# Approximation by Durrmeyer

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Approximation by Durrmeyer

ANNALES POLONICI MATHEMATICI LXIV.2 (1996) Approximation by Durrmeyer-type operators by Vijay Gupta and G. S. Srivastava (Roorkee) Abstract. We define a new kind of Durrmeyer-type summation-integral operators and study a global direct theorem for these operators in terms of the Ditzian–Totik modulus of smoothness. 1. Durrmeyer [4] introduced modified Bernstein polynomials to approximate Lebesgue integrable functions on [0, 1], later motivated by the integral modification of Bernstein polynomials by Durrmeyer; Sahai and Prasad [9] and Mazhar and Totik [8] introduced modified Lupas operators and modified Sz´ asz operators respectively to approximate Lebesgue integrable functions on [0, ∞). A lot of work has been done on these three operators (see e.g. [1], [2], [7]–[10] etc.). In a recent paper Heilmann [6] has studied the generalized operators which include all the three operators. We now give another generalization of these operators as (1.1) Mn (f, x) = ∞ X \ ∞ pn,k (x) bn,k (t)f (t) dt, 0 k=0 where pn,k (x) = (−1)k xk (k) φ (x), k! n bn,k (t) = (−1)k+1 tk (k+1) φ (t) k! n and (i) (ii) (iii) φn (x) = (1 + cx)−n/c −nx φn (x) = e φn (x) = (1 − x)n for the interval [0, ∞) with c > 0, for the interval [0, ∞) with c = 0, for the interval [0, 1] with c = −1. 1991 Mathematics Subject Classification: 41A30, 41A36. Key words and phrases: modulus of smoothness, global direct theorem, differential and integral operators. Research of the first author supported by Council of Scientific and Industrial Research, India. [153] 154 V. Gupta and G. S. Srivastava The cases (i), (ii) and (iii) mentioned above give modified Baskakov type operators, modified Sz´ asz-type operators and modified Bernstein type polynomials respectively. The case (i) for c = 1 was recently introduced by one of the authors (see e.g. [5]). By Lr1 [0, ∞) we denote the class of functions g given by Lr1 [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 remark that Lrp [0, ∞) is not contained in Lr1 [0, ∞) and L01 [0, ∞) = L1 [0, ∞). Following [3], the modulus of smoothness is given by p φ(x) = x(1 + cx), ωφ2 (f, t)p = sup k∆2hφ f kp , 0<h≤t where 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. [3]) given by ∆φh f (x) = n K 2φ (f, t2 ) = inf{kf − gkp + t2 kφ2 g′′ kp + t4 kg′′ k : g ∈ W p2 (φ, [0, ∞))}, where W p2 (φ, [0, ∞)) = {g ∈ Lp [0, ∞) : g′ ∈ ACloc [0, ∞), φ2 g′′ ∈ Lp [0, ∞)}. In the present paper, we give a global direct theorem for the operators (1.1) in terms of the Ditzian–Totik modulus of second order. Throughout the paper, we denote by C positive constants not necessarily the same at each occurrence. 2. In this section, we mention certain properties and results for the operators (1.1), which are necessary for the proof of the main result. For the cases (i) and (ii), we have ∞ X (2.1) pn,k (x) = 1, bn,k (t) = n, k=0 k=0 ∞ \ 0 ∞ X 1 pn,k (x) dx = n−c \ ∞ and bn,k (t) dt = 1, 0 and for the casep(iii) summation is from 0 to n and integration from 0 to 1. For φ(x) = x(1 + cx), we have (1) (2.2) φ2 (x)pn,k (x) = [k − nx]pn,k (x), (1) φ2 (t)bn,k (t) = [k − (n + c)t]bn,k (t). Approximation by Durrmeyer-type operators 155 Lemma 1. For m, r ∈ N0 (the set of non-negative integers), if we define Vr,n,m (x) = ∞ X \ ∞ pn+cr,k (x) bn−cr,k+r (t)(t − x)m dt, 0 k=0 then Vr,n,0 (x) = 1, Vr,n,1 (x) = (1 + r) + cx(1 + 2r) n − c(r + 1) and Vr,n,2 (x) = 2cx2 (n + 2cr 2 + 4cr + c) + 2x(n + 2cr 2 + 5cr + 2c) + r 2 + 3r + 2 . [n − c(r + 1)][n − c(r + 2)] Further , we have the recurrence relation (1) [n − c(m + r + 1)]Vr,n,m+1 (x) = φ2 (x)[Vr,n,m (x) + 2mVr,n,m−1 (x)] + [(1 + 2cx)(m + r + 1) − cx]Vr,n,m (x). By using (2.1) and (2.2) the proof of the above lemma easily follows along the lines of [6] and [1]. It may be remarked that for all x ∈ [0, ∞) (cases (i) and (ii)) and for all x ∈ [0, 1] (case (iii)), we have Vr,n,m (x) = O(n−[(m+1)/2] ). Lemma 2. If f ∈ Lrp [0, ∞) ∪ Lr1 [0, ∞), 1 < p ≤ ∞ and x ∈ [0, ∞), we have ∞ ∞ \ X pn+cr,k (x) bn−cr,k+r (t)f (r) (t) dt, (2.3) Mn(r) (f, x) = α(n, r, c) 0 k=0 where α(n, r, c) = r−1 Y l=0 n + cl . n − c(l + 1) We see that the operators defined by (2.3) are not positive. To make the operators positive, we introduce the operators Mn,r f ≡ D r Mn I r f, f ∈ Lp [0, ∞) ∪ L1 [0, ∞), where D and I are the differential and integral operators respectively. Therefore, we define the operators by Mn,r (f, x) ≡ α(n, r, c) ∞ X k=0 \ ∞ pn+cr,k (x) bn−cr,k+r (t)f (t) dt, 0 f ∈ Lp [0, ∞) ∪ L1 [0, ∞), n > (c + m)r. 156 V. Gupta and G. S. Srivastava (r) The operators Mn,r are positive and the quantity kMn f − f (r) kp , f ∈ Lrp [0, ∞), is equivalent to kMn,r f − f kp , f ∈ Lp [0, ∞). Using (2.1), we can easily prove that for n > c(r +1), kMn,r f k1 ≤ Ckf k1 for f ∈ L1 [0, ∞) and kMn,r f k∞ ≤ Ckf k∞ for f ∈ L∞ [0, ∞). Making use of the Riesz–Thorin theorem, we get (2.4) kMn,r f kp ≤ Ckf kp , f ∈ Lp [0, ∞), 1 ≤ p ≤ ∞, n > c(r + 1). Corollary 3. For every m ∈ N0 , n > c(r + 2m + 1) and x ∈ [0, ∞), we have (2.5) |Mn,r ((t − x)2m , x)| ≤ Cn−m (φ2 (x) + n−1 )m , |Mn,r ((t − x)2m+1 , x)| ≤ C(1 + 2x)n−m−1 (φ2 (x) + n−1 )m , where the constant C is independent of n. For fixed x ∈ [0, ∞), we obtain (2.6) |Mn,r ((t − x)m , x)| = O(n−[(m+1)/2] ), n → ∞. P r o o f. Since Mn,r ((t − x)m , x) = α(n, r, c)Vr,n,m (x) the estimate (2.5) follow from (2.2) along the lines of [6]; (2.6) immediately follows from (2.5). Lemma 4. Let t ∈ [0, ∞) and n > c(r + m). Then Mn,r ((1 + t)−m , x) ≤ C(1 + cx)−m , x ∈ [0, ∞), where the constant C is independent of n. P r o o f. It is easily verified that (1 + ct)−m bn−cr,k+r (t) = m−1 Y l=0 n − cr + lc bn−cr+mc,k+r (t) n + lc + kc + 1 and pn+cr,k (x) = (1 + cx)−m m Y n + cr − lc + kc l=0 n + cr − lc pn+cr−mc,k (x). Making use of these two identities and (2.1), we get Mn,r ((1 + t)−m , x) ∞ ∞ \ X pn+cr,k (x) bn−cr,k+r (t)(1 + t)−m dt = α(n, r, c) = α(n, r, c) k=0 0 ∞ X m−1 Y k=0 pn+cr,k (x) l=0 n − cr + lc n + lc + kc + 1 \ ∞ bn−cr+mc,k+r (t) dt 0 Approximation by Durrmeyer-type operators = α(n, r, c) ∞ X 157 (1 + cx)−m pn+cr−mc,k (x) k=0 × m m−1 Y n + cr − lc + kc Y l=1 n + cr − lc ≤ C(1 + cx)−m ∞ X l=0 n − cr + lc n + lc + kc + 1 pn+cr−mc,k (x) = C(1 + cx)−m . k=0 For the two monomials e0 , e1 and x ∈ [0, ∞), n → ∞, we obtain by direct computation (2.7) Mn,r (e0 , x) = 1 + O(n−1 ), (2.8) Mn,r (e1 , x) = x(1 + O(n−1 )). Lemma 5. For ∞ n∞ \u\ u\∞\o X Hn (u) = pn+cr,k (x)bn−cr,k+r (t)(u − t) dt dx, − 0 0 we have Hn (u) ≤ Cn 0 0 k=0 −1 2 φ (u), where C is independent of n and u. The proof easily follows by using (2.1) along the lines of [1, Lemma 5.2]. 3. In this section, we prove the following global direct theorem. Theorem 1. Suppose f ∈ Lp [0, ∞), 1 ≤ p < ∞, n > c(r + 5). Then kMn,r f − f kp ≤ C{ωφ2 (f, n−1/2 ) + n−1 kf kp }, where the constant C is independent of n. P r o o f. By Taylor’s expansion of g, we have (3.1) t \ g(t) = g(x) + (t − x)g′ (x) + (t − u)g′′ (u) du. x Next, since Mn,r (f, x) are uniformly bounded operators, for every g ∈ W p2 (φ, [0, ∞)), we have (3.2) kMn,r f − f kp ≤ Ckf − gkp + kMn,r g − gkp . Using (2.5), (2.8) and (3.1) and following [3], we have (3.3) kMn,r g − gkp ≤ C{kgkp + kg′ kLp [0,1] } + k(1 + 2crx)g′ kLp [0,∞) + kMn,r (R(g, t, x), ·)kp where R(g, t, x) = Tt x ≤ Cn−1 [kgkp + kφ2 g′′ kp ] + kMn,r (R(g, t, x), x)kp , g′′ (u)(t − u) du. 158 V. Gupta and G. S. Srivastava Now, we prove that (3.4) kMn,r (R(g, t, x), x)kp ≤ Cn−1 k(φ2 + n−1 )g′′ kp . We prove this for p = 1 and p = ∞. The case 1 < p < ∞ follows again by the Riesz–Thorin theorem. Using (2.5) for the case m = 1 and Lemma 4, the case p = ∞ easily follows (see e.g. [6]). For p = 1, we derive (3.4) by applying Fubini’s theorem twice, the definition of Hn (u) and Lemma 5: \ ∞ |Mn,r (R(g, t, x), x)| dx 0 ≤ α(n, r, c) \ ∞X ∞ pn+cr,k (x) 0 k=0 \ ∞ = α(n, r, c) × ∞ X 0 t \ ′′ bn−cr,k+r (t) g (u)(t − u) du dt dx \\ \\ n∞u |g′′ (u)| 0 \ ∞ 0 0 − u∞o x (u − t) 0 0 pn+cr,k (x)bn−cr,k+r (t) dt dx du k=0 \ ∞ = α(n, r, c) |g′′ (u)|Hn (u) du 0 ≤ Cn −1 2 ′′ kφ g k1 ≤ Cn−1 k(φ2 + n−1 )g′′ k1 , where C is independent of n. Hence (3.4) holds by the Riesz–Thorin theorem for 1 ≤ p < ∞. Combining the estimates (3.2), (3.3) and (3.4), we get kMn,r f − f kp = Ckf − gkp + Cn−1 {kf − gkp + kf kp + kφ2 g′′ kp + k(φ2 + n−1 )g′′ kp } ≤ C{kf − gkp + n−1 kφ2 g′′ kp + n−2 kg′′ kp + n−1 kf kp }. Next taking the infimum over all g ∈ W 2p (φ, [0, ∞)) on the right hand side, we get kMn,r f − f kp ≤ C{K 2φ (f, n−1 ) + n−1 kf kp }. This completes the proof of Theorem 1. R e m a r k. The conclusion of the above theorem is true for the space Lp [0, ∞), 1 ≤ p < ∞ (i.e. limn→∞ kMn,r f − f kp = 0 for every f ∈ Lp [0, ∞)) since the basic fact about the Ditzian–Totik modulus of smooth- Approximation by Durrmeyer-type operators 159 ness ωφ2 (f, n−1 ) is that lim ωφ2 (f, n−1 ) = 0 n→∞ for all f ∈ Lp [0, ∞) if 1 ≤ p < ∞, or for all bounded functions f ∈ C[0, ∞) which satisfy lim f (x) = L∞ < ∞ x→∞ if p = ∞ (cf. [3, p. 36]). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] M. M. D e r r i e n n i c, Sur l’approximation de fonctions int´egrables sur [0, 1] par des polynˆ omes de Bernstein modifi´es, J. Approx. Theory 31 (1981), 325–343. Z. D i t z i a n and K. I v a n o v, Bernstein type operators and their derivatives, ibid. 56 (1989), 72–90. Z. D i t z i a n and V. 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G u p t a, On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. S´er. B 42 (1991), 217–231. Department of Mathematics University of Roorkee Roorkee 247 667, India E-mail: maths%[email protected] Current address of Vijay Gupta: Department of Mathematics Institute of Engineering and Technology Rohilkhand University Bareilly 243006, India Re¸cu par la R´edaction le 27.10.1994 R´evis´e le 27.7.1995