DIRECT AND INVERSE THEOREMS FOR SZÂSZ-LUPAS TYPE OPERATORS IN SIMULTANEOUS APPROXIMATION. Naokant Deo. 1. Introduction

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1 MATEMATIQKI VESNIK 58 26), UDK originalni nauqni rad research paper DIRECT AND INVERSE THEOREMS FOR SZÂSZ-LUPAS TYPE OPERATORS IN SIMULTANEOUS APPROXIMATION Naokant Deo Abstract. In this paper we give the direct and inverse theorems for Szâsz-Lupas operators and study the simultaneous approximation for a new modification of the Szâsz operators with the weight function of Lupas operators. 1. Introduction Let f be a function defined on the interval [, ) with real values. For f [, ) andn N, the Szâsz operator S n f,x) is defined as follows: S n f,x) = s n,k x)fk/n), where s n,k x) = e nx nx) k. k! The Szâsz-type operator L n f,x) is defined by L n f,x) = s n,k x)φ n,k f), where { f), for k = φ n,k f) = n s n,k t)ft)dt, for k =1, 2,... In [1], Mazhar and Totik introduced the Szâsz-type operator and showed some approximation theorems. Lupas proposed a family of linear positive operators mapping C [, ) intoc [, ), the class of all bounded and continuous functions on [, ) namely, ) B n f)x) = n + k 1 x k p n,k x)fk/n), where p n,k x) = k 1 + x) n+k. AMS Subject Classification: 41A35 Keywords and phrases: Linear positive operators, linear combination, integral modulus of smoothness, Steklow means. 19

2 2 N. Deo Motivated by the integration of Bernstein polynomials of Derriennic [4], Sahai and Prasad [11] modified the operators B n for function integrable on [, ) as M n f)x) =n 1) p n,k x) p n,k t)ft) dt. Now we consider another modification of operators with the weight function of Lupas operators, which are defined as V n f)x) =n 1) s n,k x) p n,k t)ft) dt. 1.1) The norm. Cα on the space C α [, ) ={f C [, ) : ft) Kt α for some α>andk>} is defined by f Cα = sup ft) t α. t< To improve the saturation order On 1 ) for the operator 1.1), we use the technique of linear combination as described below: V n f,k,x) = k Cj, k)v djnf,x), where Cj, x) = k i= i j j= d j d j d i for k andc, ) = 1 and d,d 1,d 2,...,d k are k+1) arbitrary, fixed and distinct positive integers. For our convenience we shall write the operator 1.1) as where V n f,x) = W n, x, t)ft) dt, W n, x, t) =n 1) s n,k x)p n,k t). The function f is said to belong to the generalized Zygmund class Liz α, k, a, b) if there exists a constant M such that ω 2k f,η,a,b) Mη αk,η>, where ω 2k f,η,a,b) denotes the modulus of continuity of 2k-th order of fx) onthe interval [a, b]. The class Liz α, 1,a,b) is more commonly denoted by Lip*α, a, b). Let f C α [, ) and <a 1 <a 2 <a 3 <b 3 <b 2 <b 1 <. Then for m N the Steklov mean f η,m of the m-th order corresponding to f, for sufficiently small values of η> is defined by η/2 ) m { f η,m x) =η m fx)+ 1) m 1 m m η/2 i=1 } m fx) dx i, 1.2) x i i=1 where x [a 1,b 1 ]and m η fx) isthem-th order forward difference with step length η.

3 Theorems for Szâsz-Lupas type operators 21 The direct results in ordinary and simultaneous approximation for such type of modified Szâsz-Mirakyan operators were studied by many researchers see e.g. [2], [5], [6] and [12]. 2. Auxiliary results In this section, we shall give some basic results, which will be useful in proving the main results. Lemma 2.1. [9] For m N {}, let the m-th order moment for the Szâsz operator be defined by ) µ n,m x) = m k s n,k x) n x. Then we have µ n, x) =1, µ n,1 x) =and nµ n,m+1 x) =x µ n,mx)+mµ n,m 1 x) ), for n N. Consequently, i) µ n,m x) is a polynomial in x of degree [m/2]; ii) for every x [, ), µ n,m x) =O n [m+1)/2]),where[β] denotes the integral part of β. Lemma 2.2. Let the m-th moment for the Szâsz operator be defined by µ n,m x) =n 1) s n,k x) p n,k t)t x) m dt. Then i) µ n, x) =1,µ n,1 x) = 1+2x) n 2), n > 2; ii) n m 2)µ n,m+1 x) =x [ µ n,mx)+m2 + x)µ n,m 1 x) ] +m +1) 1 + 2x)µ n,m x) iii) µ n,m x) =O n [m+1)/2]) for all x [, ). Proof. By the definition of µ n,m x), we can easily obtain i). Now the proof of ii) goes as follows: xµ n,mx) =n 1) xs n,k x) p n,k t)t x) m dt mxµ n,m 1 x). Using relations t1+t)p n,k t) =k nt)p n,kt) andxs n,k x) =k nx)s n,kx), we get x [ µ n,mx)+mµ n,m 1 x) ] =n 1) k nx) s n,k x) p n,k t)t x) m dt

4 22 N. Deo =n 1) s n,k x) =n 1) s n,k x) =n 1) s n,k x) [k nt)+n t x)] p n,k t)t x) m dt t 1 + t) p n,kt)t x) m dt + nµ n,m+1 x) [ 1 + 2x)t x)+t x) 2 + x1 + x) ] p n,kt)t x) m dt + nµ n,m+1 x) = m + 1)1 + 2x)µ n,m x) m +2)µ n,m+1 x) mx1 + x)µ n,m 1 x) + nµ n,m+1 x) This leads to proof of ii). The proof of iii) easily follows from i) and ii). Lemma 2.3. If f is differentiable r times r =1, 2, 3,...) on [, ), then we have ) V n r) f x) = nr n r 1)! s n,k x) p n,k t)f r) t) dt. n 2)! Proof. By Leibnitz s theorem in 1.1) ) ) V n r) f x) =n 1) r r 1) r i n i e nx nx) k i p n,k t)ft) dt i= k=i i k 1)! ) =n 1) r i= 1) r i r n r s n,k x) p n,k+i t)ft) dt i =n 1) { r ) } r s n,k x) 1) r 1) i n r p n,k+i t)t) ft) dt. i= i Again using Leibnitz s theorem ) p n,k+i t), V r) n f p r) n 1)! r n r,k+r t) = n r 1)! ) x) = nr n r 1)! s n,k x) n 2)! i= 1) i r i integrating by parts r times, we get the required result. 1) r p r) n r,k+r t)ft) dt, Lemma 2.4. For the function f η,m x) defined in 1.2), there hold: i) f η,m C[a 1,b 1 ]; ii) r) f η,m M C[a2,b 2] rη r ω r f,η,a 1,b 1 ), r =1, 2,...,m; iii) f fη,mc[a2,b M 2] m+1ω m f,η,a 1,b 1 ); iv) f C[a2,b η,m M m+2f M 2] C[a2,b f 2] Cα, where M i are certain constants independent of f and η. For the proof of the above properties of the function f η,m x) we refer to [12, page 167].

5 Theorems for Szâsz-Lupas type operators 23 Lemma 2.5. [8, 9] There exist polynomials q i,j,r x) independent of n and k such that x r dr dx r [ e nx nx) k] = 2i+j r i,j n i k nx j qi,j,r x) [ e nx nx) k] Lemma 2.6. Let f C α [, ). Iff 2k+2) exists at a point x, ), then lim n nk+1 {V n f,k,x) fx)} = 2k+2 Qp, k, x)f p) x), p= where Qp, k, x) is a certain polynomial in x of degree p. The proof of Lemma 2.6 follows along the lines of [7]. Lemma 2.7. Let δ and γ be any two positive numbers and [a, b] [, ). Then, for any m> there exists a constant M m such that V n f,x)t γ dt M m n m. C[a,b] t x δ The proof of this result follows easily by using Schwarz inequality and Lemma 2.7 from [1]. 3. Main results Theorem 3.1. Direct Theorem) Let f C α [, ). Then, for sufficiently large n, there exists a constant M independent of f and n such that Vn f,k,.) f { max C[a2,b 2] C 1 ω 2k+2 f; n 1/2,a 1,b 1 )C 2 n k+1) } fcα, where C 1 = C 1 k) and C 2 = C 2 k, f). Proof. By linearity property Vn f,k,.) f Vn C[a2,b 2] f f 2k+2,η ),k,.) C[a2,b 2] + V n f 2k+2,η,k,.) f 2k+2,η C[a2,b 2] + f f 2k+2,η C[a2,b 2] = A 1 + A 2 + A 3, say. By property iii) of Steklov mean, we get Next, by Lemma 2.6, we get A 3 C 1 ω 2k+2 f,η,a 1,b 1 ). A 2 C 2 n k+1) f2k+2,η C[a1,b 1].

6 24 N. Deo Using the interpolation property [5] and properties of Steklov mean, A 2 C 3 n k+1){ } fcα + η 2k+2) ω 2k+2 f,η). To estimate A 1, we choose a 2,b 2 such that <a 1 <a 2 <a 3 <b 3 <b 2 <b 1 <. Also let ψt) be the characteristic function of the interval [a 2,b 2 ], then A 1 Vn ψt)ft) f 2k+2,η t)),k,.) C[a3,b 3] + Vn 1 ψt)) ft) f 2k+2,η t)),k,.) C[a3,b 3] = A 4 + A 5, say. We note that in order to estimate A 4 and A 5, it is sufficient to consider their expressions without the linear combination. It is clear that by Lemma 2.3, we obtain V n ψt)ft) f 2k+2,η t)),x) =n 1) s n,k x) p n,k t)ψt)ft) f 2k+2,η t)) dt. Hence, V n ψt)ft) f 2k+2,η t)),.) C C[a1,b 4f f C[a2,b 2k+2,η. 1] 2] Now for x [a 3,b 3 ]andt [, ) / [a 2,b 2 ] we can choose an η 1 satisfying t x η1. Therefore by Lemma 2.5 and Schwarz inequality, we have I V n 1 ψt)) ft) f 2k+2,η t)),x) n 1) 2i+j r i,j C 5 f Cα n 1) n i φ i,j,r x) x r s n,k x) k nx j p n,k t)1 ψt)) ft) f2k+2,η t) dt 2i+j r i,j C 5 η 2s fcα n 1) 1 n i 2i+j r i,j n i s n,k x) k nx j t x η 1 p n,k t) dt s n,k x) k nx j ) 1/2 p n,k t) dt p n,k t)t x) 4s dt C 5 η1 2s f { } 1/2 Cα n i s n,k x)k nx) 2j 2i+j r i,j n 1) s n,k x) ) 1/2 p n,k t)t x) 4s dt) 1/2.

7 Theorems for Szâsz-Lupas type operators 25 Hence, by Lemma 2.1 and Lemma 2.2, we have I C 6 f Cα n i+ j 2 s) C 6 n q f Cα where q = s m/2). Now choose s > such that q k + 1. Then I C 6 n k+1) fcα. Therefore by property iii) of Steklov mean, we get A 1 C 7 f f2k+2,η C[a2,b 2] + C 6n k+1) f Cα C 8 ω 2k+2 f,η,a 1,b 1 )+C 6 n k+1) f Cα Hence with η = n 1/2, the theorem follows. Theorem 3.2. Inverse Theorem) If <α<2 and f C α [, ) then in the following statements i) ii): i) Vn f,k,x) fx) = O C[a1,b n αk+1)/2),wheref C α [a, b], 1] ii) f Lizα, k +1,a 2,b 2 ). Proof. Let us choose points a,a,b,b in such a way that a 1 <a <a < a 2 <b 2 <b <b <b 1. Also suppose g C with suppg) [a,b ]andgx) =1 for x [a a,b 2 ]. It is sufficient to show that Vn fg,k,.) fg C[a,b ] = O n αk+1)/2) ii). 3.1) Using F in place of fg for all the values of r>, we get 2k+2 r F C[a,b ] 2k+2 r F V n F, k,.)) C[a,b ] + 2k+2 r V n F, k,.) C[a,b ] 3.2) By the definition of 2k+2 r, 2k+2 r V n F, k,.) C[a,b ] r r = V n F, k, x + 2k+2 ) x i dx 1...dx 2k+2 i=1 C[a,b ] r 2k+2 V 2k+2) n F, k,.) C[a,b +2k+2)r] r 2k+2{ V n 2k+2) F F η,2k+2,k,.) C[a,b +2k+2)r] + V 2k+2) n F η,2k+2,k,.) C[a, 3.3),b +2k+2)r] where F η,2k+2 is the Steklov mean of 2k + 2)-th order corresponding to F. By Lemma 3 from [1], we get 2k+2 x 2k+2 W nt, x) dt 2i+j 2k+2 i,j n 1) n i k nx qi,j,2k+2 j x) x 2k+2 s n,k x) } p n,k t) dt.

8 26 N. Deo Since p n,k t) dt = 1 n 1, by Lemma 2.1, s n,k x)k nx) 2j = n 2j ) 2j k s n,k x) n x = On j ) 3.4) Using Schwarz inequality and Lemma 2.1, we obtain V 2k+2) n F F η,2k+2,k,.) K C[a,b +2k+2)r] 1n k+1 F Fη,2k+2C[a,b ] 3.5) By Lemma 2 from [1], we get [ ] k x k W nt, x) t x) i dt =, for k>i. 3.6) By Taylor s expansion, we obtain F i) F η,2k+2 t) = 2k+1 η,2k+2 x) t x) i + F 2k+2) x)2k+2 η,2k+2 ξ)t, 3.7) i= i! 2k + 2)! where t<ξ<x. By 3.6) and 3.7), we get 2k+2 x 2k+2 V n F η,2k+2,k,.) C[a,b +2k+2)r] k Cj, k) [ ] F 2k+2) 2k+2 η,2k+2 2k + 2)! C[a,b ] x 2k+2 W d jnt, x) t x) 2k+2 dt C[a,b ]. j= Again applying Schwarz inequality for integration and summation and Lemma 3 from [1], we obtain I 2k+2 x 2k+2 W nt, x) t, x)2k+2 dt n 1) n i s n,k x) k nx j q i,j,2k+2 x) p n,k t)t x) 2k+2 dt 2i+j 2k+2 i,j 2i+j 2k+2 i,j n i qi,j,2k+2 x) x 2k+2 { n 1) s n,k x) { x 2k+2 } 1/2 s n,k x)k nx) 2j p n,k t)t x) 4k+4 dt} 1/2. 3.8) Using Lemma 2 from [1], n 1) s n,k x) p n,k t)t x) 4k+4 dt = T n,4k+4 x) =O n 2k+2)). 3.9) Using 3.4) and 3.9) in 3.8), we obtain I n i qi,j,2k+2 x) On j/2 )O n k+1)) = O1). 2i+j 2k+2 i,j x k+1

9 Theorems for Szâsz-Lupas type operators 27 Hence W n 2k+2) F η,2k+2,k,.) K C[a,b +2k+2)r] 2 F 2k+2) η,2k+2 On combining 3.2), 3.3), 3.5) and 3.1) it follows 2k+2 r F C[a 2k+2,b ] r F V n F, k,.)) C[a,b ] + K 3 r 2k+2 n k+1 F Fη,2k+2 C[a,b ] + F 2k+2) 3.1) C[a,b ]. η,2k+2) C[a,b ] Since for small values of r the above relation holds, it follows from the properties of F η,2k+2 and 3.1) that ω 2k+2 F, h, [a,b ]) K 4 {n αk+1)/2 + h 2k+2 n k+1 + η 2k+2) } F, η, [a,b ]). ω 2k+2 Choosing η is such a way that n<η 2 < 2r and following Berens and Lorentz [3], we obtain w 2k+2 F, h, [a,b ]) = Oh αk+1) ). 3.11) Since F x) =fx) in[a 2,b 2 ], from 3.11) we have w 2k+2 f,h,[a 2,b 2 ]) = Oh αk+1) ), i.e., f Lizα, k +1,a 2,b 2 ). Let us assume i). Putting τ = αk + 1), we first consider the case <τ 1. For x [a,b ], we get V n fg,k,x) fx)gx) =gx)v n ft) fx)),k,x)+ + k b1 C j, k) W dj,nt, x)fx)gt) gx)) dt + O n k+1)) j= a 1 = I 1 + I 2 + O n k+1)), 3.12) where the O-term holds uniformly for x [a,b ]. Now by assumption Vn f,k,.) f C[a1,b n = O τ/2), 1] we have I1C[a,b ] gc[a Vn f,k,.) f K,b ] C[a,b ] 5n τ/ ) By the Mean Value Theorem, we get I 2 = k b1 Cj, k) W dj,nt, x)ft) {g ξ)t x)} dt. j= a 1 Once again applying Cauchy-Schwarz inequality and Lemma 2 from [1], we get I2C[a,b ] fc[a1,b g k ) 1] Cj, k) C[a,b ] j= 1/2 max W dj,nt, x)t x) 2 dt = O n τ/2). 3.14) j k C[a,b ] Combining ), we obtain Vn fg,k,.) fg C[a,b ] = O n τ/2), for <τ 1. ).

10 28 N. Deo Now to prove the implication for <τ<2k + 2, it is sufficient to assume it for τ m 1,m) and prove if for τ m, m +1), m =1, 2, 3,...,2k + 1). Since the result holds for τ m 1,m), we choose two points x 1,y 1 in such a way that a 1 <x 1 <a <b <y 1 <b 1. Then in view of assumption i) ii) for the interval m 1,m) and equivalence of ii) it follows that f m 1) exists and belongs to the class Lip1 δ, x 1,y 1 ) for any δ>. Let g C be such that gx) =1on[a,b ]andsupp g [a,b ]. Then with χ 2 t) denoting the characteristic function of the interval [x 1,y 1 ], we have Vn f,g,k,.) fg Vn C[a gx)ft) fx)),k,.) +,b ] C[a,b ] + Vn ft)gt) gx))χt),k,.) C[a,b ] + O n k+1)). 3.15) Now Vn gx)ft) fx)),k,.) gc[a,b ] Vn f,k,.) f C[a,b ] C[a1,b 1] = O n τ/2). 3.16) Applying Taylor s expansion of f, wehave I 3 Vn ft)gt) gx))χt),k,.) = C[a,b ] [ { m 1 f m 1) ξ) f m 1) x) } V n i= f i) x) t x) i + i! m 1)! ] ) gt) gx))χt),k,. C[a,b ] where ξ lies between t and x. Since f m 1) Lip1 δ, x 1,y 1 ), f m 1) ξ) f m 1) x) K6 ξ x 1 δ K6 t x 1 δ, where K 6 is the Lip1 δ, x 1,y 1 ) constant for f m 1),wehave m 1 I 3 V f i) x) n t x) i gt) gx))χt),k,.) i= i! + K 6 g k m 1)! C[a,b ] j= Cj, k) C[a,b ] ) V dj,n t x m+1 δ χt),.) C[a,b ] = I 4 + I 5 say. 3.17) By Taylor s expansion of g and Lemma 2.6, we have I 4 = O n k+1)). 3.18) Also, by Hölder s expansion of g and Lemma 2 from [1], we have K 6 I 5 k g m 1)! C[a,b ] Cj, k) ) j= y1 max W dj,nt x) t x m+1 δ dt j k K 7 max = O x 1 y1 j k x 1 n m+1 δ)/2) = O W dj,nt x)t x) 2m+1) dt C[a,b ] m+1 δ) 2m+1) C[a,b ] n τ/2), 3.19)

11 Theorems for Szâsz-Lupas type operators 29 by choosing δ such that < δ < m + 1 δ. Combining the estimates ), we get V n fg,k,.) fg C[a n = O τ/2).,b ] This completes the proof of the Theorem 3.2. Acknowledgment. The author is extremely thankful to the referee for his valuable comments and suggestions, leading to a better presentation of the paper. The author is also thankful to Professor Mila Mršević for special attention. REFERENCES [1] Agrawal, P. N. and Thamer, K. J., Approximation of unbounded functions by a new sequence of linear positive operators, J. Math. Anal. Appl., ), [2] Agrawal,P.N.andThamer,K.J.,Degree of approximation by a new sequence of linear operators, KyungpookMath. J., 41 21), [3] Berens, H. and Lorentz, G. G., Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J., ), [4] Derriennic, M. M., Sur l approximation de functions integrable sur [, 1] par des polynomes de Bernstein modifies, J. Approx. Theory, ), [5] Goldberg, S. and Meir, V., Minimum moduli of differentiable operators, Proc. London Math. Soc., ), [6] Gupta, V., AnoteonmodifiedSzâsz operators, Bull Inst. Math. Academia Sinica, 21 3) 1993), [7] Gupta, V. and Srivastava, G. S., Simultaneous approximation by Baskakov-Szâsz type operators, Bull. Math. Dela Soc. Sci Math de Roumanie N. S.), 37 85) 1993), [8] Gupta, V. and Srivastava, G. S., On the convergence of derivative by Szâsz-Mirakyan Baskakov type operators, The Math. Student, ) 1995), [9] Kasana, H. S., Prasad, G., Agrawal, P. N. and Sahai, A., Modified Szâsz operators,conference on Mathematica Analysis and its Applications, Kuwait, Pergamon Press, Oxford, 1985, [1] Mazhar, S. M. and Totil, V., Approximation by modified Szâsz operators, Acta Sci. Math. Szeged), ), [11] Sahai, A. and Prasad, G., On simultaneous approximation by modified Lupas operators, J. Approx. Theory, ), [12] Timan,A.F.,Theory of Approximation of Functions of Real Variable English translation), Pergamon Press Long Island City, N. Y., received , in revised form ) Department of Applied Mathematics, Delhi College of Engineering, Bawana Road, Delhi-1142, India. Presently at: School of Information Science and Engineering, Graduate School of the Chinese Academy of Sciences, No.3 Nanyitiao, Zhongguancun Haidian District, Beijing-18, P. R. China. dr naokant deo@yahoo.com