Global approximation by modified Baskakov type operators

Publicacions Ma em`a iques, Vol 39 (1995), 263–271. GLOBAL APPROXIMATION BY MODIFIED BASKAKOV TYPE OPERATORS Vijay Gup a Abs ac In he p esen pape , we p o e a global di ec heo em o he modified Baskako ype ope a o s in e ms o so called Di zian- To ik modulus o smoo hness. 1. In oduc ion Mo i a ed by he in eg al modifica ion o Be ns ein polynomials by Du meye [3], Sahai and P asad [6] fi s defined and s udied modified Baskako ope a o s. Sinha e al. [7] imp o ed and co ec ed he e- sul s o [6]. Recen ly he au ho [4], in oduced ano he modifica ion o Baskako ope a o s by aking he weigh unc ion o Be a ope a o s on L1[0,∞)as (1.1) (Bn )(x)= ∞  k=0 pn,k(x)∞ 0 bn,k( ) ( )d , x ∈[0,∞) whe e pn,k(x)=n+k−1 kxk(1 + x)−n−k and bn,k( )=[B(k+1,n)]−1 k(1 + )−n−k−1, B(k+1,n) being he Be a unc ion gi en by k!(n−1)!/(n+k)!. In [4], he au ho has ob ained only local di ec heo ems in simul a- neous app oxima ion, as he ope a o s defined by (1.1) gi e be e ap- p oxima ion han he ea lie in eg al modifica ion o Baskako ope a o s Resea ch suppo ed by Council o Scien ific and Indus ial Resea ch, India unde awa d no. 9/143(163)/91-EMR-1. 264 V. Gup a s udied in [5], [6] and [7] e c., his mo i a ed us o ex end he esul s o [4] o he whole in e al [0,∞) and we s udy a global esul o he ope a o s (1.1). By L 1[0,∞), we deno e he class o unc ions ggi en by L 1[0,∞):={g:g( )∈L1[0,a] o e e y a∈(0,∞) and |g( )( )|≤M(1 + )m,Mand ma e cons an s depending on g}. We may ema k ha L p[0,∞) is no con ained in L 1[0,∞). Following [2], he modulus o smoo hness o is gi en by ω2 φ( , )p= sup 0<h≤ ∆2 hφ p,φ(x)=x(1 + x) whe e ∆2 h (x)= (x−h)−2 (x)+ (x+h),i [x−h, x +h]⊂[0,∞) 0,o he wise. This modulus o smoo hness is equi alen o he modified k- unc ional (see e.g. [2]) gi en by ¯ K2 φ( , 2)p= in { −gp+ 2φ2gp+ 4gp;g∈¯ W2 p(φ, [0,∞))} whe e ¯ W2 p(φ, [0,∞)) = {g∈Lp[0,∞):g∈ACloc[0,∞); φ2g ∈Lp[0,∞)}. In [4] he au ho was no able o ob ain global esul s. In he p esen pape , we p o e a global di ec heo em in simul aneous app oxima ion o he ope a o s (Bn )(x) defined by (1.1) in e ms o Di zian-To ik modulus o second o de . Th oughou he pape we deno e by C he posi i e cons an s no nec- essa ily he same a each occu ence. 2. Auxilia y esul s In his sec ion, we shall gi e ce ain defini ions and lemmas which will be used in he sequel. Fo e e y n∈Nand n>( +1)weha e (2.1) ∞  k=0 pn,k(x)=1,∞ 0 bn,k( )d =1 k npn,k(x)=xpn+1,k−1(x),∞ 0 bn− ,k+ ( )d =k+ +1 n− −1. Modi ied Baskako ype ope a o s 265 Lemma 2.1 [4]. Le m, ∈N0, we define T ,n,m(x)= ∞  k=0 pn+ ,k(x)∞ 0 bn− ,k+ ( )( −x)md hen T ,n,0(x)=1,T ,n,1(x)=1+ +x(1+2 ) (n− −1) , T ,n,2(x)=2(2 2+4 +n+1)x2+2(2 2+5 +2+n)x+( 2+3 +2) (n− −1)(n− −2) , and he e holds he ecu ence ela ion: (n−m− −1)T ,n,m+1(x)=φ2(x)[T(1) ,n,m(x)+2mT ,n,m−1(x)] +[(m+ +1)(1+2x)−x]T ,n,m(x),n>m+ +1. Consequen ly o each x∈[0,∞),T ,n,m(x)=0(n−[(m+1)/2]),[α]de- no es he in eg al pa o α. The p oo o his lemma easily ollows along he lines o [6], [7] using φ2(x)p n,k(x)=(k−nx)pn,k(x) and φ2( )b n,k( )=[k−(n+1) ]bn,k( ). F om he abo e lemma, we ha e (2.2) T ,n,2m(x)= m  i=0 qi,m,n(x)φ2(x) nm−i n−2i T ,n,2m+1(x)=(1+2x) m  i=0 si,m,n(x)φ2(x) nm−i n−2i−1, whe e qi,m,n(x) and si,m,n(x) a e polynomials in xo fixed deg ee wi h coefficien s ha a e bounded uni o mly o all n. Lemma 2.2. I ∈L p[0,∞)∪L 1[0,∞),1≤p≤∞,n> (1 + m) and x∈[0,∞), hen (2.3) (Bn )( )(x)=α(n, ) ∞  k=0 pn+ ,k(x)∞ 0 bn− ,k+ ( ) ( )( )d 266 V. Gup a whe e α(n, )=(n+ −1)!(n− −1)! ((n−1)!)2= −1  =0 n+ n−( +1). P oo : By using Leibni z heo em, we ha e (Bn )( )(x)=  i=0 ∞  k=i i(n+k+ −i−1)! (n−1)!(k−i)! ×(−1) −ixk−i(1 + x)−n−k− +i ×∞ 0 bn,k( ) ( )d =(n+ −1)! (n−1)! ∞  k=0 pn+ ,k(x) ×∞ 0  i=0 (−1) −i ibn,k+i( ) ( )d . Again, by he use o Leibni z heo em, we ha e b( ) n− ,k+ ( )= (n−1)! (n− −1)!  i=0 (−1)i ibn,k+i( ). Hence, (Bn )( )(x)=(n+ −1)!(n− −1)! ((n−1)!)2 ∞  k=0 pn+ ,k(x)∞ 0 (−1) b( ) n− ,k+ ( ) ( )d . On in eg a ing imes by pa s, we ge he equi ed esul . We see ha he ope a o s defined in (2.3) by B( ) n := (Bn )( ), ∈ L p[0,∞)∪L 1[0,∞) a e no posi i e. To make he ope a o s posi i e we in oduce he ope a o Bn, ≡D BnI , ∈Lp[0,∞)∪L 1[0,∞), whe e Dand Ia e diffe en ia ion and in eg a ion ope a o s espec i ely. The e o e we define he ope a o by (Bn, )(x)=α(n, ) ∞  k=0 pn+ ,k(x)∞ 0 bn− ,k+ ( ) ( )d , Modi ied Baskako ype ope a o s 267 ∈Lp[0,∞)∪L 1[0,∞), n> (1 + m). The ope a o s Bn, a e posi i e and he es ima ion (Bn )( )− ( )p. ∈L p[0,∞) is equi alen o Bn, − p, ∈Lp[0,∞). Using (2.1), we can easily p o e ha o n>( + 1), Bn, 1≤ C 1, o ∈L1[0,∞) and Bn, ≤C ∞ o ∈L∞[0,∞). Making use o Riesz-Tho in heo em, we ge (2.4) Bn, p≤C p, ∈Lp[0,∞),1≤p≤∞,n>( +1). Co olla y 2.3. Fo e e y m∈N0,n>( +2m+1)and x∈[0,∞) we ha e (2.5) |Bn, (( −x)2m,x)|≤Cn−m(φ2(x)+n−1)m, |Bn, (( −x)2m+1,x)|≤C(1+2x)n−m−1(φ2(x)+n−1)m whe e he cons an Cis independen o n. Fo fixed x∈[0,∞)we ob ain (2.6) |Bn, (( −x)m,x)|=0(n−[(m+1)/2]),n→∞. P oo : Since Bn, (( −x)m,x)=α(n, )T ,n,m(x) he es ima e (2.5) ollows om (2.2) along he lines o [5], (2.6) immedia ely ollows om (2.5). Lemma 2.4. Le ∈[0,∞)and n>( +m) hen Bn, ((1 + )−m,x)≤C(1 + x)−m,x∈[0,∞) whe e he cons an Cis independen o n. P oo : I is easily e ified ha (1 + )−mbn− ,k+ ( )= m−1  =0 n− + n+ +k+1bn− +m,k+ ( ) and pn+ ,k(x) = (1 + x)−m m  =1 n+ − +k n+ − pn+ −m,k(x). 268 V. Gup a Making use o hese wo iden i ies and (2.1) we ge Bn, ((1 + )−m,x)=α(n, ) ∞  k=0 pn+ ,k(x)∞ 0 bn− ,k+ ( )(1 + )−md =α(n, ) ∞  k=0 pn+ ,k(x) m−1  =0 n− + n+ +k+1 ×∞ 0 bn− +m,k+ ( )d =α(n, ) ∞  k=0 (1+x)−mpn+ −m,k(x) m  =1 (n+ − +k) (n+ − ) × m−1  =0 n+ − n+ +k+1 ≤C(1 + x)−m ∞  k=0 pn+ −m,k(x) =C(1 + x)−m. Fo he wo monomials e0,e1and x∈[0,∞), n→∞we ob ain by di ec compu a ion Bn, (e0,x)=1+0(n−1)(2.7) Bn, (e1,x)=x(1+0(n−1)).(2.8) Lemma 2.5. Fo Hn(u)gi en by Hn(u)=∞ 0u 0 −u 0∞ 0∞  k=0 pn+ ,k(x)bn− ,k+ ( )(u− )d dx we ha e Hn(u)≤Cn−1φ2(u), whe e Cis independen o nand u. The p oo o he abo e lemma easily ollows by using (2.1) along he lines o [1, Lemma 5.2]. 3. Di ec esul Theo em 3.1. Suppose ∈Lp[0,∞),1≤p<∞,n>( +5) hen we ha e Bn, − p≤C{ω2 φ( ,n−1/2)+n−1 p} Modi ied Baskako ype ope a o s 269 whe e he cons an Cis independen o n. P oo : By Taylo ’s expansion o g,weha e (3.1) g( )=g(x)+( −x)g(x)+ x ( −u)g(u)du. Nex , since Bn, ( ,x) a e uni o mly bounded ope a o s so o e e y g∈ ¯ W2 p(φ, [0,∞)), we ha e (3.2) Bn, − p≤C −gp+Bn, g−gp. Using (2.5), (2.8) and (3.1) and ollowing [2], we ob ain Bn, g−gp≤C{gp+gLp[0,1]}+(1+2x)gLp[1,∞) +Bn, (R(g, , x),x)p ≤Cn−1[gp+φ2gp]+Bn, (R(g, , x),x)p (3.3) whe e R(g, , x)= x( −u)g(u)du. Now, we shall p o e ha (3.4) Bn, (R(g, , x),x)p≤Cn−1(φ2+n−1)gp. We p o e his o p= 1 and p=∞. The cases 1 <p<∞ ollows again by Riesz-Tho in heo em. Using (2.5) o he case m= 1 and Lemma 2.4, he case p=∞easily ollows (see e.g. [5]). Fo p= 1, we de i e (3.4) by applying Fubini’s heo em wice, he defini ion o Hn(u) and Lemma 2.5 as ∞ 0 |Bn, (R(g, , x),x)|dx ≤α(n, )∞ 0 ∞  k=0 pn+ ,k(x)∞ 0 bn− ,k+ ( )| x ( −u)g(u)|d dx =α(n, )∞ 0 |g(u)|∞ 0u 0 −u 0∞ 0(u− ) × ∞  k=0 pn+ ,k(x)bn− ,k+ ( )d dx du =α(n, )∞ 0 |g(u)|Hn(u)du ≤Cn−1φ2g1 ≤Cn−1(φ2+n−1)g1, 270 V. Gup a whe e Cis independen o n. Hence (3.4) holds by Riesz-Tho in heo em o 1 ≤p≤∞. Combining he es ima es o (3.2), (3.3) and (3.4) we ge Bn, − p=C −gp+Cn−1{ −gp+ p+φ2gp +(φ2+n−1)gp} ≤C{ −gp+n−1φ2gp+n−2gp+n−1 p}. Nex aking he infimum o e all g∈¯ W2 p(φ, [0,∞)) on he igh hand side, we ge Bn, − p≤C{¯ K2 φ( ,n−1)+n−1 p}, his comple es he p oo o Theo em 3.1. Rema k. The conclusion o Theo em 3.1 is ue on he space Lp[0,∞), 1 ≤p<∞(i.e. lim n→∞ Bn, − p= 0 o e e y ∈Lp[0,∞)), since he mos basic ac abou ω2 φ( ,n−1) is ha lim n→∞ ω2 φ( ,n−1) = 0 o all ∈Lp[0,∞),1≤p<∞, o o all bounded unc ions ∈C[0,∞) which sa is y lim x→∞ (x)=L∞<∞,i p=∞(c . [2, p. 36]). Acknowledgemen . The au ho is g a e ul o he e e ee o many sugges ions ha g a ely imp o ed his pape . Re e ences 1. Z. Di zian and K. I ano , Be ns ein ype ope a o s and hei de i a i es, J. App ox. Theo y 56 (1989), 72–90. 2. Z. Di zian and V. To ik,“Moduli o smoo hness,” Sp inge Se ies in Compu a ional Ma hema ics 9, Sp inge -Ve lag, Be lin, Heidel- be g, New Yo k, 1987. 3. J. L. Du meye , Une o mule d’in e sion, de la ans o m´ee de Laplace: Applica ion ´a la Theo ie des Momen s, Th´ese de 3e Cycle, Facul ´e des Sciences de l’Uni e si e de Pa is, 1967. 4. V. Gup a, A no e on modified Baskako ype ope a o s, App ox. Theo y and i s Appl. 10(3) (1994), 74–78. Modi ied Baskako ype ope a o s 271 5. M. Heilmann, Di ec and con e se esul s o ope a o s o Baskako -Du meye ype, App ox. Theo y and i s Appl. 5(1) (1989), 105–127. 6. A. Sahai and G. P asad, On simul aneous app oxima ion by modified Lupas ope a o s, J. App ox. Theo y 45 (1985), 122–128. 7. R. P. Sinha, P. N. Ag awal and V. Gup a, On simul aneous app oxima ion by modified Baskako ope a o s, Bull. Soc. Ma h. Belg. Se . B 42(2) (1991), 217–231. Depa men o Ma hema ics Uni e si y o Roo kee Roo kee-247667 (U.P.) INDIA P ime a e si´o ebuda el 13 d’Oc ub e de 1994, da e a e si´o ebuda el 8 de Maig de 1995