REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 5, Número 1, 29, Páginas 31 4 SIMULTANEOUS APPROIMATION BY A NEW SEQUENCE OF SZÃSZ BETA TYPE OPERATORS ALI J. MOHAMMAD AND AMAL K. HASSAN Abstract. In this paper, we study some direct results in simultaneous approximation for a new sequence of linear positive operators M n(f(t); x) of Szãsz Beta type operators. First, we establish the basic pointwise convergence theorem and then proceed to discuss the Voronovaskaja-type asymptotic formula. Finally, we obtain an error estimate in terms of modulus of continuity of the function being approximated. 1. INTRODUCTION In [3] Gupta and others studied some direct results in simultaneous approximation for the sequence: B n (f(t); x) = q n,k (x) b n,k (t)f(t)dt, k= where x, t [, ), q n,k (x) = e nx (nx) k k! and b n,k (t) = Γ(n+k+1) Γ(n)Γ(k+1) tk (1 + t) (n+k+1). After that, Agrawal and Thamer [1] applied the technique of linear combination introduced by May [4] and Rathore [5] for the sequence B n (f(t); x). Recently, Gupta and Lupas [2] studied some direct results for a sequence of mixed Beta Szãsz type operator defined as L n (f(t); x) = b n,k (x) q n,k 1 (t)f(t) dt + (1 + x) n 1 f(). In this paper, we introduce a new sequence of linear positive operators M n (f(t); x) of Szãsz Beta type operators to approximate a function f(x) belongs to the space C α [, ) = {f C[, ) : f(t) C(1 + t) α for some C >, α > }, as follows: M n (f(t); x) = q n,k (x) b n,k 1 (t)f(t) dt + e nx f(), (1.1) We may also write the operator (1.1) as M n (f(t); x) = W n (t, x)f(t)dt where W n (t, x) = q n,k (x) b n,k 1 (t) + e nx δ(t), δ(t) being the Dirac-delta function. Key words and phrases. Linear positive operators, Simultaneous approximation, Voronovaskaja-type asymptotic formula, Degree of approximation, Modulus of continuity. 31
32 ALI J. MOHAMMAD AND AMAL K. HASSAN The space C α [, ) is normed by f Cα = sup f(t) (1 + t) α. t< There are many sequences of linear positive operators with are approximate the space C α [, ). All of them (in general) have the same order of approximation O(n 1 ) [6]. So, to know what is the different between our sequence and the other sequences, we need to check that by using the computer. This object is outside our study in this paper. Throughout this paper, we assume that C denotes a positive constant not necessarily the same at all occurrences, and [β] denotes the integer part of β. 2. PRELIMINARY RESULTS For f C [, ) the Szãsz operators are defined as S n (f; x) = q n,k (x) f ( k n), x [, ) and for m N (the set of nonnegative integers), the m-th order moment of the Szãsz operators is defined as µ n,m (x) = q n,k (x) ( k n x) m. LEMMA 2.1. [3] For m N, the function µ n,m (x) defined above, has the following properties: (i) µ n, (x) = 1, µ n,1 (x) =, and the recurrence relation is k= nµ n,m+1 (x) = x ( µ n,m (x) + mµ n,m 1(x) ), m 1; (ii) µ n,m (x) is a polynomial in x of degree at most [m/2]; (iii) For everyx [, ),µ n,m (x) = O ( n [(m+1)/2]). From above lemma, we get q n,k (x)(k nx) 2j = n 2j ( µ n,2j (x) ( x) 2j e nx) (2.1) = n 2j { O(n j ) + O(n s ) } (for any s > ) = O(n j ) (if s j). For m N, the m-th order moment T n,m (x) for the operators (1.1) is defined as: T n,m (x) = M n ((t x) m ; x) = q n,k (x) b n,k 1 (t)(t x) m dt + ( x) m e nx. LEMMA 2.2.For the function T n,m (x), we have T n, (x) = 1, T n,1 (x) = T n,2 (x) = nx2 +2nx+2x 2 (n 1)(n 2) and there holds the recurrence relation: x n 1, (n m 1)T n,m+1 (x) = xt n,m (x) + ((2x + 1)m + x) T n,m(x) (2.2) Further, we have the following consequences of T n,m (x): + mx(x + 2)T n,m 1 (x), n > m + 1. Rev. Un. Mat. Argentina, Vol 5-1
SIMULTANEOUS APPROIMATION BY A NEW SEQUENCE 33 (i) T n,m (x) is a polynomial in x of degree exactly m; (ii) For every x [, ), T n,m (x) = O ( n [(m+1)/2]). Proof: By direct computation, we have T n, (x) = 1, T n,1 (x) = T n,2 (x) = nx2 +2nx+2x 2 (n 1)(n 2) x (, ), we have T n,m(x) = q n,k(x) x n 1 and. Next, we prove (2.2). For x = it clearly holds. For b n,k 1 (t)(t x) m dt n ( x) m e nx mt n,m 1 (x). Using the relations xq n,k (x) = (k nx)q n,k(x) and t (1+t)b n,k (t) = (k (n + 1)t) b n,k (t), we get: xt n,m(x) = = (k nx) q n,k (x) q n,k (x) b n,k 1 (t)(t x) m dt + n( x) m+1 e nx mxt n,m 1(x) t(1 + t) b n,k 1(t)(t x) m dt + (n + 1)T n,m+1(x) ( x) m+1 e nx + (x + 1)T n,m(x) (x + 1)( x) m e nx mxt n,m 1(x). By using the identity t(1 + t) = (t x) 2 + (1 + 2x)(t x) + x(1 + x), we have xt n,m (x) = q n,k (x) b n,k 1 (t) (t x)m+2 dt + (1 + 2x) q n,k (x) b n,k 1 (t)(t x)m+1 dt Integrating by parts, we get + x(1 + x) q n,k (x) b n,k 1 (t)(t x)m dt + (n + 1)T n,m+1 (x) + (1 + x)t n,m (x) mxt n,m 1 (x) ( x) m e nx. xt n,m (x) = (n m 1)T n,m+1(x) (m+x+2mx)t n,m (x) mx(x+2)t n,m 1 (x) from which (2.2) is immediate. From the values of T n, (x) and T n,1 (x), it is clear that the consequences (i) and (ii) hold for m = and m = 1. By using (2.2) and the induction on m the proof of consequences (i) and (ii) follows, hence the details are omitted. From the above lemma, we have q n,k (x) b n,k 1 (t)(t x) 2r dt = T n,2r ( x) 2r e nx (2.3) = O(n r ) + O(n s ) (for any s > ) = O(n r ) (if s r). Rev. Un. Mat. Argentina, Vol 5-1
34 ALI J. MOHAMMAD AND AMAL K. HASSAN LEMMA 2.3. Let δand γ be any two positive real numbers and [a, b] (, ). Then, for any s >, we have W n (t, x)t γ dt = O(n s ). C[a,b] Making use of Schwarz inequality for integration and then for summation and (2.3), the proof of the lemma easily follows. LEMMA 2.4. [3] There exist polynomials Q i,j,r (x) independent of n and k such that x r D r (q n,k (x)) = n i (k nx) j Q i,j,r (x)q n,k (x), where D = d dx. i, j 3. MAIN RESULTS Firstly, we show that the derivative M (r) n (f(t); x) is an approximation process for f (r) (x), r = 1, 2,.... Theorem 3.1. If r N, f C α [, ) for some α > and f (r) exists at a point x (, ), then lim n M(r) n (f(t); x) = f (r) (x). (3.1) Further, if f (r) exists and is continuous on (a η, b + η) (, ), η >, then (3.1) holds uniformly in [a, b]. Proof: By Taylor s expansion of f, we have r f(t) = (t x) i + ε(t, x)(t x) r, i= where, ε(t, x) as t x. Hence M (r) n (f(t); x) = r i= := I 1 + I 2. W (r) n (t, x)(t x) i dt + W n (r) (t, x)ε(t, x)(t x) r dt Now, using Lemma 2.2 we get that M n (t m ; x)is a polynomial in x of degree exactly, for all m N. Further, we can write it as: M n(t m ; x) = Therefore, (n m 1)! nm x m + (n 1)! (n m 1)! nm 1 m(m 1) x m 1 + O(n 2 ). (3.2) (n 1)! Rev. Un. Mat. Argentina, Vol 5-1
SIMULTANEOUS APPROIMATION BY A NEW SEQUENCE 35 I 1 = r i= = f(r) (x) r! i j= i j «( x) i j «(n r 1)! n r r! W n (r) (t, x)t j dt = f (r) (x) «(n r 1)! n r f (r) (x) as n. Next, making use of Lemma 2.4 we have I 2 i, j := I 3 + I 4. n i Q i,j,r(x) x r q n,k (x) k nx j b n,k 1 (t) ε(t, x) t x r dt + (nx) r e nx ε(, x) Since ε(t, x) as t x, then for a given ε >, there exists a δ > such that ε(t, x) < ε, whenever < t x < δ. For t x δ, there exists a constant C > such that ε(t, x)(t x) r C t x γ. Now, since sup i, j Q i,j,r(x) x r := M(x) = C x (, ). Hence, I 3 C := I 5 + I 6. i, j n i q n,k (x) k nx j t x <δ + b n,k 1 (t)ε t x r dt b n,k 1 (t) t x γ dt Now, applying Schwartz inequality for integration and then for summation, (2.1) and (2.3) we led to Rev. Un. Mat. Argentina, Vol 5-1
36 ALI J. MOHAMMAD AND AMAL K. HASSAN I 5 ε C ε C i, j i, j ε C O(n r/2 ) 1 n i q n,k (x) k nx j @ b n,k 1 (t) dta1/2 11/2 @ b n,k 1 (t)(t x) 2r dta n i i, j! 1/2 q n,k (x)(k nx) 2j @ n i O(n j/2 ) = ε O(1). (since b n,k 1 (t) dt = 1) 11/2 q n,k (x) b n,k 1 (t)(t x) 2r dta Again using Schwarz inequality for integration and then for summation, in view of (2.1) and Lemma 2.3, we have I 6 C C C i, j i, j i, j O(n s ) n i q n,k (x) k nx j Z b n,k 1 (t) t x γ dt 1 1/2 Z n i q n,k (x) k nx j @ b n,k 1 (t) dta B @ n i i, j = O n r/2 s = o(1) 11/2 b n,k 1 (t)(t x) 2γ C dta 1! 1/2 1/2 q n,k (x)(k nx) 2j B Z @ q n,k (x) b n,k 1 (t)(t x) 2γ C dta n i O(n j/2 ) (for any s > ) (for s > r/2). Now, since ε > is arbitrary, it follows that I 3 = o(1). Also, I 4 as n and hence I 2 = o(1), combining the estimates of I 1 and I 2, we obtain (3.1). To prove the uniformity assertion, it sufficient to remark that δ(ε) in above proof can be chosen to be independent of x [a, b] and also that the other estimates holds uniformly in [a, b]. Our next theorem is a Voronovaskaja-type asymptotic formula for the operators M n (r) (f(t); x), r = 1, 2,.... Rev. Un. Mat. Argentina, Vol 5-1
SIMULTANEOUS APPROIMATION BY A NEW SEQUENCE 37 THEOREM 3.2. Let f C α [, ) for some α >. If f (r+2) exists at a point x (, ), then lim n n M n (r) (f(t); x) f (r) (x) = r(r + 1) f (r) (x) + ((r + 1)x + r)f (r+1) (x) (3.3) 2 + 1 2 x(x + 2) f(r+2) (x). Further, if f (r+2) exists and is continuous on the interval (a η, b + η) (, ), η >, then (3.3) holds uniformly on [a, b]. I 1 = Proof: By the Taylor s expansion of f(t), we get r+2 M n (r) (f(t); x) = i= := I 1 + I 2, M (r) n where ε(t, x) as t x. By Lemma 2.2 and (3.2), we have r+2 i=r = f(r) (x) r! i j=r i j «( x) i j M (r) n (t j ; x) M n (r) (t r ; x) + f(r+1) (x) (r + 1)! + f(r+2) (x) (r + 2)(r + 1) (r + 2)! 2 «= f (r) (n r 1)!n r (x) (n 1)! x 2 M (r) n ( (t x) i ; x ) + M n (r) ( ε(t, x)(t x) r+2 ; x ) (r + 1)( x)m (r) n (t r ; x) + M n (r) (t r+1 ; x) (t r ; x) + (r + 2)( x)m (r) (t r+1 ; x) + M (r) n «n (t r+2 ; x) j «+ f(r+1) (x) (n r 1)!n r (r + 1)( x) r! (r + 1) «ff (n r 2)!n r+1 (n r 2)!nr + (r + 1)!x + (r + 1)r r! j «(r + 1)(r + 2) (n r 1)!n x 2 r r! 2 (n 1)! «««(n r 2)!n r+1 (n r 2)!n r + (r + 2)( x) (r + 1)!x + (r + 1)!r «ff + O(n 2 ). + f(r+2) (x) (r + 2)! (n r 3)!n r+2 +. (r + 2)! x 2 + 2 (n r 3)!nr+1 (r + 2)(r + 1)(r + 1)!x Hence in order to prove (3.3) it suffices to show that ni 2 as n, which follows on proceeding along the lines of proof of I 2 as n in Theorem 3.1. The uniformity assertion follows as in the proof of Theorem 3.1. Finally, we present a theorem which gives as an estimate of the degree of approximation by M n (r) (.; x) for smooth functions. Rev. Un. Mat. Argentina, Vol 5-1
38 ALI J. MOHAMMAD AND AMAL K. HASSAN THEOREM 3.3. Let f C α [, ) for some α > and r q r + 2. If f (q) exists and is continuous on (a η, b+η) (, ), η >, then for sufficiently large n, M n (r) (f(t); x) f (r) (x) C 1n 1 C[a,b] q f (i) C[a,b] +C 2 n 1/2 ω f (q) i=r n 1/2 +O(n 2 ) where C 1, C 2 are constants independent of f and n, ω f (δ) is the modulus of continuity of f on (a η, b + η), and. C[a,b] denotes the sup-norm on [a, b]. Proof. By Taylor s expansion of f, we have q f (t) = (t x) i + f(q) (ξ) f (q) (x) (t x) q χ(t) + h(t, x)(1 χ(t)), i= where ξ lies between t, x, and χ(t) is the characteristic function of the interval (a η, b + η). Now, M (r) n (f(t); x) f (r) (x) = + @ q i= W (r) n j ff W n (r) f (q) (ξ) f (q) (x) (t, x) (t x) q χ(t) dt + := I 1 + I 2 + I 3. 1 (t,x)(t x) i dt f (r) (x) A W n (r) (t, x)h(t, x)(1 χ(t)) dt By using Lemma 2.2 and (3.2), we get I 1 = q i=r Consequently, i j=r i j I 1 C[a,b] C 1 n 1 ( q i=r «( x) i j d r (n j 1)! n j dx r (n 1)! «(n j 1)! nj 1 + j(j 1)x j 1 + O(n 2 ) f (r) (x). (n 1)! f (i) C[a,b] ) + O(n 2 ), uniformly on [a, b]. To estimate I 2 we proceed as follows: { I 2 W n (r) f (q) (t, x) (ξ) f (q) (x) } t x q χ(t) dt ω f (q)(δ) W n (r) (t, x) ( 1 + x j ) t x t x q dt δ Rev. Un. Mat. Argentina, Vol 5-1
SIMULTANEOUS APPROIMATION BY A NEW SEQUENCE 39 ω f (q)(δ) [ q (r) n,k (x) b n,k 1 (t) + ( n) r e nx (x q + δ 1 x q+1 ) ( t x q + δ 1 t x q+1 ) dt ], δ >. Now, for s =, 1, 2,..., using Schwartz inequality for integration and then for summation, (2.1) and (2.3), we have 8 < 11/2 q n,k (x) k nx j b n,k 1 (t) t x s dt q n,k (x) k nx j @ b : n,k 1 (t)dta (3.4) 1 9 1/2> = @ q n,k 1 (t)(t x) 2s dta >;! 1/2 11/2 q n,k (x)(k nx) 2j @ q n,k (x) b n,k 1 (t)(t x) 2s dta = O(n j/2 )O(n s/2 ) = O(n (j s)/2 ), uniformly on[a, b]. Therefore, by Lemma 2.4 and (3.4), we get Z q (r) n,k (x) b n,k 1 (t) t x s dt B @ = C sup i, j i, j sup x [a,b] 1 Q i,j,r(x) x r C A i, j i, j n i k nx j Q i,j,r(x) q n,k (x) (3.5) x r n i @ b n,k 1 (t) t x s dt 1 q n,k (x) k nx j b n,k 1 (t) t x s dt A n i O(n (j s)/2 ) = O(n (r s)/2 ), uniformly on [a, b]. (since sup i, j sup x [a,b] Choosing δ = n 1/2 and applying (3.5), we are led to Q i,j,r (x) xr := M(x) but fixed ) I 2 C[a,b] ω `n 1/2 h i f (q) O(n (r q)/2 ) + n 1/2 O(n (r q 1)/2 ) + O(n m ),(for any m > ) C 2 n (r q)/2 ω f (q) n 1/2. Rev. Un. Mat. Argentina, Vol 5-1
4 ALI J. MOHAMMAD AND AMAL K. HASSAN Since t [, ) \ (a η, b+η), we can choose δ > in such a way that t x δ for all x [a, b]. Thus, by Lemmas 2.3 and 2.4, we obtain I 3 i, j n i k nx j Q i,j,r(x) x r q n,k (x) Z b n,k 1 (t) h(t, x) dt + ( n) r e nx h(, x). For t x δ, we can find a constant C such that h(t, x) C t x α. Hence, using Schwarz inequality for integration and then for summation,(2.1), (2.3), it easily follows that I 3 = O(n s ) for any s >, uniformly on [a, b]. Combining the estimates of I 1, I 2 I 3, the required result is immediate. References [1] P.N. Agrawal and Kareem J. Thamer. Linear combinations of Szãsz-Baskakov type operators, Demonstratio Math.,32(3) (1999), 575-58. [2] Vijay Gupta and Alexandru Lupas. Direct results for mixed Beta-Szãsztype operators, General Mathematics 13(2), (25), 83-94. [3] Vijay Gupta, G.S. Servastava and A. Shahai. On simultaneous approximation by Szãsz-Beta operators, Soochow J. Math. 21, (1995), 1-11. [4] C.P. May. Saturation and inverse theorems for combinations of a class of exponential- type operators, Canad. J. Math. 28 (1976), 1224-125. [5] R.K.S. Rathore. Linear Combinations of Linear Positive Operators and Generating Relations in Special Functions, Ph.D. Thesis, I.I.T. Delhi (India) 1973. [6] E. Voronovskaja. Détermination de la forme asymptotique d ápproximation des fonctions par les polynômes de S.N. Bernstein, C.R. Adad. Sci. USSR (1932), 79-85. Ali J. Mohammad University of Basrah, College of Education, Dept of Mathematics, Basrah, IRAQ. alijasmoh@yahoo.com Amal K. Hassan University of Basrah, College of Science, Dept of Mathematics, Basrah, IRAQ. Recibido: 8 de noviembre de 26 Aceptado: 11 de marzo de 28 Rev. Un. Mat. Argentina, Vol 5-1