On the operators defined by Lupaş with some parameters based on q-integers

Transcription

1 Mathematics Today Vol.34A April Special Issue 0-10 ISSN , e-issn On the operators defined by Lupaş with some parameters based on q-integers Prashantkumar Patel St. Xavier s College Autonomous, Navrangpura, Ahmedabad-09 Abstract The aim of this paper, introduce q-analogue of a sequence of linear and positive operators with two parameters which was introduced by A. Lupaş in First, we estimate moments of the operators and then prove a basic convergence theorem. Next, a local approximation theorem is established. Further, we study the rate of convergence and weighted approximation theorem for these operators. Keywords: Positive linear operators; quantum integers; modulus of continuous; rate of convergence. 010 Mathematics Subject Classification: 41A5, 41A30, 41A36 1 Introduction The space of real valued continuous functions on the interval R = [0, is denoted by C R. The space of real valued bounded continuous functions on the interval R is denoted by C B R with the norm f = f x is a Banach space. x R Consider the weight function ρ λ : R [1,, ρ λ x = 1 x λ λ > 0, we define the space { } C ρλ R = f C R f x : is convergent as x ρ λ x f x endowed with the usual norm ρλ, f ρλ = x R ρ λ x. In 016, Singh et al. [1] introduced the following sequence of positive linear operators as: For f : R R, L f,x = [n] q x [n] q x k [k]q f, x 0, 1.1 [n] q where λ 0 = 1, λ k = λλ 1... λ k 1, k 1. Before proceeding further, let us give some basic definitions and notations from q-calculus. Details on q-integers can be found in [, 3]. Let q > 0, for each nonnegative integer k, the q-integer [k] q and the q-factorial [k] q! are defined as [k] q 1 q k 1 q, q 1 k, q = 1

2 Prashantkumar Patel - On the operators dened by Lupas with some parameters based on q-integers 03 and [k] q! [k] q [k 1] q...[1] q k 1 1, k = 0, respectively. For q > 0 and integers n, k, n k 0, we have [k 1] q = 1 q[k] q and [k] q q k [n k] q = [n] q. One should observe that, for q = 1, the operators 1.1 reduce to the operators introduced by Lupaş [4] as follows: L n f,x = nx nx k k k! f k, x n Some approximation properties of the operators 1. was studied in [5]. The Durrmeyer variant of the operators 1. was studied very recently in [6]. In [7] Agratini modified the operators 1. into integral form in Kantorovich sense and established their approximation properties. Recently, statistical approximation processes and some direct results of the operators L n have been studied in [8].The Jain type variant of the operators 1. was established by Patel and Mishra [9]. In this manuscript, we modified the operators 1.1 in two parameters α and β with 0 α β. Similar type of generalization of positive linear operators is known as Stancu type generalization. These type of generalization can be found in [10, 11, 1, 13, 14, 15, 16, 17] for various other operators. Motivated by this, we modified the operators 1. as follows: For 0 α β, f C ρ0 R, x R f,x = [n] q x [n] q x k [k]q α f. 1.3 Lemma 1.1 [1] The following relations hold: L 1,x = 1;L t,x = x;l t,x = qx 1 q [n] q x. Lemma 1. The following relations hold: 1,x = x; t,x = [n] q x α ; t,x = q[n] q x α 1 q[n] q x α [n]q β. Proof. We have 1,x = L 1,x = 1;

3 04 Mathematics Today Vol.34A April Special Issue 0-10 t,x = [n] q x = [n] q x k=1 = [n] q x 1 [n] q x k [k]q α [n] q x k k [k 1] q! α [n] q x [n] q x 1 k 1 k 1 [k 1] q! k=1 [n]q x1 = [n] q x = [n] q x α. [n] q x 1 k α α Similarly, t,x = [n] q x = [n] q x [n] q x [n] q x k [k] q [k] q α α k [k] q! = [n] q x [n] q x 1 [n] q x [n] x 1 q k 1 k [k] q [k] q [k 1] q! [n] q x [n] x 1 q k 1 α k [k] [k] q [k 1] q! q α [n]q x1 α [n] q x = [n] q x [n] q x 1 = q[n] q x [n] q x 1 = q[n] q x [n] q x k=1 [n] q x 1 k [k 1] q [n] q x 1 k [n] q x 1 k 1 q [k] q [n] q x 1 k [k] q = q[n] q x α 1 q[n] q x α [n]q β, α α [n] q x α α 1[n] q x α [n] q x 1[n] x q k 1 k 1 α 1[n] q x α [k 1] q! which completes the proof of the Lemma 1.. Definition 1. Let 0 < q < 1, using linear properties of the operators, for x R, we have τ,α,β x = t x,x = τ 1,α,β x = t x,x = α βx ; q 1[n] q β x [] q [n] αβ q x α.

4 Prashantkumar Patel - On the operators dened by Lupas with some parameters based on q-integers 05 Pointwise Convergence Theorem.1 Let f C ρ0 R and q n be a real sequence in 0,1 such that q n 1 and q n n 0 as n. Then, for any compact set K R, we have n Lα,β n f,x = f x uniformly in x K. Proof: The proof is based on the well-known Korovkin theorem regarding the convergence of a sequence of linear positive operators. So, it is enough to prove the conditions Now, using Lemma 1., we obtain Also, Similarly, n Lα,β n t m,x = x m, m = 0,1,. n Lα,β n 1,x = 1. [n]qn x n Lα,β n t,x = n [n] qn β α = x. [n] qn β n Lα,β n t,x q n [n] q n x α [] qn [n]qn x α = n [n]qn β = x. This completes the proof of Theorem.1. 3 Local Approximation The Peetre s K-functional is defined by K f, δ = inf g C B R { f g δ g }, where C B R = { g C B R : g, g C B R }. By [17], there exists a positive constant C > 0 such that K f, δ Cω f, δ, δ > 0, where the second-order modulus of smoothness is given by ω f, δ = 0<h δ 0x< f x h f x h f x. Also, for f C B R the usual modulus of continuity is given by ω f, δ = 0<hδ 0x< f x h f x. Theorem 3.1 Suppose that f C B R and 0 < q < 1. Then for all x R and n N, there exists an absolute constant C > 0 such that f,x f x Cω f, Proof: We are introducing the auxiliary operators as follows f,x = f,x f τ,α,β x τ x 1,α,β ω f, τ 1,α,β x [n]q x α f x.

5 06 Mathematics Today Vol.34A April Special Issue 0-10 Suppose that g C B R and x,t R. From Taylor s expansion g t = g x g xt x t x t ug udu. Applying, we get t g,x g x = g x t x,x t ug udu,x. Using Lemma 1., we obtain t g,x g x t ug udu x,x t x,x g x [ ] τ,α,β x τ 1,α,β x g. x [n] q xα [n] q β [n]q x α u g udu Since f,x f, f,x f x f g,x f gx g,x g x [n] q x α f x [ ] f g τ,α,β x τ 1,α,β x g ω f, τ 1,α,β x. Taking infimum overall g C B R, we get f,x f x K f, τ,α,β x τ 1,α,β x ω f, τ 1,α,β x. In view of K f, δ Cω f, δ, δ > 0, we have f,x f x Cω f, τ,α,β x τ x 1,α,β ω f, τ 1,α,β x, which proves the Theorem Rate of convergence In this section, we want to estimate the rate of convergence for the sequence of the operators. For any positive a, by ω a f, δ = t x δ x,t [0,a] f t f x, we denote the usual modulus of continuity of f on the closed interval [0, a]. Let ρx = 1 ϕ x, where ϕ x is a monotone increasing continuous function on the real axis and B ρ is the set of all functions f defined on the real axis satisfying the growth condition f x M f ρ x, where M f depending only on f. Then B ρ is a normed space with norm { } f x f ρ = : x R ρx is a constant, for any f B ρ. Let C ρ denote the subspace of all continuous function in B ρ, f x = 0. ρx and Cρ the subspace of all function f C ρ for which x Now, we give a rate of convergence theorem for the operator.

6 Prashantkumar Patel - On the operators dened by Lupas with some parameters based on q-integers 07 Theorem 4.1 Let f C ρ0 R, q = q n 0,1 such that q n 1 as n and ω a1 f, δ be its modulus of continuity on the finite interval [0, a 1] R, where a > 0. Then f f 6M f 1 a τ,α,β x ω a1 f, [0,a] Proof: For x [0,a] and t > a 1, since t x > 1, we have τ,α,β x. f t f x 6M f 1 a t x. 4.1 For x [0,a] and t a 1, we have f t f x ω a1 f, t x with δ > 0. Form 4.1 and 4., we can write for x [0,a] and t 0. Thus f t f x 6M f 1 a t x 1 t x ω δ a1 f,δ, 4. 1 t x ω δ a1 f,δ, f,x f x f t f x,x 6M f 1 a t x,x ω a1 f,δ 1 1 t x,x 1 δ. Hence, by Schwarz s inequality and Lemma 1., for every q 0,1 and x [0, a] f,x f x 6M f 1 a τ,α,β x ω a1 f,δ 1 1 δ τ,α,β x. By taking δ = τ,α,β x, we get the assertion of our theorem. 5 Weighted approximation The Korovkin type theorems on weighted approximation of unbounded continuous functions on unbounded sets with single weight function were first proved by Gadzhiev [19, 0]. Now, we give Gadzhiev s results in weighted spaces. Theorem 5.1 a There exists a sequence of linear positive operators A n C ρ B ρ such that A n ϕ v ϕ v ρ = 0, v = 0, 1, 5.1 n and a function f C ρ C ρ with n A n f f ρ 1. b If a sequence of linear positive operators A n C ρ B ρ such that satisfies conditions 4.1 then for every f C ρ. n A n f f ρ = 0,

7 08 Mathematics Today Vol.34A April Special Issue 0-10 Theorem 5. Let q = q n satisfies 0 < q n < 1 and let q n 1 as n. For each f C ρ 0 R, we have n Lα,β n f f ρ0 = 0. Proof: Using the theorem in [19], we see that it is sufficient to verify the following n Lα,β n t v, x x v ρ0 = 0, v = 0, 1,. 5. Since n 1,x = 1, the first condition of 4. is satisfied for v = 0. Now, n t, x x = ρ0 x 0, n t, x x 1 x α [n] qn β β [n] qn β x 0, α β [n] qn β 0 as n x 1 x and the second condition of 4. hold for r = 1. Similarly, we have n t, x x n t, x x ρ0 = x 0, 1 x q n [n] q n [n]qn β 1 x x 0, 1 x α 1 q n[n] qn [n]qn β x x 0, 1 x α [n]qn β q n 1[n] q n β[n] qn β [n]qn β α 1 q n[n] qn [n]qn β α [n]qn β q n 1[n] q n [n]qn β β[n] qn [n]qn β β [n]qn β α 1 q n[n] qn [n]qn β α [n]qn β, which implies that n t, x x ρ0 0 as n. Thus the proof is completed. We give the following theorem to approximate all functions in C ρ0 R. This type of results are given in [1] for locally integrable functions. Theorem 5.3 Let q = q n satisfies 0 < q n < 1 and let q n 1 as n. For each f C ρ0 R and ε > 0, we have n x 0, n f, x f x 1 x 1ɛ = 0.

8 Prashantkumar Patel - On the operators dened by Lupas with some parameters based on q-integers 09 Proof: For any fixed x 0 > 0, α,β n f, x f x L x 0, 1 x 1ɛ n f, x f x xx 0 n f, f 1 x 1ɛ [0,x0 ] α,β L n f, x f x x x 0 1 x 1ɛ α,β L n 1 t, x f x f x x x 0 1 x 1ɛ x x 0 1 x 1ɛ. The first term of the above inequality tends to zero from Theorem 4.1. By Lemma 1. for any fixed α,β L n 1 t, x x 0 > 0 it is easily seen that x x 0 1 x 1ɛ tends to zero as n. We can choose x 0 > 0 so large that the last part of the above inequality can be made small enough. Thus the proof is completed. Acknowledgments The authors would like to thank the referee for his/her valuable suggestions which improved the paper considerably. References [1] K. Singh, A. Gairola and Deepmala, Approximation Theorems for q-analouge of a Linear Positive Operator by A. Lupas, International Journal of Analysis and Applications, vol. 1, no. 1, pp , 016. [] V. G. Kac and P. Cheung, Quantum Calculus, New York: Universitext. Springer, 00. [3] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, Cambridge: Cambridge University Press, [4] A. Lupas, The approximation by some positive linear operators, in In: proceedings of the International Dortmund meeting on Approximation TheoryM.W. Muller et al., eds., akademie Verlag, Berlin, [5] O. Agratini, On a sequence of linear and positive operators, Facta UniversitatisNIS, vol. 14, pp , [6] N. K. Govil, V. Gupta and D. Soybas, Certain newclasses ofdurrmeyer type operators, Applied Mathematics Computation, vol. 5, pp , 013. [7] O. Agratini, On the rate of convergence of a positive approximation process, Nihonkai Mathematics Journal, vol. 11, no. 1, pp , 000. [8] S. Tarabie, On some A-statistical approximation processes, International Journal of Pure and Applied Mathematics, vol. 76, no. 3, pp , 01.

9 10 Mathematics Today Vol.34A April Special Issue 0-10 [9] P. Patel and V. N. Mishra, On new class of linear and positive operators, Vol. 8, no., pp , 015. [10] D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Revue Roumaine des Mathematiques Pures et Appliquees, vol. 13, no. 8, pp , [11] P. Patel and V. N. Mishra, A note on Simultaneous Approximation of some Integral Generalization of the Lupas operators, Asian Journal of Mathematics and Computer Research, vol. 4, no. 1, pp. 8-44, 015. [1] P. Patel and V. N. Mishra, On Simultaneous Approximation of Modified Baskakov-Durrmeyer Operators, International Journal of Analysis, vol. 015, no. Article ID , p. 10 pages, 015. [13] P. Patel and V. N. Mishra, Approximation Properties of Certain Summation Integral Type Operators, Demonstratio Mathematica, vol. 48, no. 1, pp , 015. [14] V. N. Mishra and P. Patel, On generalized integral Bernstein operators based on q-integers, Applied Mathematics and Computation, vol. 4, pp , 013. [15] V. Gupta, D. K. Verma and P. N. Agrawal, Simultaneous approximation by certain Baskakov- Durrmeyer-Stancu operators,journal of the Egyptian Mathematical Society, vol. 0, pp , 01. [16] B. Ibrahim, Approximation by Stancu-Chlodowsky polynomials, Computers & Mathematics with Applications, vol. 59, no. 1, pp. 74-8, 010. [17] V. N. Mishra and P. Patel, The Durrmeyer type modification of the q-baskakov type operators with two parameter α and β, Numerical Algorithms, vol. 67, no. 4, pp , 014. [18] R. Devore and G. Lorentz, Constructive Approximation, Berlin: Springer, [19] A. D. Gadzhiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin.s theorem Russian, Dokl. Akad. Nauk SSSR, vol. 18, pp , [0] A. D. Gadzhiev, Theorems of the type of P. P. Korovkin.s theorems Russian, presented at the international conference on the theory of approximation of functions Kaluga, Mat. Zametki, vol. 0, no. 5, pp , 1976 and Theorems of Korovkin type English, Mathematical Notes of the Academy of Sciences of the USSR, Vol. 0, no.5, pp , [1] A. D. Gadjiev, R. O. Efendiyev and E. Ibikli, On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Math. J, vol. 18, no. 1, pp , 003.