In the present paper, we prove a global direct theorem for th

Gupta, Vijay

V. Gupta

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Publicacions Matemàtiques

Global approximation by modified Baskakov type operators

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Publicacions Matema`tiques, Vol 39 (1995), 263–271.GLOBAL APPROXIMATION BYMODIFIED BASKAKOV TYPE OPERATORSVijay GuptaAbstractIn the present paper, we prove a global direct theorem for themodified Baskakov type operators in terms of so called Ditzian-Totik modulus of smoothness.1. IntroductionMotivated by the integral modification of Bernstein polynomials byDurrmeyer [3], Sahai and Prasad [6] first defined and studied modifiedBaskakov operators. Sinha et al. [7] improved and corrected the re-sults of [6]. Recently the author [4], introduced another modification ofBaskakov operators by taking the weight function of Beta operators onL1[0,∞) as(1.1) (Bnf)(x) =∞∑k=0pn,k(x)∫ ∞0bn,k(t)f(t) dt, x ∈ [0,∞)wherepn,k(x) =(n+ k − 1k)xk(1 + x)−n−kandbn,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 simulta-neous approximation, as the operators defined by (1.1) give better ap-proximation than the earlier integral modification of Baskakov operatorsResearch supported by Council of Scientific and Industrial Research, India underaward no. 9/143(163)/91-EMR-1.264 V. Guptastudied in [5], [6] and [7] etc., this motivated us to extend the resultsof [4] to the whole interval [0,∞) and we study a global result for theoperators (1.1).By Lr1[0,∞), we denote the class of functions g given byLr1[0,∞) := {g : g(r) ∈ L1[0, a] for every a ∈ (0,∞) and|g(r)(t)| ≤M(1 + t)m, M and m are constants depending on g}.We may remark that Lrp[0,∞) is not contained in Lr1[0,∞).Following [2], the modulus of smoothness of f is given byω2φ(f, t)p = sup0<h≤t‖∆2hφf‖p, φ(x) =√x(1 + x)where∆2hf(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 byK¯2φ(f, t2)p = inf{‖f − g‖p + t2‖φ2g′′‖p + t4‖g′′‖p; g ∈ W¯ 2p (φ, [0,∞))}whereW¯ 2p (φ, [0,∞)) = {g ∈ Lp[0,∞) : g′ ∈ ACloc[0,∞); φ2g′′ ∈ Lp[0,∞)}.In [4] the author was not able to obtain global results. In the presentpaper, we prove a global direct theorem in simultaneous approximationfor the operators (Bnf)(x) defined by (1.1) in terms of Ditzian-Totikmodulus of second order.Throughout the paper we denote by C the positive constants not nec-essarily the same at each occurrence.2. Auxiliary resultsIn this section, we shall give certain definitions and lemmas which willbe used in the sequel.For every n ∈ N and n > (r + 1) we have(2.1)∞∑k=0pn,k(x) = 1,∫ ∞0bn,k(t) dt = 1knpn,k(x) = xpn+1,k−1(x),∫ ∞0tbn−r,k+r(t) dt =k + r + 1n− r − 1 .Modified Baskakov type operators 265Lemma 2.1 [4]. Let m, r ∈ N0, we defineTr,n,m(x) =∞∑k=0pn+r,k(x)∫ ∞0bn−r,k+r(t)(t− x)m dtthenTr,n,0(x) = 1, Tr,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] de-notes the integral part of α.The proof of this lemma easily follows along the lines of [6], [7] usingφ2(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=0qi,m,n(x)[φ2(x)n]m−in−2iTr,n,2m+1(x) = (1 + 2x)m∑i=0si,m,n(x)[φ2(x)n]m−in−2i−1,where qi,m,n(x) and si,m,n(x) are polynomials in x of fixed degree withcoefficients that are bounded uniformly for all n.Lemma 2.2. If f ∈ Lrp[0,∞) ∪ Lr1[0,∞), 1 ≤ p ≤ ∞, n > r(1 + m)and x ∈ [0,∞), then(2.3) (Bnf)(r)(x) = α(n, r)∞∑k=0pn+r,k(x)∫ ∞0bn−r,k+r(t)f (r)(t) dt266 V. Guptawhereα(n, r) =(n+ r − 1)!(n− r − 1)!((n− 1)!)2 =r−1∏=0n+  n− ( + 1) .Proof: By using Leibnitz theorem, we have(Bnf)(r)(x) =r∑i=0∞∑k=i(ri)(n+ k + r − i− 1)!(n− 1)!(k − i)!× (−1)r−ixk−i(1 + x)−n−k−r+i×∫ ∞0bn,k(t)f(t) dt=(n+ r − 1)!(n− 1)!∞∑k=0pn+r,k(x)×∫ ∞0r∑i=0(−1)r−i(ri)bn,k+i(t)f(t) dt.Again, by the use of Leibnitz theorem, we haveb(r)n−r,k+r(t) =(n− 1)!(n− r − 1)!r∑i=0(−1)i(ri)bn,k+i(t).Hence,(Bnf)(r)(x) =(n+ r − 1)!(n− r − 1)!((n− 1)!)2∞∑k=0pn+r,k(x)∫ ∞0(−1)rb(r)n−r,k+r(t)f(t) dt.On integrating r times by parts, we get the required result.We see that the operators defined in (2.3) by B(r)n f := (Bnf)(r), f ∈Lrp[0,∞)∪L1[0,∞) are not positive. To make the operators positive weintroduce the operatorBn,rf ≡ DrBnIrf, f ∈ Lp[0,∞) ∪ L1[0,∞),where D and I are differentiation and integration operators respectively.Therefore we define the operator by(Bn,rf)(x) = α(n, r)∞∑k=0pn+r,k(x)∫ ∞0bn−r,k+r(t)f(t) dt,Modified Baskakov type operators 267f ∈ Lp[0,∞) ∪ L1[0,∞), n > r(1 +m).The operators Bn,r are positive and the estimation ‖(Bnf)(r)−f (r)‖p.f ∈ Lrp[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)mwhere the constant C is 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) thenBn,r((1 + t)−m, x) ≤ C(1 + x)−m, x ∈ [0,∞)where the constant C is independent of n.Proof: It is easily verified that(1 + t)−mbn−r,k+r(t) =m−1∏=0n− r +  n+  + k + 1bn−r+m,k+r(t)andpn+r,k(x) = (1 + x)−mm∏=1n+ r −  + kn+ r −  pn+r−m,k(x).268 V. GuptaMaking use of these two identities and (2.1) we getBn,r((1 + t)−m, x) = α(n, r)∞∑k=0pn+r,k(x)∫ ∞0bn−r,k+r(t)(1 + t)−m dt= α(n, r)∞∑k=0pn+r,k(x)m−1∏=0n− r +  n+  + k + 1×∫ ∞0bn−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∏=0n+ r −  n+  + k + 1≤ C(1 + x)−m∞∑k=0pn+r−m,k(x)= C(1 + x)−m.For the two monomials e0, e1 and x ∈ [0,∞), n→∞ we obtain by directcomputationBn,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 byHn(u) ={∫ ∞0∫ u0−∫ u0∫ ∞0} ∞∑k=0pn+r,k(x)bn−r,k+r(t)(u− t) dt dxwe have Hn(u) ≤ Cn−1φ2(u), where C is independent of n and u.The proof of the above lemma easily follows by using (2.1) along thelines of [1, Lemma 5.2].3. Direct resultTheorem 3.1. Suppose f ∈ Lp[0,∞), 1 ≤ p < ∞, n > (r + 5) thenwe have‖Bn,rf − f‖p ≤ C{ω2φ(f, n−1/2) + n−1‖f‖p}Modified Baskakov type operators 269where the constant C is independent of n.Proof: By Taylor’s expansion of g, we have(3.1) g(t) = g(x) + (t− x)g′(x) +∫ tx(t− u)g′′(u) du.Next, since Bn,r(f, x) are uniformly bounded operators so for every g ∈W¯ 2p (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) =∫ tx(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 againby Riesz-Thorin theorem.Using (2.5) for the case m = 1 and Lemma 2.4, the case p = ∞ easilyfollows (see e.g. [5]).For p = 1, we derive (3.4) by applying Fubini’s theorem twice, thedefinition of Hn(u) and Lemma 2.5 as∫ ∞0|Bn,r(R(g, t, x), x)| dx≤ α(n, r)∫ ∞0∞∑k=0pn+r,k(x)∫ ∞0bn−r,k+r(t)|∫ tx(t− u)g′′(u)| dt dx= α(n, r)∫ ∞0|g′′(u)|{∫ ∞0∫ u0−∫ u0∫ ∞0}(u− t)×∞∑k=0pn+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. Guptawhere C is independent of n. Hence (3.4) holds by Riesz-Thorin theoremfor 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 ∈ W¯ 2p (φ, [0,∞)) on the right handside, we get‖Bn,rf − f‖p ≤ C{K¯2φ(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 spaceLp[0,∞), 1 ≤ p <∞ (i.e. limn→∞ ‖Bn,rf−f‖p = 0 for every f ∈ Lp[0,∞)),since the most basic fact about ω2φ(f, n−1) is thatlimn→∞ω2φ(f, n−1) = 0 for all f ∈ Lp[0,∞), 1 ≤ p <∞,or for all bounded functions f ∈ C[0,∞) which satisfylimx→∞ f(x) = L∞ <∞, if p = ∞ (cf. [2, p. 36]).Acknowledgement. The author is grateful to the referee for manysuggestions that grately improved this paper.References1. Z. Ditzian and K. Ivanov, Bernstein type operators and theirderivatives, J. Approx. Theory 56 (1989), 72–90.2. Z. Ditzian and V. Totik, “Moduli of smoothness,” Springer Seriesin Computational Mathematics 9, Springer-Verlag, Berlin, Heidel-berg, New York, 1987.3. J. L. Durrmeyer, Une formule d’inversion, de la transforme´e deLaplace: Application a´ la Theorie des Moments, The´se de 3e Cycle,Faculte´ 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 2715. M. Heilmann, Direct and converse results for operators ofBaskakov-Durrmeyer type, Approx. Theory and its Appl. 5(1)(1989), 105–127.6. A. Sahai and G. Prasad, On simultaneous approximation bymodified Lupas operators, J. Approx. Theory 45 (1985), 122–128.7. R. P. Sinha, P. N. Agrawal and V. Gupta, On simultaneousapproximation by modified Baskakov operators, Bull. Soc. Math.Belg. Ser. B 42(2) (1991), 217–231.Department of MathematicsUniversity of RoorkeeRoorkee-247667 (U.P.)INDIAPrimera versio´ rebuda el 13 d’Octubre de 1994,darrera versio´ rebuda el 8 de Maig de 1995

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Global approximation by modified Baskakov type operators

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