(x) is a fuctio defied o [0, ], the Berstei polyomial B f (x) of f is B f (x)= f( )p, x, wherep, x = ( )(x) ( x). If the fuctio f(x) defied i the iterval (0,),>0.The Berstei polyomialb f (x; ) for this iterval is give y B (x)=b f (x;)= f( )( )(x ) ( x ) (.) Further a small modificatio of the Berstei polyomial due tokatorovich [] ad Awar&Umar [3] maes it possile to approximate Leesgue itegrale fuctio i L orm y a ewly defied polyomial + A α (f, x) = ( + ) { f(t)dt} p, x; α (.) where p, x; α =( ) x(x+α) ( x+ α) (+α) (.3) such that p, x; α =. Let the fuctio f(x) e defied o the iterval (0,),>0. To otai a modified polyomial A f (x, α; ) for this iterval, we mae the sustitutio y=x - i the polyomial A Φ (y) of the fuctio Φ y = f y,0 y ad otai i this way + A (x) = A f (x, α; ) = ( + ) { f(t)dt} p x, ; α ---- (.4) where p x ; α =(, ) (x )(x +α) ( x + α) (.5) (+α) Chlodovsy (937) has proved the theorem y assumig = is a fuctio of, which icreases to + with ad f(x) defied i the ifiite iterval 0 x<. Theorem.:- If =0() ad the fuctio f(x) is ouded i [0, + ), say f(x) M, the the B (x) f(x) holdsat ay poit of cotiuity of the fuctio f(x).

2 Degree of Approximatioof Fuctiosy Newly Defied Polyomials oa uouded iterval Theorem.:- If =0() ad M( )e α/ 0, for each α > 0, the B (x) f(x) holds at each poit of cotiuity of the fuctio f(x). I this paper our oject is to improve the aove results y taig the ew polyomial A x istead of B (x) which may e stated as follows Theorem.3:- If =0() ad the fuctio f(x) is ouded leesgue itegrale i [ 0, + ), say f(x) M, the A (x) f(x) holdsat ay poit of cotiuity of the fuctio f(x). Theorem.4:- If =0() ad M( )e β/ 0, (.6) for each β > 0,the A (x) f(x) holds at each poit of cotiuity of the fuctio f(x). II. Lemmas I order to proofour result we eed the followig Lemmas Lemma.:[3] For all values of x ϵ [ 0,] ad for α=α =0( ) + We have ( + ) { (t x) dt} Lemma.: If 0 x, the iequality, p, x; α x( x). 0 z 3 x (x ) (.) Implies + ( + ) p, x dt e z x x t x z( ) (.) Proof of lemma.: Let Φ e the geeratig fuctio of the polyomial T= x p, (x: α), which may e defied as Φ=Φ u, s = = p, x; α s=0 s! T s (x)u s s=0 s! ( x)s u s = e u( x ) ( ) x(x+α) ( x+ α) (+α) e xu = [ x + α + x x + α e u +α + x x + α x + α e u + + x x + α e u! Φ = e xu x + xe u, for α=α =0( ) ad therefore Φ = [e xu x + xe u ] (.3) To prove our result we first show that for u 3, the iequality Φ exp {x x u }.(.4) holds. For (.3) ca e writte as Φ = [xe u( x) + x)e ux ] 9 Page

3 But sice Degree of Approximatioof Fuctiosy Newly Defied Polyomials oa uouded iterval xe u ( x) + ( x)e ux = + ν= + x x + x x u ν=0 ν! [x x + ( x)( x)ν ] =+ x x u ν= ν! ( + u 3 + u 3 +.) ( 3 u ) + x x u for u 3 e x x u ase > + ad hece Φ [e x x u ] = exp{ x x u } which is (.4). ν! [x x + ( x)( x)ν ] Therefore if ѱ=ѱ u, x = e u x p, x; α (.5) the we otai for 0 u 3 ѱ ѱ u, x +Ψ u, x ad therefore, for α=α =0( ), we have ѱ exp{ x x u } (.6) ow we get our required result, we ote that for c 0 ad u 0 cѱ ( + ) ( dt)p, x; α exp [u x ] cѱ x ( + ) { f(t)dt}eu p, x; α c Now if we put c= z, we otai ( + ) ( dt)p, x; α exp [u x ] e z ѱ or + ( + ) ( dt)p, x; α x z u +x x u e z e z sice for the give rage of t x t x, we have ( + ) ( dt)p, x; α t x z u +x x u Sice. ca e writte as e z (.7) 0 z x( x) 3 But (.7)holds for 0 u 3 ad therefore for u= z x( x), we have ( + ) ( dt)p, x; α t x z x x x x +z e z 0 Page

4 Degree of Approximatioof Fuctiosy Newly Defied Polyomials oa uouded iterval ( + ) ( dt)p, x; α t x z x x this completes the proof of lemma. e z Proof of theorem.3: We have III. Proof of theorems A x f x ( + ) { f t f x dt} p x ; α, Let ϵ>0 e aritrary ad choose ifiitesimally small δ>0 such that f x f x < ϵ for x x < δ the A x f x + { f t f x dt} p x, ; α t x < δ + + { f t f x dt} p x ; α, t x δ =I +I (3.) I = + { f t f x dt} p x ; α, t x < δ <ϵ + { dt} p x ; α, t x < δ =ϵ (3.) To calculate I, we put u= x ad the we have + I = + { f t f x dt} p x ; α, t x δ M + { dt} t u δ + p, u; α M( δ ) ( + ) { (t u) dt} p, u; α M( δ u( u ) ) for all α=0( )ylemma(.), x M for all large & ( δ α=0( )sice ) = o(), < ϵ (3.3) Hece A x f x ϵ+ ϵ= ϵ this completes the proof of theorem.3. Proof of theorem.4: Proceedig as i theorem.3 we otai Page

5 Degree of Approximatioof Fuctiosy Newly Defied Polyomials oa uouded iterval A x f x ϵ +M( ) + { dt} t u δ p, u; α The secod term ca e easily estimated y meas of lemma(.), if, u u z =δ ( ) the coditio (.) satisfied if we assume, for istace, δ<x ad that is sufficietly large. Hece y (.6) we otai A x f x ϵ +M( ) exp (-z ) = ϵ +M( ) exp {-δ.[4 x = ϵ + ϵ = ϵ for large this completes the proof of the theorem.4. x ] } IV. Coclusio I this paper we have improved the results of Chlodovsy y taig the ew Modified Polyomials A x istead of Berstei Polyomials B (x). Refereces [] Chlodovsy, I (937). Sur le developmet des foctios defiies das u iterval ifii e series de polyomes de M S Berstei compositio math, 4, [] Katorovic, L A (930). Sur certais developpemets suivait les polyomes de la forme de S Berstei I, II, C R Acad. Sci. USSR, 0,563-68, [3] Awar Hai ad S Umar (980) O Geeralized Berstei Polyomials Idia J. pure appl. Math., (), Page