arXiv:1510.03408v1 [math.CA] 7 Oct 2015
Weighted Approximation theorem for Choldowsky generalization of the q-Favard-Sz´ asz operators Preeti Sharma1,⋆ , Vishnu Narayan Mishra1,2,† Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat, Surat-395 007 (Gujarat), India. 2 L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opposite - Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad-224 001, (Uttar Pradesh), India. ⋆ Email:
[email protected] † Email:
[email protected],vishnu
[email protected], v n mishra
[email protected], vishnu
[email protected]
1
Abstract. we study the convergence of these operators in a weighted space of functions on a positive semi-axis and estimate the approximation by using a new type of weighted modulus of continuity and error estimation. Keywords: q-Favard- Sz´ asz operators, error estimation.
1. Introduction and auxiliary results The classical Favard–Sz´ asz–perators are given as follows ∞ X (nx)k k −nx f( ) (1.1) Sn (f, ; ) = e k! n k=0
These operators and the generalizations have been studied by several other researcher ( see. [1]-[7]) and references there in. In 1969, Jakimovski and Leviatan [11] introduced the Favard-Sz´ asz type operator, by using Appell polynomials pk (x)(k ≥ 0) defined by ∞ X g(u)e−ux = pk (x)uk , k=0
where g(z) =
P∞
n n=0 an z is analytic function in the disc |z| < R, R > 1 and g(1) 6= 0, ∞
Pn,t (f, x) =
enx X k pk (nx)f g(1) n k=0
and they investigated some approximation properties of these operators. Atakut at el.[10] defined a choldowsky type of Favard–Sz´ asz operators as follows: nx
(1.2)
Pn∗ (f, x)
∞
e bn X k nx = pk ( )f bn , g(1) bn n k=0
with bn a positive increasing sequence with the properties lim bn = ∞ and lim bnn = 0. They also studied some n→∞ n→∞ approximation properties of the operators. Recently, A. Karaisa [12] defined Choldowsky type generalization of the Favard-Sz´ asz operators as follows: [n]q x
(1.3)
[n]q x ∞ Eq bn X Pk (q; bn ) [k]q ∗ Pn (f ; q; x) = bn , f A(1) [k]q ! [n]q k=0
2Corresponding author 1
´ 2 WEIGHTED APPROXIMATION THEOREM FOR CHOLDOWSKY GENERALIZATION OF THE Q-FAVARD-SZASZ OPERATORS
where {Pk (q; .)} ≥ 0 is a q-Appell polynomial set which is generated by A(u) P∞
e
[n]q x bn
=
u
[n] x ∞ X Pk (q; b q )uk n
[k]q !
k=0
k
and A(u) is defined by A(u) = k=0 ak u . Motivated by these results, in this paper we study weighted approximation and error estimation of these operators. During the last two decades, the applications of q-calculus emerged as a new area in the field of approximation theory. The rapid development of q-calculus has led to the discovery of various generalizations of Bernstein polynomials involving q-integers. The aim of these generalizations is to provide appropriate and powerful tools to application areas such as numerical analysis, computer-aided geometric design and solutions of differential equations. To make the article self-content, here we mention certain basic definitions of q-calculus, details can be found in [9] and the other recent articles. For each non negative integer n, the q-integer [n]q and the q-factorial [n]q ! are, respectively, defined by 1−qn 1−q , q 6= 1, [n]q = n, q = 1, and
[n]q ! =
[n]q [n − 1]q [n − 2]q ...[1]q , n = 1, 2, ..., 1, n = 0.
Then for q > 0 and integers n, k, k ≥ n ≥ 0, we have [n + 1]q = 1 + q[n]q and [n]q + q n [k − n]q = [k]q . The q-derivative Dq f of a function f is defined by f (x) − f (qx) , x 6= 0. (1 − q)x
(Dq f )(x) =
The q-analogues of the exponential function are given by exq = and Eqx
=
∞ X xn , [n]q ! n=0
∞ X
q
n(n−1) 2
n=0
xn . [n]q !
The exponential functions have the following properties: ax ax aqx x −x Dq (eax = Eqx e−x q ) = aeq , Dq (Eq ) = aEq , eq Eq q = 1.
Lemma 1. [12] The following hold: (i) Pn∗ (e0 ; q; x) = 1, −
(ii)
Pn∗ (e1 ; q; x)
(iii)
Pn∗ (e2 ; q; x)
=x+
[n]q x
q
Dq (A(1))Eq bn eq A(1) −
2
=x +
Eq
[n]q x bn
q
eq
[n]q x bn
[n]q x bn
bn [n]q , −
[qDq (A(q))+Dq (A(1)] bn x A(1) [n]q
+
[n]q x
q
Dq2 (A(1))Eq bn eq A(1)
[n]q x bn
b2n [n]2q .
where ei (x) = xi , i = 0, 1, 2. Now we give an auxiliary lemma for the Korovkin test functions. −
Lemma 2. (iii)
Pn∗ ((t
(i) 2
Pn∗ (t
− x) ; q; x) =
− x; q; x) =
[n]q x − Eq bn
[n]q x q eq bn
[n]q x
q
Dq (A(1))Eq bn eq A(1)
[n]q x bn
bn [n]q ,
[qDq (A(q))−Dq (A(1)] bn x A(1) [n]q
−
+
[n]q x
q
Dq2 (A(1))Eq bn eq A(1)
[n]q x bn
b2n [n]2q .
´ WEIGHTED APPROXIMATION THEOREM FOR CHOLDOWSKY GENERALIZATION OF THE Q-FAVARD-SZASZ OPERATORS 3
2. Weighted approximation Let Bx2 [0, ∞) be the set of functions defined on [0, ∞) satisfying the condition | f (x) |≤ Mf (1 + x2 ), where Mf is a constant depending on f only. By Cx2 [0, ∞), we denote subspace of all continuous functions belonging to f (x) Bx2 [0, ∞). Also, let Cx∗2 [0, ∞) be the subspace of all f ∈ Cx2 [0, ∞) for which lim 1+x 2 is finite. The norm on x→∞
Cx∗2 [0, ∞) if kf kx2 =
sup x∈[0,∞)
|f (x)| 1+x2 .
For any positive number a, we define ωa (f, δ) = sup
sup
| f (t) − f (x) |,
|t−x|≤δ x,t∈[0,a]
and denote the usual modulus of continuity of f on the closed interval [0, a]. We know that for a function f ∈ Cx2 [0, ∞), the modulus of continuity ωa (f, δ) tends to zero. Now, we shall discuss the weighted approximation theorem, when the approximation formula holds true on the interval [0, ∞). Theorem 1. For each f ∈ Cx∗2 [0, ∞), we have lim kPn∗ (f ; q; x) − f kx2 = 0.
n→∞
Proof. Using the theorem in [8] we see that it is sufficient to verify the following three conditions lim kPn∗ (tr ; q; x) − xr kx2 = 0, r = 0, 1, 2.
(2.1)
n→∞
Since, Pn∗ (1, x) = 1, the first condition of (2.1) is satisfied for r = 0. Now, kPn∗ (t; q; x) − xkx2
=
≤
≤ which implies that
| Pn∗ (t; q; x) − x | 1 + x2 x∈[0,∞) [n] x [n] x q bnq − bnq bn eq 1 Dq (A(1))Eq − x sup x + A(1) [n]q 1 + x2 x∈[0,∞) [n] x [n] x − q q q Dq (A(1))Eq bn eq bn bn 1 sup 2 A(1) [n]q 1 + x x∈[0,∞) sup
kPn∗ (t, x) − xkx2 = 0.
Finally, kPn∗ (t2 ; q; x) − x2 kx2
=
≤
| Pn∗ (t2 ; q; x) − x2 | 1 + x2 x∈[0,∞) [n] x [n] x [n] x [n] x − bnq q bnq − bnq q bnq 1 2 2 eq bn eq [qDq (A(q)) + Dq (A(1)] bn x Dq (A(1))Eq 2 Eq 2 sup x + + − x 1 + x2 A(1) [n]q A(1) [n]2q x∈[0,∞) sup
which implies that kPn∗ (t2 ; q; x) − x2 kx2 → 0 as [n]q → ∞. Thus proof is completed.
We give the following theorem to approximate all functions in Cx2 [0, ∞). Theorem 2. For each f ∈ Cx2 [0, ∞) and α > 0, we have Proof. For any fixed x0 > 0, | Pn∗ (f ; q; x) − f (x) | sup ≤ (1 + x2 )1+α x∈[0,∞) ≤
sup x≤x0
lim
sup
[n]q →∞ x∈[0,∞)
|Pn∗ (f ;q;x)−f (x)| (1+x2 )1+α
= 0.
| Pn∗ (f, x) − f (x) | | Pn∗ (f ; q; x) − f (x) | + sup 2 1+α (1 + x ) (1 + x2 )1+α x≥x0
kPn∗ (f ; q; x) − f kC[0,x0] + kf kx2 sup
x≥x0
| f (x) | | Pn∗ (1 + t2 ; q; x) | + sup . (1 + x2 )1+α (1 + x2 )1+α x≥x0
The first term of the above inequality tends to zero from Theorem 3. By Lemma 2 for any fixed x0 > 0 it is easily |Pn∗ (1+t2 ;q;x)| seen that supx≥x0 (1+x tends to zero as [n]q → ∞. We can choose x0 > 0 so large that the last part of the 2 )1+α above inequality can be made small enough. Thus the proof is completed.
´ 4 WEIGHTED APPROXIMATION THEOREM FOR CHOLDOWSKY GENERALIZATION OF THE Q-FAVARD-SZASZ OPERATORS
3. Error Estimation The usual modulus of continuity of f on the closed interval [0, b] is defined by ωb (f, δ) =
sup
|f (t) − f (x)|, b > 0.
|t−x|≤δ, x,t∈[0,b]
We first consider the Banach lattice, for a function f ∈ E, limδ→0+ ωb (f, q; δ) = 0, where f (x) is f inite . E := f ∈ C[0, ∞) : lim x→∞ 1 + x2
The next theorem gives the rate of convergence of the operators Pn∗ (f, x) to f (x), for all f ∈ E. Theorem 3. Let f ∈ E and ωb+1 (f, q; δ), 0 < q < 1 be its modulus of continuity on the finite interval [0, b + 1] ⊂ [0, ∞), where a > 0 then, we have p kPn∗ (f ; q; x) − f kC[0,b] ≤ Mf (1 + b2 )δn (b) + 2ωb+1 f, δn (b) . Proof. The proof is based on the following inequality
(3.1)
kPn∗ (f ; q; x)
− f k ≤ Mf (1 + b
2
)Pn∗ ((t
P ∗ (|t − x|, x) − x) , x) + 1 + n δ 2
For all (x, t) ∈ [0, b] × [0, ∞) := S. To prove (3.1), we write
ωb+1 (f, δ).
S = S1 ∪ S2 := {(x, t) : 0 ≤ x ≤ b, 0 ≤ t ≤ b + 1} ∪ {(x, t) : 0 ≤ x ≤ b, t > b + 1}. If (x, t) ∈ S1 , we can write |t − x| (3.2) |f (t) − f (x)| ≤ ωb+1 (f, |t − x|) ≤ 1 + ωb+1 (f, δ) δ
where δ > 0. On the other hand, if (x, t) ∈ S2 , using the fact that t − x > 1, we have (3.3)
|f (t) − f (x)|
≤ Mf (1 + x2 + t2 ) ≤ Mf (1 + 3x2 + 2(t − x)2 ) ≤ Nf (1 + b2 )(t − x)2
where Nf = 6Mf . Combining (3.2) and (3.3), we get (3.1). Now from (3.1) it follows that Pn∗ (|t − x|, x) ∗ 2 ∗ 2 ωb+1 (f, δ) |Pn (f ; q; x) − f (x)| ≤ Nf (1 + b )Pn ((t − x) ; q; x) + 1 + δ ! 1/2 ∗ 2 [P ((t − x) , x)] ≤ Nf (1 + b2 )Pn∗ ((t − x)2 , x) + 1 + n ωb+1 (f, δ). δ By Lemma 2 we have Pn∗ (t − x)2 ≤ δn (b). kPn∗ (f ; q; x) Choosing δ =
2
− f k ≤ Nf (1 + b )δn (b) +
p δn (b), we get the desired estimation.
! p δn (b) 1+ ωb+1 (f, δ). δ
References
[1] S.G. Gal, Approximation and geometric properties of complex FavardSz´ asz–Mirakjan operators in compact disks, Comput. Math. Appl. 56 (4) (2008) 1121–1127. [2] R.N. Mohapatra, Z. Walczak, Remarks on a class of Sz´ asz-Mirakyan type operators, East J. Approx. 15 (2) (2009) 197–206. [3] R.A. DeVore, G.G. Lorentz, Constr. Approx. Springer, Berlin, 1993. [4] J. Favard, Sur les multiplicateurs d’interpolation, J. Math. Pures Appl. (9) 23 (1944), 219–247. [5] L. Rempulska, K. Tomczak, Approximation properties of certain operators of the Sz´ aszMirakyan type, Indian J. Math. 50 (2) (2008) 283–295. [6] D.D. Stancu, A study of the remainder in an approximation formula using a Favard–Sz´ asz type operator, Stud. Univ. Babe¸s-Bolyai, Math. XXV (1980) 70–76. [7] O. Sz´ asz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45 (1950) 239–245. [8] A.D. Gadzhiev, Theorems of the type of P.P. Korovkin theorems, Math. Zametki 20(5) (1976) 781–786, English translation in Math. Notes 20(5–6) 996–998 (1976).
´ WEIGHTED APPROXIMATION THEOREM FOR CHOLDOWSKY GENERALIZATION OF THE Q-FAVARD-SZASZ OPERATORS 5
[9] V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer–Verlag, New York, (2002). ˙ uy¨ [10] C ¸ . Atakut, H. Tanberkan, S. K. Serenbay and I.B¨ ukyazici, Approximation by Chlodowsky type Jakimovski–Leviatan operators, J. Comput. Appl. Math., 259, 153–163 (2014). [11] A. Jakimovski, D. Leviatan, Generalized Sz´ asz operators for the approximation in the infinite interval, Mathematica (Cluj) 34 (1969) 97–103. [12] A. Karaisa, Choldowsky type generalization of the q-Favard -Sz´ asz operators, arXiv:1505.06904v1 [math.CA] 26 May 2015.
