Journal of Inequalities in Pure and Applied Mathematics SIMULTANEOUS APPROXIMATION FOR THE PHILLIPS-BÉZIER OPERATORS volume 7, issue 4, article 141, 2006. VIJAY GUPTA, P.N. AGRAWAL AND HARUN KARSLI School of Applied Sciences Netaji Subhas Institute of Technology Sector 3 Dwarka New Delhi 110075, India. EMail: vijay@nsit.ac.in Department of Mathematics Indian Institute of Technology Roorkee 247667, India. EMail: pnappfma@iitr.ernet.in Ankara University Faculty of Sciences Department of Mathematics 06100 Tandogan-Ankara-Turkey. EMail: karsli@science.ankara.edu.tr Received 28 July, 2006; accepted 17 November, 2006. Communicated by: F. Qi Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 204-06 Quit Abstract We study the simultaneous approximation properties of the Bézier variant of the well known Phillips operators and estimate the rate of convergence of the Phillips-Bézier operators in simultaneous approximation, for functions of bounded variation. 2000 Mathematics Subject Classification: 41A25, 41A30. Key words: Rate of convergence, Bounded variation, Total variation, Simultaneous approximation. Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 3 5 8 Title Page Contents JJ J II I Go Back Close Quit Page 2 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au 1. Introduction For α ≥ 1, the Phillips-Bézier operator is defined by Z ∞ ∞ X (α) (α) Qn,k (x) pn,k−1 (t)f (t)dt + Qn,0 (x) f (0), (1.1) Pn,α (f, x) = n k=1 0 where n ∈ N, x ∈ [0, ∞), (α) Qn,k α α (x) = [Jn,k (x)] − [Jn,k+1 (x)] , Jn,k (x) = ∞ X pn,j (x) and j=k (nx)k . k! For α = 1, the operator (1.1) reduces to the Phillips operator [1]. Some approximation properties of the Phillips operators were recently studied by Finta and Gupta [2]. The rates of convergence in ordinary and simultaneous approximations on functions of bounded variation for the Phillips operators were estimated in [3], [4] and [5]. In the present paper we extend the earlier study and here we investigate and estimate the rate of convergence for the Bézier variant of the Phillips operators in simultaneous approximations by means of the decomposition technique of functions of bounded variation. We denote the class Br,β by n (r) Br,β = f : f (r−1) ∈ C[0, ∞), f± (x) exist everywhere pn,k (x) = e−nx and are bounded on every finite subinterval of o (r) [0, ∞) and f± (x) = O(eβt ) (t → ∞), for some β > 0 , Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close Quit Page 3 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au (0) r = 1, 2, . . . . By f± (x) we mean f (x±). Our main theorem is stated as: Theorem 1.1. Let f ∈ Br,β , r = 1, 2, ... and β > 0. Then for every x ∈ (0, ∞) and n ≥ max {r2 + r, 4β} , we have n o r + α − 1 (r) (r) (r) (r) (r) Pn,α (f, x) − 1 f+ (x) + α f− (x) ≤ √ f+ (x) − f− (x) α+1 2enx √ √ n x+x/ k 2x + 1 1 2α(1 + 2x) X _ (gr,x ) + α √ 2r/2 e2βx , + 1+ 2 n x x n √ k=1 x−x/ k where gr,x is the auxiliary function defined by (r) f (r) (t) − f+ (x), x < t < ∞ 0, t=x , gr,x (t) = f (r) (t) − f (r) (x), 0 ≤ t < x − Wb is the total variation of gr,x (t) on [a, b]. In particular g0,x (t) ≡ gx (t), defined in [4]. a (gr,x (t)) Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close Quit Page 4 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au 2. Auxiliary Results In this section we give certain lemmas, which are necessary for proving the main theorem. Lemma 2.1. For all x ∈ (0, ∞), α ≥ 1 and k ∈ N ∪ {0} , we have pn,k (x) ≤ √ 1 2enx and (α) Qn,k (x) ≤ √ Vijay Gupta, P.N. Agrawal and Harun Karsli α , 2enx √ where the constant 1/ 2e and the estimation order n−1/2 (for n → ∞) are the best possible. Lemma 2.2 ([3]). If f ∈ L1 [0, ∞), f L1 [0, ∞), then Pn(r) (f, x) = n ∞ X (r−1) Z pn,k (x) ∈ A.C.loc , r ∈ N and f (r) ∈ ∞ pn,k+r−1 (t) f (r) (t)dt. 0 k=0 Title Page Contents JJ J II I Go Back Lemma 2.3 ([3]). For m ∈ N ∪ {0} , r ∈ N, if we define the m-th order moment by Z ∞ ∞ X pn,k (x) pn,k+r−1 (t) (t − x)m dt µr,n,m (x) = n Close Quit Page 5 of 15 0 k=0 then µr,n,0 (x) = 1, µr,n,1 (x) = Vijay Gupta, P.N. Agrawal and Harun Karsli r n and µr,n,2 (x) = 2nx+r(r+1) . n2 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au Also there holds the following recurrence relation nµr,n,m+1 (x) = x[µ(1) r,n,m (x) + 2mµr,n,m−1 (x)] + (m + r)µr,n,m (x). Consequently by the recurrence relation, for all x ∈ [0, ∞), we have µr,n,m (x) = O n−[(m+1)/2] . Remark 1. In particular, by Lemma 2.3, for given any number n ≥ r2 + r and 0 < x < ∞, we have (2.1) µr,n,2 (x) ≤ 2x + 1 . n Vijay Gupta, P.N. Agrawal and Harun Karsli Remark 2. We can observe from Lemma 2.2 and Lemma 2.3 that for r = 0, the summation over k starts from 1. For r = 0, Lemma 2.3 may be defined as [5, Lemma 2], with c = 0. Lemma 2.4. Suppose x ∈ (0, ∞), r ∈ N ∪ {0} , α ≥ 1 and Kr,n,α (x, t) = n ∞ X (α) Qn,k (x)pn,k+r−1 (t). k=0 Then for n ≥ r2 + r, there hold Z y α(2x + 1) (2.2) Kr,n,α (x, t)dt ≤ , n(x − y)2 0 Z ∞ α(2x + 1) (2.3) Kr,n,α (x, t)dt ≤ , n(z − x)2 z Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close 0 ≤ y < x, Quit Page 6 of 15 x < z < ∞. J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au Proof. We first prove (2.2) as follows: Z y Z Kr,n,α (x, t)dt ≤ 0 y (x − t)2 K (x, t)dt 2 r,n,α 0 (x − y) α ≤ Pn ((t − x)2 , x) (x − y)2 α(2x + 1) αµr,n,2 (x) ≤ ≤ , (x − y)2 n(x − y)2 by using (2.1). The proof of (2.3) follows along similar lines. Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close Quit Page 7 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au 3. Proof Proof of Theorem 1.1. Clearly (3.1) n o (r) (r) (r) Pn,α (f, x) − 1 f+ (x) + α f− (x) α+1 Z ∞ ∞ X (α) ≤n Qn,k (x) pn+r−1,k (x) |gr,x (t)| dt 0 k=0 1 + α+1 (r) (r) (r) (sgn α (t − x), x) . f+ (x) − f− (x) Pn,α Vijay Gupta, P.N. Agrawal and Harun Karsli (r) We first estimate Pn,α (sgnα (t − x), x) as follows: (r) Pn,α (sgn α (t − x), x) Z ∞ Z ∞ X (α) =n Qn,k (x) αpn,k+r−1 (t)dt − (1 + α) 0 k=0 = α − (1 + α)n = α − (1 + α)n ∞ X k=0 ∞ X 0 Z (α) Qn,k (x) = (1 + α)n k=0 (α) pn,k+r−1 (t)dt 0 (α) Qn,k (x) k+r−1 X Title Page x pn,k+r−1 (t)dt Contents JJ J II I Go Back k+r−1 X ! pn,j (x) j=0 k=0 ∞ X ∞ Qn,k (x) 1 − Vijay Gupta, P.N. Agrawal and Harun Karsli pn,j (x) − 1 Close Quit Page 8 of 15 j=0 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au ! r−1 = (1 + α)n pn,j (x) + √ −1 2enx j=0 k=0 # "∞ ∞ X X r − 1 (α) = (1 + α) Qn,k (x) + √ −1 pn,j (x) 2enx j=0 k=j # "∞ ∞ X X r−1 (α) α − Qn,j (x). = (1 + α) pn,j (x) [Jn,j (x)] + √ 2enx j=0 j=0 ∞ X (α) Qn,k (x) k X Vijay Gupta, P.N. Agrawal and Harun Karsli By the mean value theorem, we find that (α+1) Qn,j (x) α+1 = [Jn,j (x)] α+1 − [Jn,j+1 (x)] α = (α + 1)pn,j (x) [γn,j (x)] , where Jn,j+1 (x) < γn,j (x) < Jn,j (x). Hence by Lemma 2.1, we have (r) Pn,α (sgn α (t − x), x) "∞ # (1 + α)(r − 1) X √ ≤ (1 + α) pn,j (x) ([Jn,j (x)]α − [γn,j (x)]α ) + 2enx j=0 "∞ # X r−1 (α) ≤ (1 + α) pn,j (x)Qn,k (x) + √ 2enx j=0 α r−1 (1 + α)(r + α − 1) √ (3.2) ≤ (1 + α) √ +√ = . 2enx 2enx 2enx Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close Quit Page 9 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au (r) Next we estimate Pn (gr,x , x). By the Lebesgue-Stieltjes integral representation, we have Z ∞ (r) Pn,α (gr,x , x) = gr,x (t)Kr,n,α (x, t)dt 0 Z Z Z Z = + + + gr,x (t)Kr,n,α (x, t)dt I1 I2 I3 I4 = R1 + R2 + R3 + R4, √ √ √ √ say, where I1 = [0, x−x/ n], I2 = [x−x/ n, x+x/ n], I3 = [x+x/ n, 2x] and I4 = [2x, ∞). Let us define Z t ηr,n,α (x, t) = Kr,n,α (x, u)du. (3.3) 0 √ We first estimate R1 . Writing y = x − x/ n and using integration by parts, we have Z y R1 = gr,x (t)dt (ηr,n,α (x, t)) 0 Z y = gr,x (y)ηr,n,α (x, y) − ηr,n,α (x, t)dt (gr,x (t)). 0 |R1 | ≤ y Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close Quit By Remark 1, it follows that x _ Vijay Gupta, P.N. Agrawal and Harun Karsli Z y ηr,n,α (x, t)dt − (gr,x )ηr,n,α (x, y) + 0 x _ t ! (gr,x ) Page 10 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au ≤ x _ y α(2x + 1) λx + (gr,x ) n(x − y)2 n Z 0 y ! x _ 1 dt − (gr,x ) . (x − t)2 t Integrating by parts the last term, we have after simple computation, Wx Z y Wx α(2x + 1) 0 (gr,x ) t (gr,x ) |R1 | ≤ +2 dt . 3 n x2 0 (x − t) √ Now replacing the variable y in the last integral by x − x/ t, we get n x 2α(2x + 1) X _ |R1 | ≤ (gr,x ). nx2 √ k=1 (3.4) Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli x−x/ k √ √ Next we estimate R2 . For t ∈ [x − x/ n, x + x/ n], we have √ x+x/ n |gr,x (t)| = |gr,x (t) − gr,x (x)| ≤ _ √ (gr,x ). x−x/ n Also by the fact that Rb a dt (ηr,n (x, t)) ≤ 1 for (a, b) ⊂ [0, ∞), therefore √ x+x/ n (3.5) |R2 | ≤ _ (gr,x ) ≤ √ x−x/ n 1 n √ n x+x/ X _k √ k=1 x−x/ k Title Page Contents JJ J II I Go Back Close (gr,x ). Quit Page 11 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au √ To estimate R3 , we take z = x + x/ n, thus Z 2x R3 = Kr,n,α (x, t)gr,x (t)dt z Z 2x =− gr,x (t)dt (1 − ηr,n,α (x, t)) z = −gr,x (2x)(1 − ηr,n,α (x, 2x)) + gr,x (z)(1 − ηr,n,α (x, z)) Z 2x + (1 − ηr,n,α (x, t))dt (gr,x (t)). z Thus arguing similarly as in the estimate of R1 , we obtain √ |R3 | ≤ (3.6) n x+x/ 2α(2x + 1) X _ nx2 k=1 (gr,x ). x 2x ≤ nα nα ≤ x k=0 Title Page Contents JJ J II I Go Back ∞ Z pn,k+r−1 (t) eβt dt pn,k (x) k=0 ∞ X Vijay Gupta, P.N. Agrawal and Harun Karsli k Finally we estimate R4 as follows Z ∞ |R4 | = Kr,n,α (x, t)gr,x (t)dt ∞ X Vijay Gupta, P.N. Agrawal and Harun Karsli 2x Z pn,k (x) Close Quit ∞ pn,k+r−1 (t) eβt |t − x| dt Page 12 of 15 0 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au α ≤ x n ∞ X Z pn,k (x) pn,k+r−1 (t) (t − x)2 dt 0 k=0 × ! 21 ∞ n ∞ X Z pn,k (x) ! 12 ∞ pn,k+r−1 (t) e2βt dt . 0 k=0 For the first expression above we use Remark 1. To evaluate the second expression, we proceed as follows: n ∞ X Z pn,k (x) ∞ pn,k+r−1 (t) e2βt dt Vijay Gupta, P.N. Agrawal and Harun Karsli 0 k=0 Vijay Gupta, P.N. Agrawal and Harun Karsli ∞ X nk+r−1 =n pn,k (x) (k + r − 1)! k=0 Z ∞ tk+r−1 e−(n−2β)t dt, Title Page 0 ∞ X nk+r−1 Γ(k + r) n pn,k (x) (k + r − 1)! (n − 2β)k+r k=0 k ∞ X nr n = pn,k (x) , (n − 2β)r k=0 n − 2β Contents JJ J II I Go Back Close Quit k ∞ X n n2 x 1 nr −nx e = e2nxβ/(n−2β) ≤ 2r e4βx r (n − 2β)r n − 2β k! (n − 2β) k=0 r Page 13 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au for n > 4β. Thus α |R4 | ≤ x n ∞ X √ ≤ pn,k (x) ! 12 ∞ pn,k+r−1 (t) (t − x)2 dt 0 k=0 × (3.7) Z n ∞ X Z pn,k (x) k=0 ∞ ! 12 pn,k+r−1 (t) e2βt dt 0 α 2x + 1 r/2 2βx √ 2 e . x n Combining the estimates of (3.1)-(3.7), we get the required result. Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page Contents JJ J II I Go Back Close Quit Page 14 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au References [1] R.S. PHILLIPS, An inversion formula for semi-groups of linear operators, Ann. Math. (Ser. 2), 59 (1954), 352–356. [2] Z. FINTA AND V. GUPTA, Direct and inverse estimates for Phillips type operators, J. Math Anal Appl., 303 (2005), 627–642. [3] N.K. GOVIL, V. GUPTA AND M.A. NOOR, Simultaneous approximation for the Phillips operators, International Journal of Math and Math Sci., Vol. 2006 Art Id. 49094 92006, 1-9. [4] V. GUPTA AND G.S. SRIVASTAVA, On the rate of convergence of Phillips operators for functions of bounded variation, Ann. Soc. Math. Polon. Ser. I Commentat. Math., 36 (1996), 123–130. Vijay Gupta, P.N. Agrawal and Harun Karsli Vijay Gupta, P.N. Agrawal and Harun Karsli Title Page [5] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summationintegral type operators, Math. Comput. Modelling, 37 (2003), 1307–1315. [6] X.M. ZENG AND J.N. ZHAO, Exact bounds for some basis functions of approximation operators, J. Inequal. Appl., 6 (2001), 563–575. Contents JJ J II I Go Back Close Quit Page 15 of 15 J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006 http://jipam.vu.edu.au