Global approximation by modified Baskakov type operators

Publicacions Matem`atiques, Vol 39 (1995), 263–271. GLOBAL APPROXIMATION BY MODIFIED BASKAKOV TYPE OPERATORS Vijay Gupta Abstract In the present paper, we prove a global direct theorem for the modified Baskakov type operators in terms of so called DitzianTotik modulus of smoothness. 1. Introduction Motivated by the integral modification of Bernstein polynomials by Durrmeyer [3], Sahai and Prasad [6] first defined and studied modified Baskakov operators. Sinha et al. [7] improved and corrected the results of [6]. Recently the author [4], introduced another modification of Baskakov operators by taking the weight function of Beta operators on L1[0,∞)as (1.1) (Bnf)(x)= ∞  k=0 pn,k(x)∞ 0 bn,k(t)f(t)dt, x ∈[0,∞) where pn,k(x)=n+k−1 kxk(1 + x)−n−k and bn,k(t)=[B(k+1,n)]−1tk(1 + t)−n−k−1, B(k+1,n) being the Beta function given by k!(n−1)!/(n+k)!. In [4], the author has obtained only local direct theorems in simultaneous approximation, as the operators defined by (1.1) give better approximation than the earlier integral modification of Baskakov operators Research supported by Council of Scientific and Industrial Research, India under award no. 9/143(163)/91-EMR-1. 264 V. Gupta studied in [5], [6] and [7] etc., this motivated us to extend the results of [4] to the whole interval [0,∞) and we study a global result for the operators (1.1). By Lr 1[0,∞), we denote the class of functions ggiven by Lr 1[0,∞):={g:g(r)∈L1[0,a] for every a∈(0,∞) and |g(r)(t)|≤M(1 + t)m,Mand mare constants depending on g}. We may remark that Lr p[0,∞) is not contained in Lr 1[0,∞). Following [2], the modulus of smoothness of fis given by ω2 φ(f,t)p= sup 0<h≤t ∆2 hφfp,φ(x)=x(1 + x) where ∆2 hf(x)=f(x−h)−2f(x)+f(x+h),if [x−h, x +h]⊂[0,∞) 0,otherwise. This modulus of smoothness is equivalent to the modified k-functional (see e.g. [2]) given by ¯ K2 φ(f,t2)p= inf{f−gp+t2φ2gp+t4gp;g∈¯ W2 p(φ, [0,∞))} where ¯ W2 p(φ, [0,∞)) = {g∈Lp[0,∞):g∈ACloc[0,∞); φ2g ∈Lp[0,∞)}. In [4] the author was not able to obtain global results. In the present paper, we prove a global direct theorem in simultaneous approximation for the operators (Bnf)(x) defined by (1.1) in terms of Ditzian-Totik modulus of second order. Throughout the paper we denote by Cthe positive constants not necessarily the same at each occurrence. 2. Auxiliary results In this section, we shall give certain definitions and lemmas which will be used in the sequel. For every n∈Nand n>(r+1)wehave (2.1) ∞  k=0 pn,k(x)=1,∞ 0 bn,k(t)dt =1 k npn,k(x)=xpn+1,k−1(x),∞ 0 tbn−r,k+r(t)dt =k+r+1 n−r−1. Modified Baskakov type operators 265 Lemma 2.1 [4]. Let m,r∈N0, we define Tr,n,m(x)= ∞  k=0 pn+r,k(x)∞ 0 bn−r,k+r(t)(t−x)mdt then Tr,n,0(x)=1,T r,n,1(x)=1+r+x(1+2r) (n−r−1) , Tr,n,2(x)=2(2r2+4r+n+1)x2+2(2r2+5r+2+n)x+(r2+3r+2) (n−r−1)(n−r−2) , and there holds the recurrence relation: (n−m−r−1)Tr,n,m+1(x)=φ2(x)[T(1) r,n,m(x)+2mTr,n,m−1(x)] +[(m+r+1)(1+2x)−x]Tr,n,m(x),n>m+r+1. Consequently for each x∈[0,∞),Tr,n,m(x)=0(n−[(m+1)/2]),[α]denotes the integral part of α. The proof of this lemma easily follows along the lines of [6], [7] using φ2(x)p n,k(x)=(k−nx)pn,k(x) and φ2(t)b n,k(t)=[k−(n+1)t]bn,k(t). From the above lemma, we have (2.2) Tr,n,2m(x)= m  i=0 qi,m,n(x)φ2(x) nm−i n−2i Tr,n,2m+1(x)=(1+2x) m  i=0 si,m,n(x)φ2(x) nm−i n−2i−1, where qi,m,n(x) and si,m,n(x) are polynomials in xof fixed degree with coefficients that are bounded uniformly for all n. Lemma 2.2. If f∈Lr p[0,∞)∪L r 1[0,∞),1≤p≤∞,n>r(1 + m) and x∈[0,∞), then (2.3) (Bnf)(r)(x)=α(n, r) ∞  k=0 pn+r,k(x)∞ 0 bn−r,k+r(t)f(r)(t)dt 266 V. Gupta where α(n, r)=(n+r−1)!(n−r−1)! ((n−1)!)2= r−1  =0 n+ n−( +1). Proof: By using Leibnitz theorem, we have (Bnf)(r)(x)= r  i=0 ∞  k=ir i(n+k+r−i−1)! (n−1)!(k−i)! ×(−1)r−ixk−i(1 + x)−n−k−r+i ×∞ 0 bn,k(t)f(t)dt =(n+r−1)! (n−1)! ∞  k=0 pn+r,k(x) ×∞ 0 r  i=0 (−1)r−ir ibn,k+i(t)f(t)dt. Again, by the use of Leibnitz theorem, we have b(r) n−r,k+r(t)= (n−1)! (n−r−1)! r  i=0 (−1)ir ibn,k+i(t). Hence, (Bnf)(r)(x)=(n+r−1)!(n−r−1)! ((n−1)!)2 ∞  k=0 pn+r,k(x)∞ 0 (−1)rb(r) n−r,k+r(t)f(t)dt. On integrating rtimes by parts, we get the required result. We see that the operators defined in (2.3) by B(r) nf:= (Bnf)(r),f∈ Lr p[0,∞)∪L 1[0,∞) are not positive. To make the operators positive we introduce the operator Bn,rf≡DrBnIrf, f ∈Lp[0,∞)∪L 1[0,∞), where Dand Iare differentiation and integration operators respectively. Therefore we define the operator by (Bn,rf)(x)=α(n, r) ∞  k=0 pn+r,k(x)∞ 0 bn−r,k+r(t)f(t)dt, Modified Baskakov type operators 267 f∈Lp[0,∞)∪L 1[0,∞), n>r(1 + m). The operators Bn,r are positive and the estimation (Bnf)(r)−f(r)p. f∈Lr p[0,∞) is equivalent to Bn,rf−fp,f∈Lp[0,∞). Using (2.1), we can easily prove that for n>(r+ 1), Bn,rf1≤ Cf1, for f∈L1[0,∞) and Bn,rf≤Cf∞for f∈L∞[0,∞). Making use of Riesz-Thorin theorem, we get (2.4) Bn,rfp≤Cfp,f∈Lp[0,∞),1≤p≤∞,n>(r+1). Corollary 2.3. For every m∈N0,n>(r+2m+1)and x∈[0,∞) we have (2.5) |Bn,r((t−x)2m,x)|≤Cn−m(φ2(x)+n−1)m, |Bn,r((t−x)2m+1,x)|≤C(1+2x)n−m−1(φ2(x)+n−1)m where the constant Cis independent of n. For fixed x∈[0,∞)we obtain (2.6) |Bn,r((t−x)m,x)|=0(n−[(m+1)/2]),n→∞. Proof: Since Bn,r((t−x)m,x)=α(n, r)Tr,n,m(x) the estimate (2.5) follows from (2.2) along the lines of [5], (2.6) immediately follows from (2.5). Lemma 2.4. Let t∈[0,∞)and n>(r+m)then Bn,r((1 + t)−m,x)≤C(1 + x)−m,x∈[0,∞) where the constant Cis independent of n. Proof: It is easily verified that (1 + t)−mbn−r,k+r(t)= m−1  =0 n−r+ n+ +k+1bn−r+m,k+r(t) and pn+r,k(x) = (1 + x)−m m  =1 n+r− +k n+r− pn+r−m,k(x). 268 V. Gupta Making use of these two identities and (2.1) we get Bn,r((1 + t)−m,x)=α(n, r) ∞  k=0 pn+r,k(x)∞ 0 bn−r,k+r(t)(1 + t)−mdt =α(n, r) ∞  k=0 pn+r,k(x) m−1  =0 n−r+ n+ +k+1 ×∞ 0 bn−r+m,k+r(t)dt =α(n, r) ∞  k=0 (1+x)−mpn+r−m,k(x) m  =1 (n+r− +k) (n+r− ) × m−1  =0 n+r− n+ +k+1 ≤C(1 + x)−m ∞  k=0 pn+r−m,k(x) =C(1 + x)−m. For the two monomials e0,e1and x∈[0,∞), n→∞we obtain by direct computation Bn,r(e0,x)=1+0(n−1)(2.7) Bn,r(e1,x)=x(1+0(n−1)).(2.8) Lemma 2.5. For Hn(u)given by Hn(u)=∞ 0u 0 −u 0∞ 0∞  k=0 pn+r,k(x)bn−r,k+r(t)(u−t)dt dx we have Hn(u)≤Cn−1φ2(u), where Cis independent of nand u. The proof of the above lemma easily follows by using (2.1) along the lines of [1, Lemma 5.2]. 3. Direct result Theorem 3.1. Suppose f∈Lp[0,∞),1≤p<∞,n>(r+5) then we have Bn,rf−fp≤C{ω2 φ(f,n−1/2)+n−1fp} Modified Baskakov type operators 269 where the constant Cis independent of n. Proof: By Taylor’s expansion of g,wehave (3.1) g(t)=g(x)+(t−x)g(x)+t x (t−u)g(u)du. Next, since Bn,r(f,x) are uniformly bounded operators so for every g∈ ¯ W2 p(φ, [0,∞)), we have (3.2) Bn,rf−fp≤Cf−gp+Bn,rg−gp. Using (2.5), (2.8) and (3.1) and following [2], we obtain Bn,rg−gp≤C{gp+gLp[0,1]}+(1+2x)gLp[1,∞) +Bn,r(R(g, t, x),x)p ≤Cn−1[gp+φ2gp]+Bn,r(R(g, t, x),x)p (3.3) where R(g, t, x)=t x(t−u)g(u)du. Now, we shall prove that (3.4) Bn,r(R(g, t, x),x)p≤Cn−1(φ2+n−1)gp. We prove this for p= 1 and p=∞. The cases 1 <p<∞follows again by Riesz-Thorin theorem. Using (2.5) for the case m= 1 and Lemma 2.4, the case p=∞easily follows (see e.g. [5]). For p= 1, we derive (3.4) by applying Fubini’s theorem twice, the definition of Hn(u) and Lemma 2.5 as ∞ 0 |Bn,r(R(g, t, x),x)|dx ≤α(n, r)∞ 0 ∞  k=0 pn+r,k(x)∞ 0 bn−r,k+r(t)|t x (t−u)g(u)|dt dx =α(n, r)∞ 0 |g(u)|∞ 0u 0 −u 0∞ 0(u−t) × ∞  k=0 pn+r,k(x)bn−r,k+r(t)dt dx du =α(n, r)∞ 0 |g(u)|Hn(u)du ≤Cn−1φ2g1 ≤Cn−1(φ2+n−1)g1, 270 V. Gupta where Cis independent of n. Hence (3.4) holds by Riesz-Thorin theorem for 1 ≤p≤∞. Combining the estimates of (3.2), (3.3) and (3.4) we get Bn,rf−fp=Cf−gp+Cn−1{f−gp+fp+φ2gp +(φ2+n−1)gp} ≤C{f−gp+n−1φ2gp+n−2gp+n−1fp}. Next taking the infimum over all g∈¯ W2 p(φ, [0,∞)) on the right hand side, we get Bn,rf−fp≤C{¯ K2 φ(f,n−1)+n−1fp}, this completes the proof of Theorem 3.1. Remark. The conclusion of Theorem 3.1 is true on the space Lp[0,∞), 1 ≤p<∞(i.e. lim n→∞ Bn,rf−fp= 0 for every f∈Lp[0,∞)), since the most basic fact about ω2 φ(f,n−1) is that lim n→∞ ω2 φ(f,n−1) = 0 for all f∈Lp[0,∞),1≤p<∞, or for all bounded functions f∈C[0,∞) which satisfy lim x→∞ f(x)=L∞<∞,if p=∞(cf. [2, p. 36]). Acknowledgement. The author is grateful to the referee for many suggestions that grately improved this paper. References 1. Z. Ditzian and K. Ivanov, Bernstein type operators and their derivatives, J. Approx. Theory 56 (1989), 72–90. 2. Z. Ditzian and V. Totik,“Moduli of smoothness,” Springer Series in Computational Mathematics 9, Springer-Verlag, Berlin, Heidelberg, New York, 1987. 3. J. L. Durrmeyer, Une formule d’inversion, de la transform´ee de Laplace: Application ´a la Theorie des Moments, Th´ese de 3e Cycle, Facult´e des Sciences de l’Universite de Paris, 1967. 4. V. Gupta, A note on modified Baskakov type operators, Approx. Theory and its Appl. 10(3) (1994), 74–78. Modified Baskakov type operators 271 5. M. Heilmann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory and its Appl. 5(1) (1989), 105–127. 6. A. Sahai and G. Prasad, On simultaneous approximation by modified Lupas operators, J. Approx. Theory 45 (1985), 122–128. 7. R. P. Sinha, P. N. Agrawal and V. Gupta, On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. Ser. B 42(2) (1991), 217–231. Department of Mathematics University of Roorkee Roorkee-247667 (U.P.) INDIA Primera versi´o rebuda el 13 d’Octubre de 1994, darrera versi´o rebuda el 8 de Maig de 1995