Approximation by Bezier variant of Jakimovski-Leviatan-Paltanea operators involving Sheffer polynomials.

. In the present paper, the BØzier variant of Jakimovski-Leviatan-P…alt…anea operators involving She⁄er polynomials is introduced and the degree of approximation by these operators is investigated with the aid of Ditzian-Totik modulus of smoothness, Lipschitz type space and for functions with derivatives of bounded variations.


Introduction
Approximation theory is a crucial branch of Mathematical analysis. The fundamental property of approximation theory is to approximate a function f by another functions which have better properties than f . In 1950, Szasz [14] introduced a generalization of Bernstein polynomials on the in…nite interval [0; 1) and established the convergence properties of these operators. Subsequently, Jakimovski-Leviatan [8] generalised the Szász operators as where g(u) = P 1 k=0 a k u k ; a 0 6 = 0 is an analytic function, on the disk juj < r (r > 1); under the assumption p k (x) 0; for x 2 [0; 1): In 2008, P¼ alt¼ anea [11] de…ned a generalisation of the Phillips operators [12] based on a parameter > 0, as where s n;k (x) = e nx (nx) k k! and n;k (t) = n k (k ) e n t (nt) k 1 , which includes Szász operators for ! 1 and Phillips operators for = 1. For f 2 C[0; 1), Verma and Gupta [15] de…ned the Jakimovski-Leviatan-P¼ alt¼ anea operator as follows: where L n;k (x) = e nx g(1) p k (nx) and Q n;k (t) = n (k ) e n t (n t) k 1 and established an asymptotic formula and rate of convergence for these operators. Goyal and Agrawal [4] de…ned the Bézier variant of these operators (0.4) and established the degree of approximation using Ditzian-Totik modulus of smoothness, Lipschitz type space and for functions having a derivative of bounded variation.
Let C(z) = P 1 0 c k z k ; (c 0 6 = 0) and D(z) = P 1 k=1 d k z k ; (d 1 6 = 0) be analytic functions on the disc jzj < r; r > 1 where c k and d k are real. The She¤er type polynomials fp k (x)g are given by the generating functions of the form Under the following assumptions: (i) for t 2 [0; 1), p k (t) 0; k = 0; 1; 2; (ii) C(1) 6 = 0 and D 0 (1) = 1; Ismail [6] de…ned another generalisation of the Szász operators and the Jakimovski-Leviatan operators [8] using the She¤er polynomials as and estabilished some approximation properties of these operators. For the special case D(t) = t and C(t) = 1, we …nd p k (x) = x k k! , therefore (0.6) reduces to Szasz operators and for the case D(t) = t, the operators T n (f ; x) yield the operators P n (f ; x) de…ned in (0.1). Inspired by the work of Verma and Gupta [15], Mursaleen et al. [9] de…ned the Jakimovski-Leviatan-P¼ alt¼ anea operators by means of She¤er polynomials, and integral modi…cation of the operators given by (0.6), as where L n;k (x) = e nxD(1) C(1) p k (nx) and Q n;k (t) = n (k ) e n t (n t) k 1 and established some convergence properties of these operators with the help of the Korovkin-type theorem, rate of convergence by using Ditzian-Totik modulus of smoothness and approximation properties for the functions having derivatives of bounded variation.
Since the Bézier curves have important applications in computer aided graphics and applied mathematics, Zeng and Piriou [16] initiated the study of a Bézier variant of Bernstein operators. Zeng [17] introduced the Szasz-Bézier operators and discussed the rate of convergence of these operators for the functions of bounded variations. Subsequently several researchers de…ned the Bézier variants of some other sequences of positive linear operators and studied their approximation properties (see, e.g., [1,2,4,5,7,13]).  (1) C(1) p k (nx) and J n;k (x) = P 1 j=k L n;j (x) with the following properties: (1) J n;k (x) J n;k+1 (x) = L n;k (x) , k = 0; 1; 2; ; (i) if = 1, the operators M n; (f ; x) include the operators given by (0.7), (ii) if = 1 and D(t) = t, the operators M n; (f ; x) reproduce the operators de…ned in [15], (iii) if C(t) = 1, D(t) = t, = 1 and = 1; the operators M n; (f ; x) reduce to the well known Phillips operators [12]. The organization of the paper as follows: In Section 1, the Bézier variant of Jakimovski-Leviatan-P¼ alt¼ anea operators involving She¤er polynomials has been introduced. In Section 2, some auxiliary results such as moments, central moments and lemmas have been presented. In Section 3, the rate of convergence by using Ditzian-Totik modulus of smoothness and Lipschitz type space have been discussed. In Section 4, the approximation result for the functions having derivatives of bounded variation has been discussed.

Auxiliary Results
Lemma 1.1. The r th order moments M n; (t r ; x), for r = 0; 1; 2; are given by the following identities: : As a consequence of the above lemma, we obtain Lemma 1.2. The central moments M n; ((t x) r ; x); r = 1; 2, are given by the following equalities: : In what follows, we denote M n; ((t x) 2 ; x) = n; (x): For su¢ ciently large n and > 2, one has be the family of all continuous and bounded functions de…ned on [0; 1): Proof. The proof of this lemma is readily follow with the help of Lemma 1.1(i). Hence, the details are omitted.
Proof. For 0 u; v 1 and 1, the following inequality holds in view of the inequality (1.4), we have From (1.5) and (1.6), we get the desired result.

Main Results
For t 0, x > 0; and 0 < 1; the Lipschitz type space [10] is de…ned as: where K is some positive constant.
In the next theorem, we investigate the rate of convergence of the operators M n; ( ; x) for the function f 2 Lip K ( ).
The Peetre's K-functional is de…ned by where W := fg : g 2 AC loc ; k g 0 k < 1; kg 0 k < 1g and g 2 AC loc means that g is a locally absolutely continuous function in [0; 1): From [3], it is known that ! (f ; ) K (f ; ), i.e. there exists a constant > 0; such that In the next theorem, Ditzian-Totik …rst order modulus of smoothness is used to establish a direct approximation theorem.
where C is a constant and independent on f and n: Proof. Let x 2 [0; 1) be arbitrary but …xed. For g 2 W , we have the following representation Applying the operator M n; (f ; x) on both sides of the above equation, we obtain In view of Lemma 1.5, we have  ; i.e. there exists some k 1 > 0; such that Hence, applying Cauchy-Schwarz inequality in equation (2.4), we have jM n; (g; x) g(x)j kg 0 kM n; jt xj; x : Case-II: If x 2 (1=n; 1], then from (2.5), we obtain M n; ((t x) 2 ; x) x n 1 + 1 + D 00 (1) . Hence, there exists some constant k 2 > 0, such that and for any x; t 2 (0; 1), Now, combining equations (2.4) and (2.7) and using Cauchy-Schwarz inequality, for any x 2 (1=n; 1), we have jM n; (g; x) g(x)j 2k g 0 k 1 (x)M n; jt xj; x Again, combining equations (2.4), (2.6) and (2.8), for x 2 [0; 1) we have Hence, using Lemma 1.4 and above equation, we get Finally, taking the in…mum on the right side of the above equation over all g 2 W ; and using the relation (2.3), we get Now taking C = 0 , the proof of the theorem is completed.

Functions with Derivatives of Bounded Variation
Let DBV 2 [0; 1); be the class of all functions f de…ned on [0; 1) with jf (t)j C(1 + t 2 ); C > 0 and having a derivative f 0 equivalent to a function of bounded variation on every …nite subinterval of [0; 1). Then we observe that for all functions f 2 DBV 2 [0; 1), there holds the following representation where g is a function of bounded variation on every …nite subinterval of (0; 1): In view of the Dirac-delta function, the alternate form of the operator M n; (f ; x) can be written as To establish the rate of convergence of the operators given by (3.1) for f 2 DBV 2 [0; 1); the following lemma is needed: Lemma 3.1. Let x 2 (0; 1) and > 2. Then for su¢ ciently large n; we have Proof. (i) Using (3.1) and Remark 1.3, we have In the same way, assertion (ii) can be easily proved.