Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure and Applied Mathematics ON A HYBRID FAMILY OF SUMMATION INTEGRAL TYPE OPERATORS VIJAY GUPTA AND ESRA ERKUŞ School of Applied Sciences Netaji Subhas Institute of Technology Sector-3, Dwarka New Delhi , India. vijaygupta21@hotmail.com Çanakkale Onsekiz Mart University Faculty of Sciences and Arts Department of Mathematics Terzioğlu Kampüsü 172, Çanakkale, TURKEY. erkus@comu.edu.tr volume 7, issue 1, article 23, 26. Received 21 November, 25; accepted 1 December, 25. Communicated by: A. Lupaş Abstract Home Page c 2 Victoria University ISSN (electronic):

2 Abstract The present paper deals with the study of the mixed summation integral type operators having Szász and Baskakov basis functions in summation and integration respectively. Here we obtain the rate of point wise convergence, a Voronovskaja type asymptotic formula, an error estimate in simultaneous approximation. We also study some local direct results in terms of modulus of smoothness and modulus of continuity in ordinary and simultaneous approximation. 2 Mathematics Subject Classification: 41A25; 41A3. Key words: Linear positive operators, Summation-integral type operators, Rate of convergence, Asymptotic formula, Error estimate, Local direct results, K-functional, Modulus of smoothness, Simultaneous approximation. 1 Introduction Auxiliary Results Direct Results Local Approximation References Page 2 of 23

3 1. Introduction The mixed summation-integral type operators discussed in this paper are defined as (1.1) where S n (f, x) = = (n 1) W n (x, t) = (n 1) δ(t) being Dirac delta function, and W n (x, t)f(t)dt b n,ν 1 (t)f(t)dt e nx f(), x [, ), b n,ν 1 (t) e nx δ(t), nx (nx)ν = e ν! ( ) n ν 1 b n,ν (t) = t ν (1 t) n ν ν are respectively Szász and Baskakov basis functions. It is easily verified that the operators (1.1) are linear positive operators, these operators were recently proposed by Gupta and Gupta in [3]. The behavior of these operators is very similar to the operators studied by Gupta and Srivastava [5], but the approximation Page 3 of 23

4 properties of the operators S n are different in comparison to the operators studied in [5]. The main difference is that the operators (1.1) are discretely defined at the point zero. Recently Srivastava and Gupta [8] proposed a general family of summation-integral type operators G n,c (f, x) which include some well known operators (see e.g. [4], [7]) as special cases. The rate of convergence for bounded variation functions was estimated in [8], Ispir and Yuksel [6] considered the Bézier variant of the operators G n,c (f, x) and studied the rate of convergence for bounded variation functions. We also note here that the results analogous to [6] and [8] cannot be obtained for the mixed operators S n (f, x) because it is not easier to write the integration of Baskakov basis functions in the summation form of Szász basis functions, which is necessary in the analysis for obtaining the rate of convergence at the point of discontinuity. We propose this as an open problem for the readers. In the present paper we study some direct results, for the class of unbounded functions with growth of order t γ, γ >, for the operators S n we obtain a point wise rate of convergence, asymptotic formula of Voronovskaja type, and an error estimate in simultaneous approximation. We also estimate local direct results in terms of modulus of smoothness and modulus of continuity in ordinary and simultaneous approximation. Page 4 of 23

5 2. Auxiliary Results We will subsequently need the following lemmas: Lemma 2.1. For m N = N {}, if the m-th order moment is defined as U n,m (x) = ( ν ) m n x, ν= then U n, (x) = 1, U n,1 (x) = and Consequently nu n,m1 (x) = x [ U (1) n,m(x) mu n,m 1 (x) ]. U n,m (x) = O ( n [(m1)/2]). Lemma 2.2. Let the function µ n,m (x), m N, be defined as Then µ n,m (x) = (n 1) µ n, (x) = 1, µ n,1 (x) = 2x n 2, also we have the recurrence relation: b n,ν 1 (t)(t x) m dt ( x) m e nx. µ n,2(x) = nx(x 2) 6x2 (n 2)(n 3), (n m 2)µ n,m1 (x) = x [ µ (1) n,m(x) m(x 2)µ n,m 1 (x) ] [m 2x(m 1)] µ n,m (x); n > m 2. Page 5 of 23

6 Consequently for each x [, ) we have from this recurrence relation that µ n,m (x) = O ( n [(m1)/2]). Remark 1. It is easily verified from Lemma 2.2 that for each x (, ) (2.1) S n (t i, x) = (n i 2)! (nx) i (n i 2)! i(i 1) (nx) i 1 O(n 2 ). Lemma 2.3. [5]. There exist the polynomials Q i,j,r (x) independent of n and ν such that x r D r [] = n i (ν nx) j Q i,j,r (x), where D d dx. i,j Lemma 2.4. Let n > r 1 and f (i) C B [, ) for i {, 1, 2,..., r} (cf. Section 3). Then S (r) n (f, x) = n r (n 2) (n r 1) ν= b n r,νr 1 (t)f (r) (t)dt. Page 6 of 23

7 3. Direct Results In this section we consider the class C γ [, ) of continuous unbounded functions, defined as f C γ [, ) {f C[, ) : f(t) Mt γ, for some M >, γ > }. We prove the following direct estimates: Theorem 3.1. Let f C γ [, ), γ > and f (r) exists at a point x (, ), then (3.1) lim S n (r) n (f(t), x) = f (r) (x). Proof. By Taylor s expansion of f, we have r f (i) (x) f(t) = (t x) i ε(t, x)(t x) r, i! i= where ε(t, x) as t x. Thus, using the above, we have S (r) n (f, x) = = W n (r) (t, x)f(t)dt r f (i) (x) i! i= = R 1 R 2, say. W n (r) (t, x)(t x) i dt W n (r) (t, x)ε(t, x)(t x) r dt Page 7 of 23

8 First to estimate R 1, using a binomial expansion of (t x) m and applying (2.1), we have r f (i) (x) i ( i i ν r R 1 = )( x) W i! ν x r n (t, x)t ν dt i= ν= = f [ ] (r) (x) d r (n r 2)!n r x r terms containing lower powers of x r! dx r [ ] (n r 2)!n = f (r) r (x) f (r) (x) as n. Next using Lemma 2.3, we obtain R 2 (n 1) i,j n i Q i,j,r(x) x r ν nx j = R 3 R 4, say. ( n) r e nx ε(, x) ( x) r b n,ν 1 (t) ε(t, x) (t x) r dt Since ε(t, x) as t x for a given ε > there exists a δ > such that ε(t, x) < ε whenever < t x < δ. Further if s max {γ, r}, where s is any integer, then we can find a constant M 1 > such that ε(t, x)(t x) r Page 8 of 23

9 M 1 t x s, for t x δ. Thus R 3 M 2 (n 1) i,j n i ν nx {ε j t x <δ b n,ν 1 (t) t x r dt } b n,ν 1 (t)m 1 t x s dt t x δ = R 5 R 6, say. Applying the Schwarz inequality for integration and summation respectively, and using Lemma 2.1 and Lemma 2.2, we obtain R 5 εm 2 (n 1) n i εm 2 i,j ν nx j ( i,j n i ( ( (n 1) ) 1 ( 2 ) 1 b n,ν 1 (t)dt b n,ν 1 (t)(t x) 2r 2 dt ) 1 2 (ν nx) 2j b n,ν 1 (t)(t x) 2r dt ) 1 2 Page 9 of 23

10 εm 2 i,j n i O ( n j/2) O ( n r/2) = εo(1). Again using the Schwarz inequality, Lemma 2.1 and Lemma 2.2, we get R 6 M 3 (n 1) M 3 = i,j i,j i,j n i ( ( n i (n 1) ν nx j (ν nx) 2j ) 1 2 n i O ( n j/2) O ( n s/2) = O ( n (r s)/2) = o(1). t x δ b n,ν 1 (t)(t x) 2s dt b n,ν 1 (t) t x s dt Thus due to the arbitrariness of ε > it follows that R 3 = o(1). Also R 4 as n and therefore R 2 = o(1). Collecting the estimates of R 1 and R 2, we get (3.1). Theorem 3.2. Let f C γ [, ), γ >. If f (r2) exists at a point x (, ), ) 1 2 Page 1 of 23

11 then lim n [ S n (r) (f, x) f (r) (x) ] n r(r 3) = f (r) (x) [x(2 r) r] f (r1) (x) x 2 2 (2 x)f (r2) (x). Proof. By Taylor s expansion of f, we have f(t) = r2 i= n [ S n (r) (f(t), x) f (r) (x) ] = n f (i) (x) (t x) i ε(t, x)(t x) r2 i! where ε(t, x) as t x. Applying Lemma 2.2 and the above Taylor s expansion, we have [ r2 f (i) (x) r2 E 1 = n i= f (i) (x) i! i= i! [ n = E 1 E 2, say. i j= ( i )( x) i j j W n (r) (t, x)(t x) i dt f (r) (x) ] W n (r) (t, x)ε(t, x)(t x) r2 dt W n (r) (t, x)t j dt nf (r) (x) = f (r) (x) n [ S n (r) (t r, x) r! ] r! f (r1) (x) (r 1)! n [ (r 1)( x)s n (r) (t r, x) S n (r) (t r1, x) ] ] Page 11 of 23

12 f [ (r2) (x) (r 2)(r 1) (r 2)! n x 2 S n (r) (t r, x) 2 ] (r 2)( x)s n (r) (t r1, x) S n (r) (t r2, x). Therefore, using (2.1) we have [ ] n E 1 = nf (r) r (n r 2)! (x) 1 n f [ { } (r1) (x) n r (n r 2)! (r 1)( x)r! (r 1)! { n r1 (n r 3)! (r 1)!x r(r 1) nr (n r 3)! f [ (r2) (x) (r 2)(r 1)x 2 (r!) nr (n r 2)! (r 2)! 2 { n r1 (n r 3)! (r 2)( x) { } n r2 (n r 4)! (r 2)! x 2 2 (r 1)(r 2) nr1 (n r 4)! (r 1)!x O ( n 2)]. }] r! (r 1)!x r(r 1) nr (n r 3)! r! In order to complete the proof of the theorem it is sufficient to show that E 2 as n, which can easily be proved along the lines of the proof of Theorem 3.1 and by using Lemma 2.1, Lemma 2.2 and Lemma 2.3. } Page 12 of 23

13 Theorem 3.3. Let f C γ [, ), γ > and r m r 2. If f (m) exists and is continuous on (a η, b η) (, ), η >, then for n sufficiently large S (r) n (f, x) f (r) M4 n 1 m f (i) M5 n 1/2 w ( f (r1), n 1/2) O ( n 2), i=1 where the constants M 4 and M 5 are independent of f and n, w(f, δ) is the modulus of continuity of f on (a η, b η) and denotes the sup-norm on the interval [a, b]. Proof. By Taylor s expansion of f, we have f(t) = m i= (t x) i f (i) (x) i! (t x) m ζ(t) f m (ξ) f m (x) m! h(t, x) (1 ζ(t)), where ζ lies between t and x and ζ(t) is the characteristic function on the interval (a η, b η). For t (a η, b η), x [a, b], we have f(t) = m i= (t x) i f (i) (x) i! For t [, ) \ (a η, b η), x [a, b], we define h(t, x) = f(t) (t x) m f m (ξ) f m (x). m! m i= (t x) i f (i) (x). i! Page 13 of 23

14 Thus S (r) n (f, x) f (r) (x) = Using (3.1), we obtain m f (i) (x) 1 = i! Hence = i= m i= f (i) (x) i! { m f (i) (x) i! i= { { W (r) n = 1 2 3, say. j= W n (r) (t, x)(t x) i dt f (r) (x) (t, x) f m (ξ) f m (x) m! W n (r) (t, x)h(t, x)(1 ζ(t))dt } } (t x) m ζ(t)dt } i ( i i j r )( x) W j x r n (t, x)t j dt f (r) (x) i ( [ i i j r (n j 2)! )( x) (nx) j j x r j= j(j 1) 1 M 4 n 1 uniformly in x [a, b]. Next 2 (n j 2)! (nx) j 1 O ( n 2)] f (r) (x). m i=r ( f (i) ) O n 2, W (r) n (t, x) f m (ξ) f m (x) t x m ζ(t)dt m! Page 14 of 23

15 w ( f (m), δ ) m! W (r) n (t, x) Next, we shall show that for q =, 1, 2,... (n 1) ( 1 ) t x t x m dt. δ ν nx j b n,ν 1 (t) t x q dt = O ( n (j q)/2). Now by using Lemma 2.1 and Lemma 2.2, we have (n 1) ( ν nx j b n,ν 1 (t) t x q dt (ν nx) 2j ) 1 2 ( (n 1) = O ( n j/2) O ( n q/2) = O ( n (j q)/2), uniformly in x. Thus by Lemma 2.3, we obtain (n 1) s (r) n,ν(x) M 6 i,j = O ( n (r q)/2), n i [(n 1) b n,ν 1 (t) t x q dt b n,ν 1 (t) (t x) 2q dt ] ν nx j b n,ν 1 (t) t x q dt ) 1 2 Page 15 of 23

16 uniformly in x, where M 6 = sup we get for any s >, i,j sup x [a,b] Q i,j,r (x) x r. Choosing δ = n 1/2, 2 w ( f (m), n 1/2) [ O(n (r m)/2 ) n 1/2 O ( n (r m 1)/2) O ( n s)] m! M 5 w ( f (m), n 1/2) n (m r)/2. Since t [, ) \ (a η, b η), we can choose a δ > in such a way that t x δ for all x [a, b]. Applying Lemma 2.3, we obtain 3 (n 1) i,j n i ν nx j Q i,j,r (x) s x r n,ν (x) t x δ b n,ν 1 (t) h(t, x) dt n r e nx h(, x). If β is any integer greater than or equal to {γ, m}, then we can find a constant M 7 such that h(t, x) M 7 t x β for t x δ. Now applying Lemma 2.1 and Lemma 2.2, it is easily verified that 3 = O (n q ) for any q > uniformly on [a, b]. Combining the estimates 1 3, we get the required result. Page 16 of 23

17 4. Local Approximation In this section we establish direct local approximation theorems for the operators (1.1). Let C B [, ) be the space of all real valued continuous bounded functions f on [, ) endowed with the norm f = sup f(x). The K- x functionals are defined as K(f, δ) = inf { f g δ g : g W 2 }, where W 2 = {g C B [, ) : g, g C B [, )}. By [1, pp 177, Th. 2.4], there exists a constant M such that K(f, δ) Mw 2 (f, ) δ, where δ > and the second order modulus of smoothness is defined as w 2 (f, ) δ = sup sup f(x 2h) 2f(x h) f(x), <h δ x [, ) where f C B [, ). Furthermore, let w (f, δ) = sup sup <h δ x [, ) f(x h) f(x) be the usual modulus of continuity of f C B [, ). Our first theorem in this section is in ordinary approximation which involves second order and ordinary moduli of smoothness: Theorem 4.1. Let f C B [, ). Then there exists an absolute constant M 8 > such that ) ( ) x(1 x) x S n (f, x) f(x) M 8 w 2 (f, w f,, n 2 n 2 Page 17 of 23

18 for every x [, ) and n = 3, 4,.... Proof. We define a new operator S n : C B [, ) C B [, ) as follows ( ) nx (4.1) S n (f, x) = S n (f, x) f(x) f. n 2 Then by Lemma 2.2, we obtain S n (t x, x) =. Now, let x [, ) and g W 2. From Taylor s formula we get (4.2) g(t) = g(x) g (x)(t x) t x (t u)g (u)du, t [, ) ( S n (g, x) g(x) = S t ) n (t u)g (u)du, x x ( t ) = S n (t u)g (u)du, x On the other hand, (4.3) t x x nx/(n 2) x (t u)g (u)du (t x)2 g ( ) n n 2 x u g (u)du. Page 18 of 23

19 and (4.4) nx/(n 2) x ( ) n n 2 x u g (u)du ( ) 2 nx n 2 x g 4x2 (n 2) 2 g 4x(1 x) (n 2) 2 g. Thus by (4.2), (4.3), (4.4) and by the positivity of S n, we obtain S n (g, x) g(x) S ( n (t x) 2, x ) g 4x(1 x) (n 2) 2 g. Hence in view of Lemma 2.2, we have ( ) S n (g, x) g(x) 2nx (n 6)x 2 4x(1 x) (4.5) (n 2)(n 3) (n 2) 2 g ( n n 3 1 ) 4x(1 x) g n 2 n 2 18 n 2 x(1 x) g. Again applying Lemma 2.2 S n (f, x) (n 1) b n,ν 1 (t) f(t) dt e nx f() f. Page 19 of 23

20 This means that S n is a contraction, i.e. S n f f, f C B [, ). Thus by (4.2) (4.6) S n f S nf 2 f 3 f, f C B [, ). Using (4.1), (4.5) and (4.6), we obtain S n (f, x) f(x) S n (f g, x) (f g)(x) ( S n (g, x) g(x) nx f(x) f n 2) 4 f g 18 ( n 2 x(1 x) g nx f(x) f n 2) { } ( ) x(1 x) x 18 f g n 2 g w f,. n 2 Now taking the infimum on the right hand side over all g W 2 and using (4.1) we arrive at the assertion of the theorem. The following error estimation is in terms of ordinary modulus of continuity in simultaneous approximation: Theorem 4.2. Let n > r 3 4 and f (i) C B [, ) for i {, 1, 2,..., r}. Then S (r) n (f, x) f (r) (x) ( ) n r (n r 2)! f 1 (r) n r (n r 2)! Page 2 of 23

21 ( ) [n (r 1)(r 2)] x2 2 [n r(r 2)] x r(r 1) 1 n r 3 w ( f (r), (n r 2) 1/2) where x [, ). Proof. Using Lemma 2.4 and taking into account the well known property w ( f (r), λδ ) (1 λ)w ( f (r), δ ), λ, we obtain S (r) (4.7) n (f, x) f (r) (x) nr (n r 1)! b n r,νr 1 (t) [ f (r) (t) f (r) (x) ] dt ν= [ ] n r (n r 2)! 1 f (r) (x) nr (n r 1)! ν= b n r,νr 1 (t) ( 1 δ 1 t x ) w ( f (r), δ ) dt [ n r (n r 1)! f 1] (r). Page 21 of 23

22 Further, using Cauchy s inequality, we have (4.8) (n r 1) { ν= (n r 1) b n r,νr 1 (t) t x dt ν= b n r,νr 1 (t) (t x) 2 dt } 1 2. By direct computation (4.9) (n r 1) = ν= b n r,νr 1 (t) (t x) 2 dt n (r 1)(r 2) 2n 2r(r 2) (n r 3)(n r 2) x2 (n r 3)(n r 2) x r(r 1) (n r 3)(n r 2). Thus by combining (4.7), (4.8) and (4.9) and choosing δ 1 = n r 2, we obtain the desired result. Page 22 of 23

23 References [1] R.A. DEVORE AND G.G. LORENTZ, Constructive Approximation, Springer-Verlag, Berlin Heidelberg, New York, [2] Z. FINTA AND V. GUPTA, Direct and inverse estimates for Phillips type operators, J. Math Anal Appl., 33(2) (25), [3] V. GUPTA AND M.K. GUPTA, Rate of convergence for certain families of summation-integral type operators, J. Math. Anal. Appl., 296 (24), [4] V. GUPTA, M.K. GUPTA AND V. VASISHTHA, Simultaneous approximation by summation integral type operators, J. Nonlinear Functional Analysis and Applications, 8(3) (23), [5] V. GUPTA AND G.S. SRIVASTAVA, On convergence of derivatives by Szász-Mirakyan-Baskakov type operators, The Math Student, 64 (1-4) (1995), [6] N. ISPIR AND I. YUKSEL, On the Bezier variant of Srivastava-Gupta operators, Applied Mathematics E Notes, 5 (25), [7] C.P. MAY, On Phillips operators, J. Approx. Theory, 2 (1977), [8] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summation integral type operators, Math. Comput. Modelling, 37 (23), Page 23 of 23

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