MATEMATIQKI VESNIK 58 26), 19 29 UDK 517.984 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 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.
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
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].
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].
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.
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.
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
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/2. 3.13) 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 3.12 3.14), we obtain Vn fg,k,.) fg C[a,b ] = O n τ/2), for <τ 1. ).
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)
Theorems for Szâsz-Lupas type operators 29 by choosing δ such that < δ < m + 1 δ. Combining the estimates 3.15 3.19), 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., 225 1998), 66 672. [2] Agrawal,P.N.andThamer,K.J.,Degree of approximation by a new sequence of linear operators, KyungpookMath. J., 41 21), 65 73. [3] Berens, H. and Lorentz, G. G., Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J., 21 1972), 693 78. [4] Derriennic, M. M., Sur l approximation de functions integrable sur [, 1] par des polynomes de Bernstein modifies, J. Approx. Theory, 31 1981), 325 343. [5] Goldberg, S. and Meir, V., Minimum moduli of differentiable operators, Proc. London Math. Soc., 23 1971), 1 15. [6] Gupta, V., AnoteonmodifiedSzâsz operators, Bull Inst. Math. Academia Sinica, 21 3) 1993), 275 278. [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), 73 85. [8] Gupta, V. and Srivastava, G. S., On the convergence of derivative by Szâsz-Mirakyan Baskakov type operators, The Math. Student, 64 1 4) 1995), 195 25. [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, 29 41. [1] Mazhar, S. M. and Totil, V., Approximation by modified Szâsz operators, Acta Sci. Math. Szeged), 49 1985), 257 269. [11] Sahai, A. and Prasad, G., On simultaneous approximation by modified Lupas operators, J. Approx. Theory, 45 1985), 122 128. [12] Timan,A.F.,Theory of Approximation of Functions of Real Variable English translation), Pergamon Press Long Island City, N. Y., 1963. received 7.4.24, in revised form 29.1.26) 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. E-mail: dr naokant deo@yahoo.com