V. Gupta, A. Aral and M. Ozhavzali. APPROXIMATION BY q-szász-mirakyan-baskakov OPERATORS

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1 F A S C I C U L I M A T H E M A T I C I Nr V. Gupta, A. Aral and M. Ozhavzali APPROXIMATION BY -SZÁSZ-MIRAKYAN-BASKAKOV OPERATORS Abstract. In the present paper we propose the analogue of well known Szász-Mirakyan-Baskakov operators see e.g. [14], [7]. We apply -derivatives, and -Beta functions to obtain the moments of the -Szász-Mirakyan-Baskakov operators. Here we estimate some direct approimation results for these operators. Key words: -Szász-Mirakyan-Baskakov operators, -binomial coefficients, -derivatives, -integers, -Beta functions, -integral. AMS Mathematics Subject Classification: 41A25, 41A3. 1. Introduction Recently Mahmudov [12] and Aral [2] see also [4] proposed the -analogues of the well known Szász-Mirakyan operators and estimated some approimation results. The operators studied in [12] are different from those studied in [4]. King type generalization of the -Szász operators defined in [2] can be found in [1]. Also some approimation properties of the another Szász-Mirakyan type operators were presented in [15]. The most commonly used integral modifications of the Szász-Mirakyan operators are Szász-Mirakyan-Kantorovich and Szász-Mirakyan-Durrmeyer operators. -analogue of some Durrmeyer type operators were studied in [13] and [9]. Very recently Gupta and Aral [8] proposed analogue of Szász-Mirakyan- Beta operators and established some approimation properties. In the year 1983, Prasad-Agrawal-Kasana [14] proposed the integral modification of Szász-Mirakyan operators by taking the weight functions of Baskakov operators, but there were so many gaps in the results obtained in [14]. In the year 1993 Gupta [7] filled the gaps and improved the results The work was done joıntly, while the first author visited Kırıkkale University during September 21, supported by The Scientific and Tecnological Research Council of Turkey

2 36 V. Gupta, A. Aral and M. Ozhavzali of [14]. To approimate Lebesgue integrable functions on the interval [,, the Szász-Mirakyan-Baskakov operators are defined as 1 G n f, = n 1 s n,k p n,k tftdt, [, where n nk n k 1 s n,k = e, p n,k t = k! k k 1 nk. First we recall some notations of -calculus, which can also be found in [6] and [1]. Throughout the present article be a real number satisfying the ineuality < < 1. For n N, := 1 n 1, { [n] [n 1]! := [1], n = 1, 2,... 1, n = and { n 1 1 n := j= 1 j, n = 1, 2,... 1, n =. The -derivative D f of a function f is given by 2 D f = f f, if. 1 The -improper integrals considered in the present paper are defined as see [11] a f d = 1 a f a n n, a > and 3 f d = 1 n= n= n n f A A, A > provided the sums converge absolutely. There are two analogues of the eponential function e see [1] as e z = z k [k]! = z, z < 1 1, < 1

3 and E z = 1 1 j z = j= Approimation by kk 1/2 < 1, where 1 = j= 1 j. The -Gamma integral is defined by [1] 4 Γ t = 1 1 zk t 1 E d, t > which satisfies the following functional euation: Γ t 1 = [t] Γ t, Γ 1 = 1. The Beta function see [16] is defined as 5 B t, s = K A, t where K, t = 1 1 t 1 1 integer n and t [k]! = 1 1 z, t 1 1 ts d, 1 1 t. In particular for any positive K, n = nn 1 2, K, = 1 6 B t, s = Γ t Γ s Γ t s. Based on -eponential function Mahmudov [12], introduced the following -Szász-Mirakyan operators as 1 k S n, f, = kk 1/2 [k] 7 f E [k]! k 2 = s n,k f [k] k 2, s n,k = k Lemma 1 [12]. We have [k]! S n, 1, = 1, S n, t, =, kk 1/2 1 E. S n, t 2, = 2 2.

4 38 V. Gupta, A. Aral and M. Ozhavzali In the present article, we introduce the analogue of the Szász-Mirakyan -Baskakov operators, obtain its moments using -Beta function and estimate some direct results in terms of modulus of continuity. 2. -Operators and moments For every n N,, 1, the analogue of 1 can be defined as 8 G n f t, := [n 1] s n,k k p n,k t f t d t where and s n,k = k [k]! p n,k t := [ n k 1 k kk 1/2 1 E ] kk 1/2 t k 1 t nk for [, and for every real valued continuous function f on [,. These operators satisfy linearity property. As a special case when = 1 the above operators reduce to the Szász-Mirakyan-Baskakov operators 1 discussed in [14] and [7]. and and Remark 1. We have D s n,k [k] = k 2 2 k 2 s [n] n,k, t 1 t D p n,k Proof. Using -derivative, we have Also we have t = [n] 2 p n,k t [k] k 1 t. D E = E 1 D = E E E E = E. D s n,k k 1 = [k] kk 1/2 1 [k]! E E k [k]! kk 1/2

5 Approimation by [k] = k k kk 1/2. E [k]! Remark 2. Following euality is obvious: Therefore s n,k = k [k]! kk 1/2 1 E = k k kk 1/2 1 1 [k]! E = k 1 1 s n,k. s n,k = k 1 1 s n,k. Lemma 2. If we define the central moment as T n,m = G n t m, := [n 1] then we have s n,k k p n,k t tm d t [m 1] T n,m 1 [m 2] T n,m1 = { 1 T n,m1 The following eualities hold: i T n, = G n 1, = 1, ii T n,1 = G n t, = iii T n,2 = G n t 2, = T n,m 1 1 D T n,m } [n] 2 [n 2] 1 Proof. Using Remark 1, we have D T n,m = [n 1] [n 2], for n > 1, [n] [n 2] [n 3] [n]12 5 [n 2] [n 3] [2] 3 [n 2] [n 3], for n > 3, = [n 1] D s n,k k [k] k 2 2 p n,k t tm d t. p n,k t tm d t s n,k 2k 2

6 4 V. Gupta, A. Aral and M. Ozhavzali Using Remark 1 and Remark 2, we have D T n,m = [n 1] s n,k 2k 2 [k] k 2 t t 2 p [n] n,k t tm d t = [n 1] s n,k [k] 2k 2 k 1 t p [n] n,k t tm d t [n 1] s n,k 2k 2 t 2 p n,k t tm d t = [n 1] s n,k 2k [n 1] Using -integration by parts we have D T n,m = [n 1] 2 t m1 t m2 D p n,k t s n,k k t 2 p n,k t tm d t. s n,k 2k [n 1] 2 = 1 1 [n 1] 2 [m 1] t m [m 2] t m1 p n,k t d t s n,k k t 2 p n,k t tm d t s n,k k 2 [m 1] t m [m 2] t m1 p n,k t d t d t [n] 1 T n,m1 1 1 T n,m 1 = 1 1 [m 1] T n,m

7 1 1 1 [m 2] T n,m1 Approimation by [n] 1 T n,m1 1 1 T n,m. This completes the proof of recurrence relation. The moments i-iii can be obtained easily by the above recurrence relation keeping in mind that T n, = 1, which follows from 5 and 6. Remark 3. In case 1, we get the central moments discussed in [14] and [7] as G 1 n G 1 n 1, = G n 1, = 1, G 1 n t, = G n t, = 1 2 n 2, t 2, = G n t 2, = n 62 2n 3 2. n 2n 3 3. Direct results Let C B [, be the space of all real-valued continuous bounded functions f on [,, endowed with the norm f = sup [, f. The Peetre s K-functional is defined by K 2 f; δ = inf g C 2 B [, { f g δ g }, where C 2 B [, := {g C B [, : g, g C B [, }. By [[5], p. 177, Theorem 2.4] there eists an absolute constant M > such that 9 K 2 f; δ Mω 2 f; δ, where δ > and the second order modulus of smoothness is defined as ω 2 f; δ = sup sup f 2h 2f h f, <h δ [, where f C B [, and δ >. Also we set 1 ωf; δ = sup sup f h f, <h δ [,

8 42 V. Gupta, A. Aral and M. Ozhavzali [n] 2 δ n = 6 2 [n 2] [n 3] 2 1 [n 2] [n] [n 2] [n 3] [n 2] [2] α n = 3, [n 2] [n 3] [n] 2 1 [n 2] 1 [n 2]. Lemma 3. Let f C B [,. Then, for all g CB 2 [,, we have 11 Ĝ ng; g δ n αn 2 g, where 12 Ĝ nf; = G 1 nf; f f [n 2] 2 1. Proof. From 12 we have Ĝ nt ; = G 1 13 nt ; [n 2] 2 1 = G nt; G n1; 1 [n 2] 2 1 =. Let [, and g CB 2 [,. Using the Taylor s formula we can write by 13 that t gt g = t g Ĝ ng; g = Ĝ nt g ; Ĝ n t ug udu, = g Ĝ nt ; G n [n] 2 1 [n 2] [n 2] t t [n] 2 [n 2] 2 t ug udu; t ug udu; 1 u g udu [n 2]

9 = G n t Approimation by t ug udu; [n] 2 1 [n 1] [n 1] [n] 2 [n 2] 1 u g udu. [n 2] On the other hand, since t t ug udu t t u g u du g t t u du t 2 g and [n] 2 1 [n 2] [n 2] [n] 2 [n 2] 1 u [n 2] g udu [n] g [n 2] [n 2] [n] 1 2 = 2 1 g := α [n 2] [n 2] n 2 g we conclude that t Ĝ ng; g = Ĝ n t ug udu; [n] 2 1 [n 2] [n 2] [n] 1 2 u g udu [n 2] [n 2] G nt 2 g [n] 1 2 ; 2 1 g [n 2] [n 2] = δ n α 2 n g.

10 44 V. Gupta, A. Aral and M. Ozhavzali Theorem 1. Let f C B [,. Then, for every [,, there eists a constant L > such that G nf; f Lω 2 f; δ n α 2 n ωf; α n. Proof. From 12, we can write that G nf; f Ĝ nf; f f f [n] [n 2] Ĝ nf g; f g f f [n] [n 2] Ĝ nf g; f g f f [n] [n 2] Ĝ ng; g Ĝ ng; g. Now, taking into account boundedness of Ĝ n and the ineuality 11, we get G nf; f 4 f g f f [n 2] 2 1 δ n αn 2 g 1 4 f g ω f; [n 2] [n 2] δ n α 2 n g. Now, taking infimum on the right-hand side over all g CB 2 [, and using 9, we get the following result G nf; f 4K 2 f; δ n α 2 n ωf; α n 4Mω 2 f; δ n α 2 n ωf; α n = Lω 2 f; δ n α 2 n ωf; α n where L = 4M >. Theorem 2. Let < α 1 and f C B [,. Then, if f Lip M α, i.e. the condition 14 fy f M y α,, y [,,

11 holds, then, for each [,, we have Approimation by G nf; f Mδ α 2 n, where δ n is the same as in Theorem 1, M is a constant depending on α and f. Proof. Let f C B [, Lip M α with < α 1. By linearity and monotonicity of G n G nf; f = G nf; G nf; G n f y f ; MG n y ;. Using the Hölder ineuality with p = 2 α, = 2 2 α we find that { } G nf; f M [G n y αp ; ] 1 p [G n1 ; ] 1 [ ] = M G n y 2 α 2 ; = Mδ α 2 n. Theorem 3. Let f be bounded and integrable on the interval [,, second derivative of f eists at a fied point [, and = n, 1 such that n 1 as n, then lim [n] n n [G n n f, f] = 1 2f 2 2 f. Proof. In order to prove this identity we use Taylor s epansion f t f = t f t f ε t where ε is bounded ε is bounded and lim t ε t =. By applying the operator G nf to the above relation we obtain G n n f, f = f G n n t, 1 2 f G n n t 2, G n n ε t t 2, = f α n 1 2 f δ n G n n ε t t 2,, where α n and δ n defined as in 1. Using Cauchy-Schwarz ineuality we have n G n n ε t t 2, G n n ε 2 t 1 2 [n] 2 n G n n t 4 1 2,.

12 46 V. Gupta, A. Aral and M. Ozhavzali Using Lemma 1, we can show that Also, since lim n [n]2 n G n n t 4, = lim α n = 1 2 and lim δ n = 2 2 n n we have desired result. 4. Error estimation The usual modulus of continuity of f on the closed interval [, b] is defined by ω b f, δ = sup ft f, b >. t δ,t [,b] It is well known that, for a function f E, where E := lim ω bf, δ =, δ { } f f C[, : lim is finite. 1 2 The net theorem gives the rate of convergence of the operators G nf; to f, for all f E. Theorem 4. Let f E and let ω b1 f, δ b > be its modulus of continuity on the finite interval [, b 1] [,. Then for fied, 1, we have G nf; f C[,b] N f 1 b 2 δ n b 2ω b1 f, δ n b. Proof. The proof is based on the following ineuality 15 G nf; f N f 1 b 2 G nt 2 ; 1 G n t ; ω b1 f, δ δ for all, t [, b] [, := S. To prove 15 we write S = S 1 S 2 := {, t : b, t b 1} {, t : b, t > b 1}.

13 If, t S 1, we can write 16 ft f ω b1 f, t Approimation by t ω b1 f, δ δ where δ >. On the other hand, if, t S 2, using the fact that t > 1, we have 17 ft f M f 1 2 t 2 M f t 2 N f 1 b 2 t 2 where N f = 6M f. Combining 16 and 17, we get 15. Now from 15 it follows that G nf; f N f 1 b 2 G nt 2 ; 1 G n t ; ω b1 f, δ δ By Lemma 2 we have N f 1 b 2 G nt 2 ; [ G n t 2 ; ] 1/2 1 ω b1 f, δ. δ G nt 2 ; δ n b G nf; f N f 1 b 2 δn b δ n b 1 ω b1 f, δ. δ Choosing δ = δ n b, we get the desired estimation. Acknowledgements. The authors are thankful to referee for valuable suggestions for better presentation of the paper. References [1] Agratini O., Doğru O., Weighed approimation by -Szász-King type operators, Taiwanese J. Math., 1421, [2] Aral A., A generalization of Szász Mirakyan operators based on -integers, Math. Comput. Model., 4728, [3] Aral A., Gupta V., On the Durrmeyer type modification of the -Baskakov type operators, Nonlinear Analysis, 7221, [4] Aral A., Gupta V., The -derivative and applications to -Szász Mirakyan operators, Calcolo, 43326,

14 48 V. Gupta, A. Aral and M. Ozhavzali [5] DeVore R.A., Lorentz G.G., Constructive Approimation, Springer, Berlin, [6] Gasper G., Rahman M., Basic Hypergeometrik Series, Encyclopedia of Mathematics and its Applications, Vol 35, Cambridge University Press, Cambridge, UK, 199. [7] Gupta V., A note on modified Szász operators, Bull. Inst. Math. Acad. Sinica, , [8] Gupta V., Aral A., Convergence of the -analogue of Szász Beta operators, Applied Mathematics and Computation, 21621, [9] Gupta V., Heping W., The rate of convergence of -Durrmeyer operators for < < 1, Math. Methods Appl. Sci., , [1] Kac V.G., Cheung P., Quantum Calculus, Universitet, Springer-Verlag, New York, 22. [11] Koornwinder T.H., -Special Functions, a Tutorial, in: M. Gerstenhaber, J. Stasheff Eds, Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemp. Math., , Amer. Math.Soc [12] Mahmudov N.I., On -parametric Szász-Mirakjan operators, Mediterr J. Math., 7321, [13] Mahmudov N.I., Kaffaoglu H., On -Szász-Durrmeyer operators, Central Eur. J. Math., 8221, [14] Prasad G., Agrawal P.N., Kasana H.S., Approimation of functions on [, ] by a new seuence of modified Szász operators, Math. Forum, , [15] Radu C., Taraibe S., Veteleanu A., On the rate of convergence of a new Szász-Mirakjan operators, Stud. Univ. Babes-Bolyai Math., , [16] De Sole A., Kac V.G., On integral representation of -gamma and -beta functions, AttiAccad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl., 16125, Vijay Gupta School of Applied Sciences Netaji Subhas Institute of Technology Sector 3 Dwarka, New Delhi 1178 India vijaygupta21@hotmail.com Ali Aral and Muzeyyen Ozhavzali Kirikalle University Faculty of Science and Arts Department of Mathematics Yahşihan, Turkey aliaral73@yahoo.com or thavzalimuzeyyen@hotmail.com Received on and, in revised form, on