MOMENTS OF A q BASKAKOV BETA OPERATORS IN CASE 0 < q < Introduction

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1 Journal of Classical Analysis Volume 2, Number 1 213, 9 22 doi:1.7153/jca-2-2 MOMENTS OF A BASKAKOV BETA OPERATORS IN CASE < < 1 A. R. GAIROLA, P.N.AGRAWAL, G.DOBHAL AND K. K. SINGH Abstract. In this paper we obtain the estimates of the central moments for the recently defined -analogue of Baskakaov-beta operators. We obtain the evaluation for the rate of convergence in term of the first modulus of smoothness and Voronovskaja-type theorem for these operators. 1. Introduction Recently, Phillips 19] proposed the following generalization of celebrated Bernstein polynomials operators based on -integers B n, f,x= n k= f k] p n,k ;x, f C,1] where p n,k ;x= ] n k x k n k 1 r= 1 r x. These operators have been studied by several authors cf. 12], 19] 25]. The Bernstein polynomials were suitably modified by Durrmeyer to approximate Lebesgue integrable functions cf. 7], 15]. Derriennic 5] introduced a -analogue of the Durrmeyer operators wherein she established some approximation properties of the -Durrmeyer operators. Similar to these modification, the -analogue of some well known positive linear operators e.g. Bernstein, Baskakov and Szász operators were introduced and studied by several authors, many of which have been introduced by Gupta see 2], 8], 1], 11]. Motivated by these modifications the authors in 3] introduced the -Baskakov-beta operators B n, f,x as follows: Let N be the set of positive integer and f C B, the class of the continuous and bounded functions on,. For any n N, the operator B n, : C B, C, is defined by B n, f,x= k= b n,k ;x /A k p n,k ;u f ud u, kk 1/2 x k ] n + k 1 kk 1/2 where b n,k ;x= B k + 1,n1 + x n+k+1, p x k n,k;x= and k 1 + x n+k 1 + x n = n 1 j= 1 + j x. Mathematics subject classification 21: 41A25, 41A3. Keywords and phrases: Baskakov-beta operators, -integer, modulus of continuity. c D l,zagreb Paper JCA-2-2 9

2 1 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH The operators B n, are linear positive and reproduce constant functions. Some approximation properties was established by the authors in 3]. In order to make the paper self contained we recall some definitions and properties of -calculus. see 14], 2]. Let be a real number satisfying < < 1. For n N, we define { 1 n = 1, 1 n, = 1. { n 2]...1], n = 1,2,...! = 1, n =. The -binomial coefficients are defined by And by ] n k =!, k n. k]!n k]! a + b n n 1 = j= a + j b we denote the -rising factorial. The -analogue E x of classical exponential function which we shall use in this paper is given by E x = j j 1/2 x j j= j]!. For further properties see 14]. The -Jackson integrals and -improper integrals are givenby13] and17]. and /A a f xd x =1 a f xd x =1 n= n= f a n n n n f A A, A > respectively, whenever the sums converge absolutely. For,1 and any arbitrary real function f : R R, the -derivative D f t is defined as { f x f x ; x 1 x D f x= lim x D f t; x =. The -derivative of the product is given by the formula D f xgx = f xd gx + gxd f x

3 MOMENTS OF A -BASKAKOV-BETA OPERATORS IN CASE < < 1 11 Analogous to the well known gamma and beta function the -gamma and -beta functions are introduced. We define -gamma function Γ t= 1/1 x t 1 E x d x. The -beta function is given by B t,s=ka,t /A x t x t+s d x, where Kx,t= 1 1+x xt x t 1 + x1 t. In particular, for any positive integer n Kx,n= nn 1 2, Kx, =1 and B t,s= Γ tγ s Γ t + s. Γ t and B t,s are the -analogues of the gamma and beta functions. In the limit 1 they reduce to Γt and Bt,s respectively and also satisfy certain well known properties of classical Γt and Bt,s functions. The space C B, is endowed with the norm f = sup{ f x : x,}. The first order modulus of smoothness of f C B, is defined by ω φ f, δ= sup sup <h δ x, f x + h f x. In what follows, we shall denote x1 + x by φx and CB 2, will be used for the space of all twice continuously differentiable functions for which f is bounded. Further, throughout this paper C is a constant different at each occurrence. 2. Moment Estimates In this section we shall use the identities see 3] k φ 2 xd p n,k ;x] = k] k x p n,k ;x and k φ 2 xd b n,k ;x] = k] k n + 1]x b n,k ;x, freuently, where D denotes the -derivative operator.

4 12 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH LEMMA 1. 3] Let us define A n,m x=b n, t m,x. Then, we have A n,1 x= A n, x=1, n 2] n + 1]x and 3 2]+2] 2 n + 1]x +n + 1]n + 2]x 2 A n,2 x= 6. n 2]n 3] Further, there holds the recurrence relation: m+1 n m 2]A n,m+1 x= n + 1]x +m + 1] A n,m x+φ 2 xd A n,m x 1 COROLLARY 1. 3] Since, B n, f,x are linear and preserve constants, it follows that n + 1] B n, t x,x= 2 n 2] 1 1 x + n 2]. And B n, t x 2 1,x = n + 1] 2 5 n 3] x n 2]n 3] + n + 1]n + 2] 2 4 n + 1]n 3]+ 6 n 2]n 3] x 2]. Moreover, there holds the ineuality B n, t x 2,x 8 φ 2 x+ 1 6 n 2] n 3] = 16 6 n 2] δ n 2 x, where δn 2 x=max{φ 2 1 x, n 3] }. LEMMA 2. Let Tn,mx=B n, t x m,x be the mth -central moments of the operators B n,, then there holds the recurrence relation φ 2 xd T n,m x = m+1 n m 2] T n,m+1 x+ x m 1 n + 1]x +x m x 1 m + 1]1 + 2 m x + m 1 x + 2m 1 x 2 m + 2]x m 1 ] T n,mx m]φ 2 x+n + 1] m 1x 2 m 1 x 2 m 1 +m + 1]x m 11 + m x + m 1 x ] +m + 1] m x 2 m 11 + m 1 x T n,m 1 x.

5 Proof. We have MOMENTS OF A -BASKAKOV-BETA OPERATORS IN CASE < < 1 13 φ 2 xd T n,m x = = E 1 = m] k= k= + φ 2 x φ 2 x k= = E 1 + E 2 say. k= = m]φ 2 xt n,m 1 x. /A /A φ 2 x k p n,k ;td b n,k ;xt x m d t k p n,k ;tb n,k ;x /A φ 2 xb n,k ;x And in view of k φ 2 xd b n,k ;x] = E 2 = k= n + 1]x = E 3 + E 4 say. k b n,k ;x k= D t x m d t k p n,k ;tt x m D b n,k ;x d t /A k p n,k ;tt x m 1 d t k] k n + 1]x b n,k ;x we get /A b n,k ;x k] k p n,k ;tt x m d t /A k p n,k ;tt x m d t It is obtained easily that t x m = x m 1t ] x m 1 +t x m. Therefore, E 4 = n + 1]x x m 1T n,m 1 x+t n,mx. Next, Again, E 3 = k= + /A k b n,k ;x k= = E 5 + E 6 say. E 6 = 1 k b n,k ;x k= k] k 1 t k p n,k ;tt x m d t /A b n,k ;x k 1 t k p n,k ;tt x m d t /A k p n,k ;tt m xt x m d t

6 14 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH + 1 k= b n,k ;x /A m x k p n,k ;tt x m d t = E 7 + E 8 say. ] Clearly, E 8 = m 1 x x m 1T n,m 1 x+t n,mx. Next, E 7 = 1 k= b n,k ;x /A = 1 x m+1 1Tn,m x+t n,m+1 ]. x k p n,k ;tt x m+1 d t In order to simplify E 5 we take the transformation t = z which is valid in the case of -integration. E 5 = = k= k= b n,k ;x b n,k ;x /A /A k] k z z x m p n,k ;zd z φ 2 zd p n,k ;zz x m d z. 2 Next, we write the function φ 2 z=z+z 2 as φ 2 z= m x+ m 1 x+1z m x + m 1 x1+ m 1 x+ z m xz m+1 x and substitute in 2. We obtain three terms corresponding 2 to the three terms in 2, namely E 9, E 1 and E 11. m x + m 1 x + 1 E 9 = /A k= k b n,k ;x k.d p n,k ;zz x m+1 d z. Now, we integrate by parts and then make the inverse transformation z = 1 t which gives m x + m 1 x + 1 E 9 = k b n,k ;x k= ] /A } {p n,k ;zz x m+1 = m + 1] 2 m x + m 1 x + 1 /A p n,k ;zd z x m+1 d z k= b n,k ;x k

7 MOMENTS OF A -BASKAKOV-BETA OPERATORS IN CASE < < 1 15 /A p n,k ;zz x m d z m x + m 1 x + 1 = m + 1] /A p n,k ;tt x m d z k= b n,k ;x k ] = m + 1] m x + m 1 x + 1 x m 1T n,m 1 x+t n,m x. And E 1 = m 1 x1 + m 1 x k= = m + 1] m x1 + m 1 x b n,k ;x k= /A b n,k ;x k+1 D pn,k ;z z x m+1 d z /A = m + 1] m x1 + m 1 x x m 1T n,m 1 x+t n,m x k p n,k ;tt x m d t ]. Similarly, E 11 = m + 2] ] x m+1 1T n,m x+t n,m+1 x. Combining the estimates E 1 E 11 the lemma is established. Now, we are in a position to state the following lemma LEMMA 3. Let m N, < < 1. There exists a constant C = Cm > independent of x and n such that for any x, we have B n, t x m 1,x C m+1/2. Proof. The proof follows by induction on m and the Lemma 2. LEMMA 4. Suppose the functions A n,m x are expressed as A n,m x=a m + am 1 x + am 2 x2 +...a m m xm, where a m i are corresponding coefficients. Then the recurrence relations hold 2m+2 n m 2]a m+1 m+1 =n + m + 1]am m; 3

8 16 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH 2m+1 n m 2]a m+1 m =n + m]a m m 1 +2m + 1]am m ; 4 2m r+1 n m 2]a m+1 m r =n + m r]a m m r 1 +2m r]am m r, 1 r < m 5 m+1 n m 2]a m+1 =m + 1]a m. 6 Proof. Using A n,m x= m i= am i xi in 1 and euating the coefficients of various powers of x the we get the recurrence relations 2m+2 n m 2]a m+1 m+1 m]+ = m n + 1] a m m ; 2m+1 n m 2]a m+1 m = m 1]+ m 1 n + 1] a m m 1 + m]+ m m + 1] a m m ; 2m r+1 n m 2]a m+1 m r = m r 1]+ m r 1 n + 1] a m m r 1 + m r]+ m r m + 1] a m m r, 1 r < m m+1 n m 2]a m+1 =m + 1]a m. Now, using = k n k]+k] in these relations 3 6 are established. LEMMA 5. Let m N, < < 1, n> 4. There exists a constant C independent of x and n and ˆ,1 such that 2 B n, t x 4 16,x C 6 n 2] δ n 2 x, ˆ]. Proof. The uantities A n,k x, k =,1,2 are obtained in Lemma 1. Now,using 1 we obtain A n,3 x = n + 1]x 1 +3] 3 2]+2] 2 n + 1]x +n + 1]n + 2]x 2 +φx/2] 2 n + 1]+2]n + 1]n + 2]x 1 ] 1 9 n 2]n 3]n 4] and A n,4 x = n + 1]x 1 +3]n + 1]x 1 +4] 3 2]+2] 2 n + 1] ] { +n + 1]n + 2] + φx/ 2] 2 n + 1]+2]n + 1]n + 2]x 1 +2] 2 n + 1]+2] 2 3]n + 1]+2]x 1 2]n + 1]n + 2] 2 +2] 2 1 n + 1]+2] 2 n + 1] 2 +3]n + 1]n + 2] +3]x 2 2 2]n + 1]n + 2] 3 +n + 1] 2 n + 2] 1 } ] +2] 2 n + 1]

9 MOMENTS OF A -BASKAKOV-BETA OPERATORS IN CASE < < n 2]n 3]n 4]n 5] We write B n, t x 4,x= 4 4 k= k 1 k x 4 k A n,k x and substitute A n,k, k =,1, The coefficient of x 4 in the uantity C 16 6 n 2] δ n 2x Bn, t x 4,x is obtained as 256C 8 n k] n + 1]+1n + 1]n + 2]n 2]n 5] 5 k=2 2 n 2] 1 5 k=2 2 n 2] 5 k=2 n k] n k] n + k]n 2]n 4]n 5]+3] k=1 Now the coefficient of the terms of O 5 in numerator is k=1 1 n + k] which is 1/4 positive iff Further for sufficiently large n, remaining terms in the 6 numerator are positive. Therefore the coefficient of x 4 becomes positive for, ˆ, 1/4 where ˆ = Similarly the coefficients of x i, i = 1,2,3 are positive for large 6 n. In case 1, we have n as is continuous function of, the lemma is established easily in this case again. This completes the proof. LEMMA 6. For the coefficients a m i defined in lemma 4 there holds 1 r a m m r = O. Proof. For r = 1, we need to prove a m m 1 = O 1. In case m = 1, we have a 1 = 1 n 2] and using recurrence relation 2m+1 n m 2]=n+m]a m m 1 +2m+1]am m with a m m = n+1]...n+m], the lemma is proved for r = 1. Suppose the lemma is mm+1 n 2]...n m 1] true for a certain r, then from the recurrence relation 5, we get n + m r]a m m r+1 = 2m r+1 n m 2]a m+1 m r 2m r + 1]a m m r r+1 = 2m r n m 2]O 2m r + 1]O r+1 a m m r+1 = O 1. r This proves the lemma.

10 18 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH 3. Applications of the moments The following theorem due to Pop 21] will be used in our asymptotic results. THEOREM 1. Let I R be an interval, a I, n N and the function f : I R, f is n times derivable in a. According to Taylor s expansion theorem for the function f around a, we have f x= r k= x a k f k a+x a r μx a, k! where μ is a bounded function and lim t x μx a=. If f r is a continuous function on I, then for any δ > μx a r! δ 2 x a 2 ]ω 1 f r,δ. THEOREM 2. Let r and s 1. Then, for < < 1, D r Bn, t r+s,x r + s]! n + 1]...n + r + s] = s]! r+sr+s+1 n 2]...n r s 1] xs + r + s 1]! n + 1]...n + r + s] s 1]! r+s + j=2 M j x r+s j O where M j s are independent of n. r+s 1 j= r+s j 1 2 j + 1] r+s2 +3r+s+1 n 2]...n r s 2] 1 j, Proof. Using Lemma 1 we get following coefficients and in general a = 1, a1 = 1 n 2], x s 1 a 2 = 2] 6 n 2]n 3], a2 1 = 2]2 n + 1] 6 n 2]n 3], n + 1]n + 2] a2 2 = 6 n 2]n 3] a m = 1 mm+1/2 From previous Lemma 6, it is known that m j] j= n 2 j] m 3. ar+s r+s n + 1]...n + r + s] = r+sr+s+1 n 2]...n r s 1]

11 MOMENTS OF A -BASKAKOV-BETA OPERATORS IN CASE < < 1 19 Simplifying recurrence relation 4, we get 2m+1 n m 2]a m+1 m =n + m]a m m 1 +2m + 1]am m. Now, putting m = 1,2,... we obtain from above recurrence relation a m+1 m = n + 1]n + 2]...n + m] m +3] m m + 1] m2 +3m+1 n 2]...n m 1] = m i=1 n + i]m j= m j 2 j + 1] m2 +3m+1 m+2 k=2 n k]. Similarly simplification of 5 gives 2m n m 2]a m+1 m 1 =n + m 1]am m 2 +2m]am m 1. Now, in view of Lemma 6, it follows that a m+r m+r j = O 1 j, j = 2,3,... This completes the Theorem. Following is a Voronovskaya-type theorem for monomials. THEOREM 3. Let r and s 1 and n be a seuence in,1 such that n 1 as n, then lim D r n Bn, t r+s,x ] r + s! x s r + sr + s! = x s 1 s! s 1! Proof. It results from Theorem 2 and the limit r + s 1]! n + 1]...n + r + s] 2] 2 r+s +5] r+s r + 2s + 1] n s 1]! r+s2 +3r+s+1 n 2]...n r s 2] r + s 1!r + sr + s + 2 3] =. s 1! THEOREM 4. For f CB 2,,, ˆ] we have ] 2 B B n, t x n, f,x k,x f k x k! ω f 4, 3 n 2] δ 16 nx 6 n 2] δ n 2 x. k= Proof. From finite Taylor s expansion for the function f around x, given in Theorem 1,wehave 2 B n, t x B n, f,x k,x f k x k= k! = B n, t x 2 μt x,x

12 2 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH 1 2! ω f,δ ] Bn, t x 2,x + δ 2 Bn, t x 4,x 1 2! ω f,δ 2 ] 16 6 n 2] δ n x+δ 6 n 2] δ n 2 x. Choosing δ 2 = 16 6 n 2] δ n 2 x the theorem follows. THEOREM 5. If f CB 2,, and n be a seuence in,1 such that lim n n = 1, then we have ] lim B n, f,x f x = d x1 + x f n dx x. Proof. Throughout the proof it will be assumed that = n. The proof results from Theorem 4 and the limits lim n B n,n t x j,x, j =,1,2. We have lim n B n,n 1,x=and n + 1] lim B n, n n t x,x = lim n 2 n 2] 1 x + 1 n 2] 2]1 + n x = lim n n x + 2] n 2] n 2] + = 1 + 2x, where we have used the relation =2]+ 2 n 2]. Writing B n,n t x 2,x=c + c 1 x + c 2 x 2, where c is are the coefficients given in Cor. 1. Then, lim B n,t x 2,x = lim c + c 1 x + c 2 x 2 n n Clearly, lim n c =, n + 1] 2 5 n 3] lim c 1x = lim n n 6 x n 2]n 3] = lim 2]+ 2 n 2] n = 2x. 2] 2 4] 6 n 2]n 3] + 2] n 2] x

13 And simplification yields MOMENTS OF A -BASKAKOV-BETA OPERATORS IN CASE < < 1 21 lim n c 2x 2 = lim n = lim n = 2x 2. n + 1]n + 2] 2 4 n + 1]n 3]+ 6 n 2]n 3] x 2 6 n 2]n 3] 4] 23] 2 + 4] n 2] 2 n 3] + 4] 2 6 n 2]n 3] x 2 Acknowledgement. The authors gratefully acknowledge Professor, Ovidiu T. Pop for providing the proof of Theorem 1. The work of K. K. Singh was supported by the Council of Scientific and Industrial Research, New Delhi, India. REFERENCES 1] A. ARAL AND V. GUPTA, On the Durrmeyer type modification of the -Baskakov type operators, Nonlinear Anal , ] A. ARAL, V. GUPTA, The -derivative and applications to -Szász Mirakyan operators, CALCOLO 43 26, ] A. R. GAIROLA AND G. DOBHAL, Uniform approximation by -analogue of Baskakov-beta type operators, in communication. 4] A. R. GAIROLA AND P. N. AGRAWAL, Direct and inverse theorems for the Bézier variant of certain summation-integral type operators, Turk. J. Math , ] M. M. DERRIENNIC, Modified Bernstein polynomials and Jacobi polynomials in -calculus, Rend. Circ. Mat. Palermo Serie II Suppl.76 25, ] Z. DITZIAN AND V. TOTIK, Moduli of Smoothness, Springer, New York, ] J. L. DURRMEYER, Une formula d inversion, de la transformee de Laplace: Application a la theorie des Moments, These de 3e Cycle, Faculte des Sciences de l universite de Paris, ] V. GUPTA AND W. HEPING, The rate of convergence of -Durrmeyer operators for < < 1, Math. Meth.Appl.Sci.31 28, ] V. GUPTA AND A. AHMAD, Simultaneous approximation by modified beta operators, IstanbulUniv. Fen. Fak. Mat. Der , ] V. GUPTA, Some approximation properties of -Durrmeyer operators, Appl. Math. Comput. 197, 1 28, ] V. GUPTA, On certain -Durrmeyer type operators, Appl. Math. Comput , ] A. II INSKII, S. OSTROVSKA, Convergence of generalized Bernstein polynomials, J. Approx. Theory 116, 1 22, ] F. H. JACKSON, On a -definite integrals, Quarterly J. Pure Appl. Math , ] V. G. KAC AND P. CHEUNG, Quantum Calculus, Universitext, Springer-Verlag, New York, ] L. V. KANTOROVICH, Sur certain developments suivant les polynomes de la forms de S. Bernstein, Dokl. Akad. Nauk. SSSR , , ] H. KARSLI AND V. GUPTA, Some approximation properties of -Chlodowsky operators, Appl. Math. Comput , ] T. H. KOORNWINDER, -Special Functions, a tutorial, in: M. Gerstenhaber, J. Stasheff eds.: Deformation and Qauntum groups with Applications of Mathematical Physics, Contemp. Math , Amer. Math. Soc ] S. OSTROVSKA, -Bernstein polynomials and their iterates, J. Approx. Theory 123, 2 23,

14 22 A. R. GAIROLA,P.N.AGRAWAL,G.DOBHAL AND K. K. SINGH 19] G. M. PHILLIPS, Bernstein polynomials based on the -integers, Annals of Numerical Mathematics , ] G. M. PHILLIPS, Interpolation and Approximation by Polynomials, CMS Books in Mathematics, Vol. 14. Springer: Berlin, ] O. T. POP, About some linear and positive operators defined by infinite sum, Demonstratio Mathematica 39, 2 26, ] H. WANG, Voronovskaya-type formulas and saturation of convergence for -Bernstein polynomials for < < 1, J. Approx. Theory , ] H. WANG, Properties of convergence for -Bernstein polynomials, J. Math. Anal. Appl. 34, 2 28, ] H. WANG, F.MENG, The rate of convergence of -Bernstein polynomials for < < 1, J. Approx. Theory 136, 2 25, ] H. WANG, X. WU, Saturation of convergence of -Bernstein polynomials in the case > 1, J. Math. Anal. Appl 337, 1 28, ] A. ZYGMUND, Trigonometric Series, Dover, New York, Received May 13, 211 A. R. Gairola Department of Computer Applications Graphic Era University Dehradun, India ashagairola@gmail.com G. Dobhal Department of Computer Applications Graphic Era University Dehradun, India girish.dobhal@gmail.com P. N. Agrawal Department of Mathematics Indian Institute of Technology Roorkee, India pna iitr@yahoo.co.in K. K. Singh Department of Mathematics Indian Institute of Technology Roorkee, India kks iitr@gmail.com Journal of Classical Analysis jca@ele-math.com