Simultaneous approximation by a new sequence of Szãsz Beta type operators

0
1
10
3 weeks ago
PDF Preview
Full text
(1)REVISTA DE LA ´ MATEMATICA ´ UNION ARGENTINA Volumen 50, N´ umero 1, 2009, P´ aginas 31–40. SIMULTANEOUS APPROXIMATION BY A NEW SEQUENCE ˜ OF SZASZ–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 Mn (f (t); x) of Sz˜ asz–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: Z∞ ∞ X bn,k (t) f (t) dt, Bn (f (t); x) = qn,k (x) k=0. −nx. 0. k. Γ(n+k+1) k t (1 + t)−(n+k+1) . where x, t ∈ [0, ∞), qn,k (x) = e k!(nx) and bn,k (t) = Γ(n)Γ(k+1) After that, Agrawal and Thamer [1] applied the technique of linear combination introduced by May [4] and Rathore [5] for the sequence Bn (f (t); x). Recently, Gupta and Lupas [2] studied some direct results for a sequence of mixed Beta– ∞ R∞ P bn,k (x) qn,k−1 (t)f (t) dt + (1 + Sz˜asz type operator defined as Ln (f (t); x) = k=1. 0. x)−n−1 f (0). In this paper, we introduce a new sequence of linear positive operators Mn (f (t); x) of Sz˜asz–Beta type operators to approximate a function f (x) belongs to the space Cα [0, ∞) = {f ∈ C[0, ∞) : |f (t)| ≤ C(1 + t)α for some C > 0, α > 0}, as follows: Mn (f (t); x) =. ∞ X. k=1. qn,k (x). Z∞. bn,k−1 (t)f (t) dt + e−nx f (0),. We may also write the operator (1.1) as Mn (f (t); x) = Wn (t, x) =. ∞ P. (1.1). 0. R∞. Wn (t, x)f (t) dt where. 0. qn,k (x) bn,k−1 (t) + e−nx δ(t), δ(t) being the Dirac-delta function.. k=1. Key words and phrases. Linear positive operators, Simultaneous approximation, Voronovaskaja-type asymptotic formula, Degree of approximation, Modulus of continuity.. 31.

(2) 32. ALI J. MOHAMMAD AND AMAL K. HASSAN. The space Cα [0, ∞) is normed by kf kCα = sup |f (t) | (1 + t)−α . 0≤t<∞. There are many sequences of linear positive operators with are approximate the space Cα [0, ∞). 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 [0, ∞) the Sz˜asz operators are defined as Sn (f ; x) =. ∞ P. qn,k (x) f. k=1. k n.  ,. x ∈ [0, ∞) and for m ∈ N 0 (the set of nonnegative integers), the m-th order moment ∞ m P qn,k (x) nk − x . of the Sz˜asz operators is defined as µn,m (x) = k=0. LEMMA 2.1. [3] For m ∈ N 0 , the function µn,m (x) defined above, has the following properties: (i) µn,0 (x) = 1 , µn,1 (x) = 0 , and the recurrence relation is  nµn,m+1 (x) = x µ′n,m (x) + mµn,m−1 (x) , m ≥ 1; (ii) µn,m (x) is a polynomial in x of degree at most [m/2]; (iii) For everyx ∈ [0, ∞),µn,m (x) = O n−[(m+1)/2] . From above lemma, we get ∞ X. qn,k (x)(k − nx)2j = n2j µn,2j (x) − (−x)2j e−nx. k=1. = n2j. . O(n−j ) + O(n−s ). = O(nj ). . (2.1). (for any s > 0) (if s ≥ j).. 0. For m ∈ N , the m-th order moment Tn,m (x) for the operators (1.1) is defined as: m. Tn,m (x) = Mn ((t − x) ; x) =. ∞ X. k=1. qn,k (x). Z∞. bn,k−1 (t)(t − x)m dt + (−x)m e−nx .. 0. LEMMA 2.2.For the function Tn,m (x), we have Tn,0 (x) = 1 , Tn,1 (x) = Tn,2 (x) =. 2. nx +2nx+2x (n−1)(n−2). 2. x n−1 ,. and there holds the recurrence relation:. ′ (n − m − 1)Tn,m+1 (x) = xTn,m (x) + ((2x + 1)m + x) Tn,m (x). + mx(x + 2)Tn,m−1 (x), Further, we have the following consequences of Tn,m (x): Rev. Un. Mat. Argentina, Vol 50-1. (2.2) n > m + 1..

(3) 33. SIMULTANEOUS APPROXIMATION BY A NEW SEQUENCE. (i) (ii). Tn,m (x) is a polynomial in x of degree exactly m;  For every x ∈ [0, ∞) , Tn,m (x) = O n−[(m+1)/2] .. Proof: By direct computation, we have Tn,0 (x) = 1, Tn,1 (x) = 2. nx +2nx+2x (n−1)(n−2). Tn,2 (x) = x ∈ (0, ∞), we have ′ Tn,m (x). =. ∞ X. 2. x n−1. and. . Next, we prove (2.2). For x = 0 it clearly holds. For. ′ qn,k (x). k=1. Z∞. bn,k−1 (t)(t − x)m dt − n (−x)m e−nx − mTn,m−1 (x).. 0. ′ Using the relations xqn,k (x) = (k−nx) qn,k (x) and t (1+t) b′n,k (t) = (k − (n + 1) t) × bn,k (t), we get: ′ xTn,m (x). =. ∞ X. (k − nx) qn,k (x). k=1. =. ∞ X. Z∞. bn,k− 1 (t)(t − x)m dt + n(−x)m+1 e−nx − mxTn,m−1 (x). 0. qn,k (x). k=1. Z∞. t(1 + t) b′n,k−1 (t)(t − x)m dt + (n + 1)Tn,m+1 (x) − (−x)m+1 e−nx. 0. + (x + 1)Tn,m (x) − (x + 1)(−x)m e−nx − mxTn,m−1 (x).. By using the identity t(1 + t) = (t − x)2 + (1 + 2x)(t − x) + x(1 + x), we have ′ xTn,m (x) =. ∞ X. qn,k (x). k=1. Z∞. b′n,k−1 (t) (t − x)m+2 dt + (1 + 2x). ∞ X. qn,k (x). k=1. 0. + x(1 + x). ∞ X. qn,k (x). k=1. Z∞. Z∞. b′n,k−1 (t)(t − x)m+1 dt. 0. b′n,k−1 (t)(t − x)m dt + (n + 1)Tn,m+1 (x). 0. + (1 + x)Tn,m (x) − mxTn,m−1 (x) − (−x)m e−nx .. Integrating by parts, we get ′ (x) = (n − m − 1) Tn,m+1 (x) − (m + x + 2mx) Tn,m (x) − mx(x + 2) Tn,m−1 (x) xTn,m. from which (2.2) is immediate. From the values of Tn,0 (x) and Tn,1 (x), it is clear that the consequences (i) and (ii) hold for m = 0 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 ∞ X. k=1. qn,k (x). Z∞. bn,k−1 (t)(t − x)2r dt = Tn,2r − (−x)2r e−nx. (2.3). 0. = O(n−r ) + O(n−s ) = O(n. −r. ). (for any s > 0) (if s ≥ r).. Rev. Un. Mat. Argentina, Vol 50-1.

(4) 34. ALI J. MOHAMMAD AND AMAL K. HASSAN. LEMMA 2.3. Let δand γ be any two positive real numbers and [a, b] ⊂ (0, ∞). Then, for any s > 0, we have. Z. = O(n−s ). Wn (t, x) tγ dt . |t−x|≥δ 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 Qi,j,r (x) independent of n and k such that P d . ni (k − nx)j Qi,j,r (x) qn,k (x), where D = dx xr Dr (qn,k (x)) = 2i + j ≤ r i, j ≥ 0 3. MAIN RESULTS (r). Firstly, we show that the derivative Mn (f (t); x) is an approximation process for f (r) (x) , r = 1 , 2 , . . .. Theorem 3.1. If r ∈ N , f ∈ Cα [0, ∞) for some α > 0 and f (r) exists at a point x ∈ (0, ∞), then lim Mn(r) (f (t); x) = f (r) (x).. n→∞. (3.1). Further, if f (r) exists and is continuous on (a − η, b + η) ⊂ (0, ∞) , η > 0, then (3.1) holds uniformly in [a, b]. Proof: By Taylor’s expansion of f , we have f (t) =. r X f (i) (x) i=0. i!. (t − x)i + ε(t, x)(t − x)r ,. where, ε(t, x) → 0 as t → x. Hence ∞. Mn(r). (f (t); x) =. Z r X f (i) (x) i=0. i!. 0. Wn(r) (t, x)(t. i. − x) dt +. Z∞. Wn(r) (t, x) ε(t, x) (t − x)r dt. 0. := I1 + I2 .. Now, using Lemma 2.2 we get that Mn (tm ; x)is a polynomial in x of degree exactly , for all m ∈ N 0 . Further , we can write it as: Mn (tm ; x) =. (n − m − 1)! nm m (n − m − 1)! nm−1 x + m(m − 1) xm−1 + O(n−2 ). (3.2) (n − 1)! (n − 1)!. Therefore, Rev. Un. Mat. Argentina, Vol 50-1.

(5) 35. SIMULTANEOUS APPROXIMATION BY A NEW SEQUENCE. « Z∞ i „ r X f (i) (x) X i (−x)i−j Wn(r) (t, x) tj dt j i! j=0 i=0 0 « „ « „ (r) f (x) (n − r − 1)! r (n − r − 1)! r = n r! = f (r) (x) n → f (r) (x) as n → ∞. r! (n − 1) ! (n − 1) !. I1 =. Next, making use of Lemma 2.4 we have. |I2 | ≤. X. 2i + j ≤ r i, j ≥ 0. |Qi,j,r (x)| ni xr. ∞ X. j. qn,k (x) |k − nx|. k=1. Z∞. bn,k−1 (t) |ε(t, x)| |t − x|r dt. 0. + (nx)r e−nx |ε(0, x)| := I3 + I4 .. Since ε(t, x) → 0 as t → x, then for a given ε > 0, there exists a δ > 0 such that |ε(t, x)| < ε, whenever 0 < |t − x| < δ. For |t − x| ≥ δ, there exists a constant C > 0 such that |ε(t, x)(t − x)r | ≤ C |t − x|γ . |Qi,j,r (x)| := M (x) = C ∀ x ∈ (0, ∞). Hence, Now, since sup xr 2i + j ≤ r i, j ≥ 0. I3 ≤ C. X. 2i + j ≤ r i, j ≥ 0. ni. ∞ X. k=1. . j qn,k (x) |k − nx| . Z. r. bn,k−1 (t) ε |t − x| dt. |t−x|<δ. +. Z. |t−x|≥δ. := I5 + I6 .. .  γ bn,k−1 (t) |t − x| dt. Now, applying Schwartz inequality for integration and then for summation, (2.1) and (2.3) we led to Rev. Un. Mat. Argentina, Vol 50-1.

(6) 36. ALI J. MOHAMMAD AND AMAL K. HASSAN. I5 ≤ ε C. X. n. i. ∞ X. k=1. 2i + j ≤ r i, j ≥ 0. j. 0. qn,k (x) |k − nx| @. Z∞ 0. 11/2 0. bn,k−1 (t) dtA. @. Z∞ 0. 11/2. 2r. bn,k−1 (t)(t − x) dtA. (since. Z∞. bn,k−1 (t) dt = 1). 0. ≤ εC. X. ni. ∞ X. qn,k (x)(k − nx)2j. k=1. 2i + j ≤ r i, j ≥ 0. X. ≤ ε C O(n−r/2 ). !1/2 0 @. ∞ X. qn,k (x). k=1. Z∞ 0. 11/2. bn,k−1 (t)(t − x)2r dtA. ni O(nj/2 ) = ε O(1).. 2i + j ≤ r i, j ≥ 0. Again using Schwarz inequality for integration and then for summation, in view of (2.1) and Lemma 2.3, we have. I6 ≤ C. X. ni. 2i + j ≤ r i, j ≥ 0. ≤C. X. ni. X. i. n. 2i + j ≤ r i, j ≥ 0 X −s ≤ O(n ). Z. qn,k (x) |k − nx|j. bn,k−1 (t) |t − x|γ dt. k=1. |t−x|≥δ. ∞ X. 11/2 0 0∞ Z B qn,k (x) |k − nx|j @ bn,k−1 (t) dtA @. k=1. 2i + j ≤ r i, j ≥ 0. ≤C. ∞ X. ∞ X. 0. 2j. qn,k (x)(k − nx). k=1. ni O(nj/2 ). 2i + j ≤ r i, j ≥ 0 “ ” = O nr/2−s = o(1). !1/2. Z. 11/2. |t−x|≥δ. 0. ∞. BX qn,k (x) @ k=1. Z. |t−x|≥δ. C bn,k−1 (t)(t − x)2γ dtA 2γ. bn,k−1 (t)(t − x). 11/2. C dtA. (for any s > 0). (for s > r/2).. Now, since ε > 0 is arbitrary, it follows that I3 = o(1). Also, I4 → 0 as n → ∞ and hence I2 = o(1), combining the estimates of I1 and I2 , 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 (r) Mn (f (t); x) , r = 1 , 2 , . . . . Rev. Un. Mat. Argentina, Vol 50-1.

(7) SIMULTANEOUS APPROXIMATION BY A NEW SEQUENCE. 37. THEOREM 3.2. Let f ∈ Cα [0, ∞) for some α > 0. If f (r+2) exists at a point x ∈ (0, ∞) , then “ ” r(r + 1) (r) f (x) + ((r + 1)x + r)f (r+1) (x) (3.3) lim n Mn(r) (f (t); x) − f (r) (x) = n→0 2 1 + x(x + 2) f (r+2) (x). 2 Further, if f (r+2) exists and is continuous on the interval (a − η, b + η) ⊂ (0, ∞), η > 0, then (3.3) holds uniformly on [a, b].. Proof: By the Taylor’s expansion of f (t), we get Mn(r) (f (t); x) =. r+2 (i) X f (x) i=0. i!. := I1 + I2 ,.   Mn(r) (t − x)i ; x + Mn(r) ε(t, x)(t − x)r+2 ; x. where ε(t, x) → 0as t → x. By Lemma 2.2 and (3.2), we have I1 =. « r+2 (i) i „ X f (x) X i (−x)i−j Mn(r) (tj ; x) j i! i=r j=r. ” f (r+1) (x) “ f (r) (x) (r) r (r + 1)(−x)Mn(r) (tr ; x) + Mn(r) (tr+1 ; x) Mn (t ; x) + r! (r + 1)! „ « (r+2) f (x) (r + 2)(r + 1) 2 (r) r + x Mn (t ; x) + (r + 2)(−x)Mn(r) (tr+1 ; x) + Mn(r) (tr+2 ; x) (r + 2)! 2 „ « (n − r − 1)!nr (r) = f (x) (n − 1)! «  „ f (r+1) (x) (n − r − 1) !nr r! + (r + 1)(−x) (r + 1) (n − 1) ! «ff „ r+1 (n − r − 2) !nr (n − r − 2) !n (r + 1)!x + (r + 1)r r ! + (n − 1) ! (n − 1) !  „ « f (r+2) (x) (r + 1)(r + 2) 2 (n − r − 1) !nr x r! + (r + 2)! 2 (n − 1)! „„ « „ « « (n − r − 2) !nr+1 (n − r − 2)!nr + (r + 2)(−x) (r + 1) !x + (r + 1) !r (n − 1) ! (n − 1) ! «ff „ (n − r − 3) !nr+2 (r + 2)! 2 (n − r − 3) !nr+1 . x + (r + 2)(r + 1)(r + 1) !x + O(n−2 ). + (n − 1) ! 2 (n − 1) ! =. Hence in order to prove (3.3) it suffices to show that nI2 → 0 as n → ∞, which follows on proceeding along the lines of proof of I2 → 0 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 ap(r) proximation by Mn (.; x) for smooth functions. Rev. Un. Mat. Argentina, Vol 50-1.

(8) 38. ALI J. MOHAMMAD AND AMAL K. HASSAN. THEOREM 3.3. Let f ∈ Cα [0, ∞) for some α > 0 and r ≤ q ≤ r + 2. If f (q) exists and is continuous on (a − η, b + η) ⊂ (0, ∞) , η > 0, then for sufficiently large n, ‚ ‚ ‚ (r) ‚ (r) ‚Mn (f (t); x) − f (x)‚. C[a,b]. ≤ C1 n−1. q ‚ ‚ X ‚ (i) ‚ ‚f ‚ i=r. C[a,b]. “ ” +C2 n−1/2 ωf (q) n−1/2 +O(n−2 ). where C1 , C2 are constants independent of f and n, ωf (δ) is the modulus of continuity of f on (a − η, b + η), and k . kC[a,b] denotes the sup-norm on [a, b] . Proof. By Taylor’s expansion of f , we have f (t) =. q X f (i) (x). i!. i=0. (t − x)i +. f (q) (ξ) − f (q) (x) (t − x)q χ(t) + h(t, x)(1 − χ(t)), q!. where ξ lies between t, x, and χ(t) is the characteristic function of the interval (a − η, b + η). Now, 0 1 Z∞ q (i) X f (x) Mn(r) (f (t); x) − f (r) (x) = @ Wn(r) (t, x)(t − x)i dt − f (r) (x)A i! i=0 0. +. Z∞. Wn(r) (t, x). . f. (q). (ξ) − f q!. (q). 0. (x). ff Z∞ (t − x) χ(t) dt + Wn(r) (t, x)h(t, x)(1 − χ(t)) dt q. 0. := I1 + I2 + I3 .. By using Lemma 2.2 and (3.2), we get. I1 =. „ « q i „ X (n − j − 1)! nj j dr f (i) (x) X i x (−x)i−j r j i! dx (n − 1)! j=r i=r +. « (n − j − 1)! nj−1 j(j − 1)xj−1 + O(n−2 ) − f (r) (x). (n − 1)!. Consequently,. kI1 kC[a,b] ≤ C1 n. −1. q . X (i) f i=r. C[a,b]. !. + O(n−2 ), uniformly on [a, b].. To estimate I2 we proceed as follows: ( ) Z∞

(9)

(10)

(11)

(12) f (q) (ξ) − f (q) (x)

(13)

(14)

(15) (r)

(16) q |I2 | ≤ |t − x| χ(t) dt

(17) Wn (t, x)

(18) q! 0. ωf (q) (δ) ≤ q!.  Z∞

(19)

(20)  |t − x|

(21) (r)

(22) q |t − x| dt W (t, x) 1 +

(23) n

(24) δ 0. Rev. Un. Mat. Argentina, Vol 50-1.

(25) SIMULTANEOUS APPROXIMATION BY A NEW SEQUENCE. 39. "∞

(26)

(27) Z∞   ωf (q) (δ) X

(28) (r)

(29) q q+1 dt ≤

(30) qn,k (x)

(31) bn,k−1 (t) |t − x| + δ −1 |t − x| q! k=1 0  + (−n)r e−nx (xq + δ −1 xq+1 ) , δ > 0.. Now, for s = 0 , 1 , 2 , . . . , using Schwartz inequality for integration and then for summation, (2.1) and (2.3), we have ∞ X. qn,k (x) |k − nx|j. k=1. Z∞. bn,k−1 (t) |t − x|s dt ≤. ∞ X. k=1. 0. 80 ∞ 1 1/2 < Z j @ qn,k (x) |k − nx| bn,k−1 (t)dtA : 0. (3.4) 11/2 9 0∞ > Z = ×@ qn,k−1 (t)(t − x)2s dtA > ; 0. 11/2 !1/2 0 ∞ Z∞ ∞ X X 2s A 2j @ qn,k (x)(k − nx) qn,k (x) bn,k−1 (t)(t − x) dt. ≤. k=1. = O(n. k=1. j/2. )O(n. −s/2. 0. ). = O(n(j−s)/2 ), uniformly on[a, b].. Therefore, by Lemma 2.4 and (3.4), we get ∞ ˛ ∞ ˛ Z∞ X X ˛ (r) ˛ bn,k−1 (t) |t − x|s dt ≤ ˛qn,k (x) ˛. k=1. k=1. 0. ni |k − nx|j. X. 2i + j ≤ r i, j ≥ 0. |Qi,j,r (x) | qn,k (x) xr. ×. Z∞. (3.5). bn,k−1 (t) | t − x|s dt. 0. 0. B B sup ≤B B @ 2i + j ≤ r i, j ≥ 0 =C. X. 1. sup x∈[a,b]. C |Qi,j,r (x)| C C C xr A. X. i. 2i + j ≤ r i, j ≥ 0. 0. n @. ∞ X. qn,k (x) |k − nx|. j. k=1. Z∞ 0. s. bn,k−1 (t) |t − x| dt A. ni O(n(j−s)/2 ) = O(n(r−s)/2 ), uniformly on [a, b] .. 2i + j ≤ r i, j ≥ 0 (since. sup 2i + j ≤ r i, j ≥ 0. sup x∈[a,b]. |Qi,j,r (x)| xr. := M(x) but fixed ). Choosing δ = n−1/2 and applying (3.5), we are led to ` ´ ωf (q) n−1/2 h. i O(n(r−q)/2 ) + n1/2 O(n(r−q−1)/2 ) + O(n−m ) , (for any m > 0) “ ” ≤ C2 n−(r−q)/2 ωf (q) n−1/2 .. kI2 kC[a,b] ≤. 1. q!. Rev. Un. Mat. Argentina, Vol 50-1.

(32) 40. ALI J. MOHAMMAD AND AMAL K. HASSAN. Since t ∈ [0, ∞) \ (a−η, b+η), we can choose δ > 0 in such a way that |t − x| ≥ δ for all x ∈ [a, b]. Thus, by Lemmas 2.3 and 2.4 , we obtain |I3 | ≤. ∞ X. k=1. X. ni |k − nx|j. 2i + j ≤ r i, j ≥ 0. |Qi,j,r (x)| qn,k (x) xr. Z. bn,k−1 (t) |h(t, x) | dt + (−n)r e−nx |h(0, x)| .. |t−x|≥δ. α. For |t − x| ≥ δ, we can find a constant C such that |h(t, x)| ≤ C |t − x| . Hence, using Schwarz inequality for integration and then for summation ,(2.1), (2.3), it easily follows that I3 = O(n−s ) for any s > 0, uniformly on [a, b]. Combining the estimates of I1 , I2 I3 , the required result is immediate. References [1] P.N. Agrawal and Kareem J. Thamer. Linear combinations of Sz˜ asz-Baskakov type operators, Demonstratio Math.,32(3) (1999), 575-580. [2] Vijay Gupta and Alexandru Lupas. Direct results for mixed Beta-Sz˜ asztype operators, General Mathematics 13(2), (2005), 83-94. [3] Vijay Gupta, G.S. Servastava and A. Shahai. On simultaneous approximation by Sz˜ asz-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), 1224-1250. [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. D´ etermination de la forme asymptotique d’´ approximation des fonctions par les polynˆ omes 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 2006 Aceptado: 11 de marzo de 2008. Rev. Un. Mat. Argentina, Vol 50-1.

(33)

Nuevo documento

Evolución estructural organizacional de la dirección general de comunicación social y relaciones públicas Caso: Universidad Autónoma de Nuevo León, México
La investigación se evalúa con el criterio de implicaciones prácticas, porque los hallazgos que se obtengan del estudio, aportarán datos prácticos que harán evidente las áreas de
Diseño de mobiliario con materiales reciclados
A continuación se va a calcular cuántos de cartón serían necesarios para la fabricación de la silla MOCA a fin de conocer un poco más exactamente el precio del material necesario:
Diseño de materiales para el desarrollo del pensamiento crítico en edades tempranas a través del área de ingles
DISEÑO / METODOLOGÍA Una vez analizado el diseño de materiales y el desarrollo del pensamiento crítico, se ha dado el paso hacia la práctica creando un material propio y original,
Diseño de materiales curriculares para una sección bilingüe
Sin embargo, como sabemos, los cambios en todo lo relacionado con las tecnologías son muy rápidos y si en 2003 Llorente veía dificultades en su uso, tan solo tres años después ya
Diseño de materiales curriculares para "Paseo por la alamenda de Cervantes" para la educación primaria
Diseño de materiales curriculares para: “Paseo por la Alameda de Cervantes Soria” para Educación Primaria” TERCER CICLO La zona posterior a la Fuente de Tres Caños será la destinada
Diseño de material multimedia como herramienta didáctica en la diabetes gestacional
De esta forma se consigue un material más visual, claro y práctico sobre la correcta forma de realizar las técnicas, además de mostrar a los profesionales de enfermería como los
Diseño de material educativo sobre la copa menstrual y esferas vaginales para mujeres jóvenes
Salud recomienda “iniciar la práctica de ejercicios de suelo pélvico para reducir el riesgo de incontinencia urinaria en el futuro, instruyendo a las mujeres sobre cómo realizar
Diseño de material didáctico en el marco de proyecto "Conozco y reconozco"
Es totalmente obvio que es necesario saber cuáles son las capacidades para la comprensión y representación de las nociones espaciales, temporales y sociales de un niño de 9 y 10 años
Diseño de los estabilizadores verticales y horizontales de un vehículo aéreo no tripulado (UAV)
The main purpose to have a bigger chord at the root respect the tip is to have a more stable structure reducing the bending stress at the root and this allow the vertical tail to have
"El diseño de las práctica en la Educación Artística de Primaria"
Esta conquista de la capacidad de reproducción figurativa, constituye por parte del niño un proceso de percepción mediante la identificación “fácil” de la realidad, esquemático se

Etiquetas

Documento similar

Tilted electric-field and hydrostatic pressure effects on donor impurity states in cylindrical GaAs quantum disks
Summing up, using the effective mass approximation and a variational procedure we report the effect of tilted applied electric field and hydrostatic pressure on noncorrelated electron
0
0
3
Fully dynamic and memory-adaptative spatial approximation trees
These data structures are hybrid schemes, with the full features of dynamic spatial approximation trees and able of using the available memory to improve the query time.. It has been
0
0
12
Is it always good to let universities select their students?
In particular, the assumptions guarantee that there will always be some type-B individuals attending universities, and that there is always the possibility of any given type of students
0
0
17
On the Use of Boundary Integral Equations and Linear Operators in Room Acoustics
Some general methods of solving linear operator and integral equations are reviewed and discussed, such as approximation methods and finite basis methods.. In addition, some of the
0
0
8
On the Krall-type discrete polynomials
In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal
0
0
18
Chapter 4. Probability Distribution. IT. I Quarter
The Poisson approximation to the Binomial distribution has mean λ=np where n is the number of trials and p is the probability of success for a Binomial distribution and the same
0
0
18
Bézier variant of modified Srivastava–Gupta operators
In this paper, we introduce the B´ ezier variant of the operators defined by Yadav and give a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness and the
0
0
16
Fractional type integral operators of variable order
Once the basic properties are established, the second step was to study the boundedness of the Hardy–Littlewood maximal operator and the fractional Hardy– Littlewood maximal operator on
0
0
16
An extension of best $L^2$ local approximation
In [4] the authors proved the existence of the best local approximation under weaker conditions, more precisely they assumed existence of L2 lateral derivatives of order q − 1 and L2
0
0
12
Best simultaneous approximation on small regions by rational functions
Now, following the same patterns of the proof of existence for best rational approximation to a single function see [11, Theorem 2.1], we can find a subsequence vr0 which P2 converges
0
0
14
Vector valued T(1) Theorem and Littlewood Paley theory on spaces of homogeneous type
Singular integral operators associated to kernels valued on Hilbert spaces are studied in the setting of spaces of homogeneous type.. By following the work of David and Journ´ e
0
0
29
Accurate approximation of a generalized Mathieu series
approximation, bounds, convergence acceleration, estimate, EulerMaclaurin, Hermite’s summation, inequality, Mathieu series.... In the recently published paper [13] further bounds for
0
0
10
Maximal operators associated with generalized Hermite polynomial and function expansions
We study the weak and strong type boundedness of maximal heat–diffusion operators associated with the system of generalized Hermite polynomials and with two different systems of
0
0
25
Approximation in weighted Lp spaces
Wheeden, Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform, Trans.. Ky, Moduli of mean smoothness and approximation with Ap -weights, Annales
0
0
13
Approximation and shape preserving properties of the truncated Baskakov operator of max product kind
In a similar manner, by Corollary 4.4 and by Remark 1 after Corollary 4.5, we can produce many subclasses of functions for which the order of uniform approximation given by the
0
0
19
Show more