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Asymptotic Formulas for Generalized Szász–Mirakyan Operators
Applied Mathematics and Computation 263 (2015) 233–239
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic Formulas for Generalized Szász–Mirakyan Operators Tuncer Acar∗ Kirikkale University, Department of Mathematics, TR-71450, Yahsihan, Kirikkale, Turkey
a r t i c l e
i n f o
a b s t r a c t
MSC: Primay 41A25 41A36
In the present paper, we consider the general Szász–Mirakyan operators and investigate their asymptotic behaviours. We obtain quantitative Voronovskaya and quantitative Grüss type Voronovskaya theorems using the weighted modulus of continuity. The particular cases are presented for classical Szász–Mirakyan operators.
Keywords: Szász–Mirakyan operators Quantitative Voronovskaya theorem Grüss–Voronovskaya-type theorem Weighted modulus of continuity
© 2015 Elsevier Inc. All rights reserved.
1. Introduction In the paper [8], the authors introduced the sequence of linear Bernstein-type operators defined for f C[0, 1] by Bn ( f ◦ τ −1 ) ◦ τ , Bn being the classical Bernstein operators and τ being any function satisfying some certain conditions. The results obtained there showed that approximation with these new construction of Bernstein operators are sensitive and present better convergence results with the suitable selection of τ . In [6], the authors introduced a general sequence of linear Szász–Mirakyan type operators by ∞ ρ Sn f ; x = e−nρ(x) f k=0
ρ −1
k (nρ (x))k , n k!
(1.1)
where ρ is a function being continuously differentiable on R+ := [0, ∞) and satisfying the conditions ρ (0) = 0, infx∈R+ ρ (x) ≥ 1. This function ρ not only characterizes the operators but also characterizes the Korovkin set {1, ρ , ρ 2 } in a weighted function space. The authors studied degree of weighted convergence and some shape-preserving properties. The most of the results given in [6] are related to uniform convergence of the operators. Therefore, in this paper we continue to study further approximation properties of the operators (1.1). We present Voronovskaya type theorems in quantitative form. The quantitative Voronovskaya theorem and its general forms were extensively studied in [11], authors presented the quantitative Voronovskaya theorem for any positive linear operators acting on compact intervals, using the least concave majorant of the modulus of continuity. The advantage of the quantitative Voronovskaya theorems is to describe the rate of convergence and to present an upper bound for the error of approximation, simultaneously. For some other quantitative versions of Voronovskaya’s theorem, we can refer the readers to [2,4,10,12,18,19]. All of mentioned theorems were given for the operators acting on compact intervals, in particular, Bernstein polynomials and their derivatives. But for the operators acting on unbounded intervals, the usual modulus of continuity does not work. Therefore, quantitative Voronovskaya theorems for general sequence of linear positive ∗
Corresponding author. Tel.: +9003183574242. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.amc.2015.04.060 0096-3003/© 2015 Elsevier Inc. All rights reserved.
234
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
operators via weighted modulus of continuity studied in [3]. For quantitative Voronovskaya type theorems in weighted spaces we can refer the readers to [1,20]. We also study Grüss–Voronovskaya-type results for the operators (1.1). The application of Grüss inequality in approximation theory was given in the paper [5]. In the recent note [13], the authors applied the Grüss inequality to the operators on compact intervals and thus obtained a new approach for the Bernstein polynomials using the . In [9], Gal and Gonska obtained the Voronovskaya-type theorem with the aid of Grüss inequality for least concave majorant ω Bernstein operators for the first time and called it Grüss–Voronovskaya-type theorem. Here, we shall give Grüss–Voronovskayatype theorem with the weighted modulus of continuity. As a corollary of our main results, quantitative Voronovskaya theorem for classical Szász–Mirakyan operators and Voronovskaya theorem for classical Szász–Mirakyan operators and generalized Szász– Mirakyan operators are captured. For the classical Voronovskaya theorem for Szá sz–Mirakyan operators and their modifications, one can consult the papers [7,16,17] and references therein. 2. Auxiliary results For our main results we shall need following moments, central moments and a weighted modulus of continuity. Since the moments are similar to the corresponding results for the Szász–Mirakyan operators we give the lemmas without proofs. Lemma 1. We have
ρ ρ Sn 1; x = 1, Sn (ρ ; x) = ρ (x) , ρ (x) ρ , Sn ρ 2 ; x = ρ 2 (x) + n
Lemma 2. If we define the central moment operator by ρ
ρ
Mn,m (x) = Sn
(ρ (t) − ρ (x))m ; x ,
then we have ρ
ρ
Mn,0 (x) = 1, ρ
Mn,2 (x) =
Mn,1 (x) = 0,
ρ (x) n
,
ρ
Mn,3 (x) =
(2.1)
ρ (x) n2
,
(2.2)
ρ
3ρ 2 (x) ρ (x) + 3 , n2 n
(2.3)
ρ
15ρ 3 (x) 25ρ 2 (x) ρ (x) + + 5 , n3 n4 n
(2.4)
Mn,4 (x) = Mn,6 (x) = for all n, m ∈ N.
Throughout the paper we consider the weight function ρ satisfying the following assumptions: (i) ρ is a continuously differentiable function on R+ and ρ (0) = 0. (ii) inf ρ (x) ≥ 1. x≥0
Let ϕ (x) = 1 + ρ 2 (x) and Bϕ R+ = f : |f (x)| ≤ Mf ϕ (x) , where Mf is constant which may depend only on f. Cϕ R+ denotes + + + ∗ the subspace of all continuous functions in Bϕ R . By Cϕ R , we denote the subspace of all functions f ∈ Cϕ R for which + f (x) limx→∞ ϕ( be the space of functions f ∈ Cϕ R+ such that f/ϕ is uniformly continuous. Bϕ R+ is the x) is finite. Also let Uϕ R f (x) linear normed space with the norm f ϕ = supx≥0 ϕ( x) . + For each f ∈ Cϕ R and for every δ > 0 the weighted modulus of continuity defined in [14] is given by
ωρ f, δ =
sup x,y≥0 |ρ( y)−ρ(x)|≤δ
|f ( y) − f (x)| ϕ ( y) + ϕ (x)
and it was shown that: Lemma 3. ([14]) For every f ∈ Uϕ R+ , limδ → 0 ωρ ( f, δ ) = 0 and
|f ( y) − f (x)| ≤ (ϕ ( y) + ϕ (x)) 2 +
|ρ ( y) − ρ (x)| ωρ f, δ . δ
Remark 1. If ρ (x) = x, then ωρ is equivalent with 2 given in [15]
2 f ; δ
f x + h − f (x) . = sup 2 1 + x2 x,y≥0 1 + h |h|≤δ
(2.5)
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
235
3. Main results We give quantitative form of Voronovskaya theorem and a new pointwise convergence result, called Grüss-type-Voronovskaya theorem in quantitative mean.
Theorem 1. If the function ρ satisfies the conditions (i) and (ii) and f /(ρ )2 , f .ρ /(ρ )3 ∈ Cϕ (R+ ), then we have for x ∈ R+ that
ρ (x) D2 f ◦ ρ −1 (ρ (x)) 2 f fρ ρ ρ 2 ≤ 6 ρ (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ 3 , δn (x)
ρ n Sn f ; x − f (x) −
ρ
where δn (x) =
ρ
√ 4
ρ
15ρ 3 (x)+25ρ 2 (x)+ρ(x) √ . n
Proof. By the Taylor expansion of f◦ρ −1 at the point x ∈ R+ we can write
f ◦ ρ −1
h (u, x) =
D2 f ◦ ρ −1 (ρ (x))(ρ (u) − ρ (x))2 + h (u, x)(ρ (u) − ρ (x))2 , 2
+ where
(ρ (u)) = f ◦ ρ −1 (ρ (x)) + D f ◦ ρ −1 (ρ (x))(ρ (u) − ρ (x))
(3.1)
D2 f ◦ ρ −1 ρ ξ − D2 f ◦ ρ −1 (ρ (x)) 2
and ξ is a number between x and u. If we apply the operator to both sides of equality (3.1), we immediately have
D2 f ◦ ρ −1 (ρ (x)) ρ ρ Mn,2 (x) Sn f ; x − f (x) − 2 ρ ≤ Sn |h (u, x)| (ρ (u) − ρ (x))2 ; x
and using Lemma 2 we write
ρ Sn f ; x − f (x) −
ρ (x) D2 f ◦ ρ −1 (ρ (x))
ρ
≤ Sn
2n
|h (u, x)| (ρ (u) − ρ (x))2 ; x .
ρ In order to complete the proof, we estimate the Sn |h (u, x)| (ρ (u) − ρ (x))2 ; x . Since
f ◦ ρ −1
(ρ (t)) =
ρ (t) (t) 2 − f (t) 3 ρ (t) ρ (t) f
and by (2.5) we can write
f ◦ ρ −1 1 = 2 =
1 2
ρ ξ − f ◦ ρ −1 (ρ (x))
2
ξ ρ ξ f (x) ρ (x) 2 − f ξ 3 − 2 + f (x) ρ ξ ρ ξ ρ (x) ρ (x) 3
f
ξ ρ ξ f (x) ρ (x) 2 − 2 + f (x) 3 − f ξ 3 ρ ξ ρ (x) ρ (x) ρ ξ
f
|ρ (u) − ρ (x)| ≤ (ϕ (u) + ϕ (x)) 2 + δ
ωρ
f 2 ;
ρ
δ + ωρ
f .ρ 3 ;
δ
ρ
.
On the other hand, in the case of |ρ (u) − ρ (x)| < δ since ϕ (t) + ϕ (x) δ 2 + 2ρ 2 (x) + 2ρ (x)δ + 2 we have
|h (u, x)| ≤ 3 δ + 2ρ (x) + 2ρ (x)δ + 2 2
2
f
ωρ 2 , δ + ωρ ρ
fρ 3 , δ
(3.2)
ρ
ρ(x) 2 2 δ + 2ρ 2 (x) + 2ρ (x)δ + 2 we get and in the case of |ρ (u) − ρ (x)| < δ since ϕ (u) + ϕ (x) ≤ ρ(u)− δ
ρ (u) − ρ (x) |h (u, x)| ≤ 3 δ
4
δ + 2ρ (x) + 2ρ (x)δ + 2 2
2
f
ωρ 2 , δ + ωρ ρ
fρ 3 , δ
ρ
.
(3.3)
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T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
Combining (3.2) and (3.3) and choosing δ < 1 we deduce
|h (u, x)| ≤ 6 ρ (x) + ρ (x) + 2 2
ρ (u) − ρ (x) 4
δ
Hence, using (2.2) and (2.4) we deduce
ρ n Sn f ; x − f (x) − ≤ 6n
ρ (x) + ρ (x) + 2 1
ρ
× Mn,2 (x) +
f
ωρ 2 , δ + ωρ ρ
+1
fρ 3 , δ
ρ
.
ρ (x) D2 f ◦ ρ −1 (ρ (x))
2
2
f
ωρ 2 , δ + ωρ ρ
ρ
fρ 3 , δ
ρ
Mn,6 (x) f fρ = 6 ρ 2 (x) + ρ (x) + 2 ωρ 2 , δ + ωρ 3 , δ
δ4
ρ 25ρ 2 (x)
ρ
15ρ (x) ρ (x) + + 4 n2 n3 n f fρ ≤ 6 ρ 2 (x) + ρ (x) + 2 ωρ 2 , δ + ωρ 3 , δ ×
×
1
ρ (x) +
δ4
ρ
1
ρ (x) +
If we choose δn (x) =
3
√ 4
δ4
ρ ρ 2 15ρ (x) + 25ρ (x) + ρ (x) 3
n2
15ρ 3 (x)+25ρ 2 (x)+ρ(x) √ , n
then we have
ρ (x) D2 f ◦ ρ −1 (ρ (x)) 2 f fρ ρ ρ 2 ≤ 6 ρ (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ 3 , δn (x)
ρ n Sn f ; x − f (x) −
ρ
ρ
which completes the proof.
Corollary 1. The followings hold: i): Let f ∈ Cϕ R+ . If we choose ρ (x) = x in Theorem 1, we have quantitative Voronovskaya theorem for classical Szász– Mirakyan operators as:
n Sn f ; x − f (x) − xf (x) ≤ 12 1 + x 3 2 f ; 2
δn (x) ,
√ 4 where δn (x) = 15x3 + 25x2 + x/ n. 2 3 , f .ρ / ρ ∈ Uϕ (R+ ). If we take limit with n → in Theorem 1, we have the Voronovskaya theorem ii): Let f / ρ obtained in [6]:
ρ lim n Sn f ; x − f (x) =
2
n→∞
iii): Let f / ρ
2
, f .ρ / ρ
3
ρ (x) D2 f ◦ ρ −1 (ρ (x))
∈ Uϕ R
+
. If we take the limit with n → with the selection of ρ (x) = x in Theorem 1, we have
the Voronovskaya theorem for classical Szász–Mirakyan operators as:
xf (x)
. lim n Sn f ; x − f (x) = 2
n→∞
Theorem 2. If f, g,
f .ρ , g .ρ , f , g ∈ Cϕ (R+ ) such that ( fg) .ρ3 , ( fg )2 ∈ Cϕ (R+ ), then we have at any point x ∈ R+ that (ρ )3 (ρ )3 (ρ )2 (ρ )2 (ρ ) (ρ )
ρ ρ (x) fg (x) ρ (x) ρ ρ n Sn fg; x − Sn f ; x Sn (g; x) − x f x − g ( ) ( ) (ρ (x)) n (ρ (x))2 fg fg .ρ ρ ρ 2 ≤ 6 ρ (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ 3 , δn (x)
ρ
ρ
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
+ 6 f ϕ
+ 6 g ϕ
237
g g .ρ ρ 2 (x) + ρ (x) + 2 1 + ρ (x) 2 ωρ 2 , δnρ (x) + ωρ 3 , δnρ (x) ρ ρ
ρ
2
(x) + ρ (x) + 2 1 + ρ (x) 2
ωρ
f ◦ρ −1
2
ϕ
f f .ρ ρ ρ 2 , δn (x) + ωρ 3 , δn (x)
ρ
+ nIn f In (g) , ϕ(x) where In f =
ρ
ρ ρ ρ(x) ρ 1 2Mn,2 (x) + 2ϕ( x) Mn,3 (x) + ϕ(x) Mn,4 (x) and In (g) is the analogues one.
Proof. For x ∈ R+ and n ∈ N, we can write
ρ g (x) f (x) ρ ρ ρ Sn fg; x − Sn f ; x Sn (g; x) − Mn,2 (x) (ρ (x))2 g (x) f (x)ρ (x) g (x) f (x)ρ (x) ρ ρ − Mn,2 (x) − Mn,2 (x) 3 (ρ (x)) (ρ (x))3 ρ Mn,2 (x) ρ = Sn fg; x − f (x) g (x) − fg ◦ ρ −1 (ρ (x)) 2 ρ Mn,2 (x) ρ −1 g◦ρ − f (x) Sn (g; x) − g (x) − (ρ (x)) 2 ρ Mn,2 (x) ρ −1 f ◦ρ − g (x) Sn f ; x − f (x) − (ρ (x)) 2 ρ ρ + g (x) − Sn (g; x) . Sn f ; x − f (x) so using (2.2) we can write
ρ ρ (x) ρ ρ g (x) f (x) − Sn fg; x − Sn f ; x Sn (g; x) − n (ρ (x))2
ρ (x) fg (x) (ρ (x))
≤ |I1 | + |I2 | + |I3 | + |I4 | . By Theorem 1, we have the estimates
|I1 | ≤ 6 ρ
2
(x) + ρ (x) + 2 1 + ρ (x)
|I2 | ≤ 6 f ϕ
fg
ρ
ωρ 2 , δn (x) + ωρ ρ
fg .ρ ρ 3 , δn (x)
ρ
g g .ρ ρ 2 (x) + ρ (x) + 2 1 + ρ (x) 2 ωρ 2 , δnρ (x) + ωρ 3 , δnρ (x) ρ ρ
|I3 | ≤ 6 g ϕ ρ
2
(x) + ρ (x) + 2 1 + ρ (x) 2
ωρ
f f .ρ ρ ρ 2 , δn (x) + ωρ 3 , δn (x) .
ρ
ρ
On the other hand, by the assumptions of theorem we can write
1 ρ ρ ρ Sn f ; x − f (x) = f ◦ ρ −1 (ρ (x)) Mn,1 (x) + Sn ρ ξ (ρ (t) − ρ (x))2 ; x f ◦ ρ −1 2 hence using (2.1) we have
ρ Sn f ; x − f (x) 1 ρ ξ (ρ (t) − ρ (x))2 ; x ≤ Sn f ◦ ρ −1 2 −1 1 ρ 2 2 ρ ≤ f ◦ 2 Sn 1 + ρ ξ (ρ (t) − ρ (x)) ; x , ϕ
where ξ is a number between t and x. If t < ξ < x, then 1 + ρ 2 (ξ ) 1 + ρ 2 (x). In this case we get
−1 ϕ (x) ρ f ◦ρ ϕ ρ Sn f ; x − f (x) ≤ Mn,2 (x) 2
,
238
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
or if x < ξ < t, then 1 + ρ 2 (ξ ) 1 + ρ 2 (t). In this case we get
ρ Sn f ; x − f (x) f ◦ ρ −1 ϕ ρ Sn 1 + ρ (t)2 (ρ (t) − ρ (x))2 ; x ≤ 2 f ◦ ρ −1 ϕ ρ ρ ρ = 1 + ρ 2 (x) Mn,2 (x) + 2ρ (x) Mn,3 (x) + Mn,4 (x) . 2
Therefore, for two cases of ρ (ξ ) we obtain
ρ Sn f ; x − f (x) f ◦ ρ −1 ϕ (x) ϕ ≤ 2 2ρ (x) ρ ρ × 2Mn,2 (x) + M (x) + ϕ (x) n,3 := In f
1 ρ Mn,4 (x) . ϕ (x)
Hence, we get
ρ ρ (x) fg (x) ρ (x) ρ ρ n Sn fg; x − Sn f ; x Sn (g; x) − g (x) f (x) − (ρ (x)) n (ρ (x))2 fg fg .ρ ρ ρ , δ x ≤ 6 ρ 2 (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ ( ) n 3
ρ
2 + 6 f ϕ ρ 2 (x) + ρ (x) + 2 1 + ρ (x) 2 + 6 g ϕ ρ 2 (x) + ρ (x) + 2 1 + ρ (x)
+ nIn f In (g) ,
ωρ
ρ
g g .ρ ρ ρ 2 , δn (x) + ωρ 3 , δn (x)
ρ f
ρ
ωρ 2 , δn (x) + ωρ ρ
ρ
f .ρ ρ 3 , δn (x)
ρ
which is desired. Corollary 2. The followings hold: i): Let f, g, f , g ∈ Cϕ R+ such that fg ∈ Cϕ R+ . If we choose ρ (x) = x in Theorem 2, we have quantitative Grüss-type Voronovskaya theorem for classical Szász–Mirakyan operators as:
x n Sn fg; x − Sn f ; x Sn (g; x) − g (x) f (x) n ≤ 6 x2 + x + 2 1 + x 2 fg ; δn (x) 2 + 6 f ϕ x2 + x + 2 1 + x 2 g , δn (x) 2 + 6 g ϕ x2 + x + 2 1 + x 2 f , δn (x) + nIn f In (g) ,
where δn (x) =
In f =
√ 4 15x3 + 25x2 + x/ n and In (f) reduces to
f 2 1 + x 2 2
2x 2x2 1 + + n 1 + x2 1 + x2 n 2
x 3x2 + 3 n2 n
.
ii): Let f, g, f , g ∈ Uϕ R+ such that fg ∈ Uϕ R+ . If we take the limit with n → in Theorem 2, we have Grüss type ρ Voronovskaya theorem for Sn as:
ρ
ρ ρ lim n Sn fg; x − Sn f ; x Sn (g; x) =
n→∞
ρ (x) fg (x) ρ (x) g . x f x − ( ) ( ) (ρ (x)) (ρ (x))2
iii): Let f, g, f , g ∈ Uϕ R+ such that fg ∈ Uϕ R+ . If we take the limit with n → with the selection of ρ (x) = x in Theorem 2, we have Grüss type Voronovskaya theorem for classical Szász–Mirakyan operators as:
lim n Sn fg; x − Sn f ; x Sn (g; x) = xg (x) f (x) .
n→∞
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
239
Acknowledgment The authors are thankful to referees for making valuable suggestions leading to the better presentation of the paper. This paper is dedicated to my wife Özlem Acar and our newborn son Ya˘gız Acar. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
T. Acar, Quantitative q-Voronovskaya and q-Grüss–Voronovskaya-type results for q-Szász Operators, Georgian Math. J., (in press). T. Acar, A. Aral, On pointwise convergence of q-Bernstein operators and their q-derivatives, Numer. Funct. Anal. Optim. 36 (3) (2015) 287–304. T. Acar, A. Aral, I. Rasa, The new forms of Voronovskaya Theorem in weighted spaces, Positivity, (In Press). T. Acar, A. Aral, I. Rasa, Modified Bernstein–Durrmeyer operators, General Math. 22 (1) (2014) 27–41. A.M. Acu, H. Gonska, I. Rasa, ¸ Grüss-type and Ostrowski-type inequalities in approximation theory, Ukranian Math. J. 63 (6) (2011) 843–864. A. Aral, D. Inoan, I. Rasa, ¸ On the generalized Szász–Mirakyan operators, Results Math. 3-4 (2014) 441–452. P.L. Butzer, H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodovsky polynomials and the Szász–Mirakyan operator, Comment. Math. 49 (1) (2009) 33–58. D. Cardenas-Morales, P. Garrancho, I. Rasa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62 (1) (2011) 158–163. S.G. Gal, H. Gonska, Grüss and Grüss–Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables, arXiv:1401.6824v1. H. Gonska, R. P˘alt˘anea, General Voronovskaya and asymptotic theorems in simultaneous approximation, Mediterr. J. Math. 7 (1) (2010) 37–49. H. Gonska, P. Pitul, I. Rasa, ¸ On Peano’s form of the taylor remainder,Voronovskaja’s theorem and the commutator of positive linear operators, in: Proc. International Conf. Numer. Anal. Approx. Theory, 2006, pp. 55–80. Cluj-Napoca. H. Gonska, I. Rasa, Asymptotic behaviour of differentiated Bernstein polynomials, Mat. Vesnik. 61 (1) (2009) 53–60. H. Gonska, G. Tachev, Grüss-type inequalities for positive linear operators with second order moduli, Mat. Vesnik 63 (4) (2011) 247–252. A. Holhos, Quantitative estimates for positive linear operators in weighted space, General Math. 16 (4) (2008) 99–110. N. Ispir, On modified Baskakov operators on weighted spaces, Turk. J. Math. 25 (3) (2001) 355–365. M. Orkcu, q-Szász–Mirakyan-Kantorovich operators of functions of two variables in polynomial weighted spaces, Abstr. Appl. Anal. (2013). Art. ID 823803. M. Orkcu, O. Dogru, Weighted statistical approximation by Kantorovich type q-Szász–Mirakjan operators, Appl. Math. Comput. 217 (20) (2011) 7913–7919. G.T. Tachev, Voronovskaja’s theorem revisited, J. Math. Anal. Appl. 343 (1) (2008) 399–404. G.T. Tachev, New estimates in Voronovskaja’s theorem, Numer. Algor. 59 (1) (2012) 119–129. G. Ulusoy, T. Acar, Approximation By Modified Szász-Durrmeyer Operators, Period. Math. Hungar., (in press).
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic Formulas for Generalized Szász–Mirakyan Operators Tuncer Acar∗ Kirikkale University, Department of Mathematics, TR-71450, Yahsihan, Kirikkale, Turkey
a r t i c l e
i n f o
a b s t r a c t
MSC: Primay 41A25 41A36
In the present paper, we consider the general Szász–Mirakyan operators and investigate their asymptotic behaviours. We obtain quantitative Voronovskaya and quantitative Grüss type Voronovskaya theorems using the weighted modulus of continuity. The particular cases are presented for classical Szász–Mirakyan operators.
Keywords: Szász–Mirakyan operators Quantitative Voronovskaya theorem Grüss–Voronovskaya-type theorem Weighted modulus of continuity
© 2015 Elsevier Inc. All rights reserved.
1. Introduction In the paper [8], the authors introduced the sequence of linear Bernstein-type operators defined for f C[0, 1] by Bn ( f ◦ τ −1 ) ◦ τ , Bn being the classical Bernstein operators and τ being any function satisfying some certain conditions. The results obtained there showed that approximation with these new construction of Bernstein operators are sensitive and present better convergence results with the suitable selection of τ . In [6], the authors introduced a general sequence of linear Szász–Mirakyan type operators by ∞ ρ Sn f ; x = e−nρ(x) f k=0
ρ −1
k (nρ (x))k , n k!
(1.1)
where ρ is a function being continuously differentiable on R+ := [0, ∞) and satisfying the conditions ρ (0) = 0, infx∈R+ ρ (x) ≥ 1. This function ρ not only characterizes the operators but also characterizes the Korovkin set {1, ρ , ρ 2 } in a weighted function space. The authors studied degree of weighted convergence and some shape-preserving properties. The most of the results given in [6] are related to uniform convergence of the operators. Therefore, in this paper we continue to study further approximation properties of the operators (1.1). We present Voronovskaya type theorems in quantitative form. The quantitative Voronovskaya theorem and its general forms were extensively studied in [11], authors presented the quantitative Voronovskaya theorem for any positive linear operators acting on compact intervals, using the least concave majorant of the modulus of continuity. The advantage of the quantitative Voronovskaya theorems is to describe the rate of convergence and to present an upper bound for the error of approximation, simultaneously. For some other quantitative versions of Voronovskaya’s theorem, we can refer the readers to [2,4,10,12,18,19]. All of mentioned theorems were given for the operators acting on compact intervals, in particular, Bernstein polynomials and their derivatives. But for the operators acting on unbounded intervals, the usual modulus of continuity does not work. Therefore, quantitative Voronovskaya theorems for general sequence of linear positive ∗
Corresponding author. Tel.: +9003183574242. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.amc.2015.04.060 0096-3003/© 2015 Elsevier Inc. All rights reserved.
234
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
operators via weighted modulus of continuity studied in [3]. For quantitative Voronovskaya type theorems in weighted spaces we can refer the readers to [1,20]. We also study Grüss–Voronovskaya-type results for the operators (1.1). The application of Grüss inequality in approximation theory was given in the paper [5]. In the recent note [13], the authors applied the Grüss inequality to the operators on compact intervals and thus obtained a new approach for the Bernstein polynomials using the . In [9], Gal and Gonska obtained the Voronovskaya-type theorem with the aid of Grüss inequality for least concave majorant ω Bernstein operators for the first time and called it Grüss–Voronovskaya-type theorem. Here, we shall give Grüss–Voronovskayatype theorem with the weighted modulus of continuity. As a corollary of our main results, quantitative Voronovskaya theorem for classical Szász–Mirakyan operators and Voronovskaya theorem for classical Szász–Mirakyan operators and generalized Szász– Mirakyan operators are captured. For the classical Voronovskaya theorem for Szá sz–Mirakyan operators and their modifications, one can consult the papers [7,16,17] and references therein. 2. Auxiliary results For our main results we shall need following moments, central moments and a weighted modulus of continuity. Since the moments are similar to the corresponding results for the Szász–Mirakyan operators we give the lemmas without proofs. Lemma 1. We have
ρ ρ Sn 1; x = 1, Sn (ρ ; x) = ρ (x) , ρ (x) ρ , Sn ρ 2 ; x = ρ 2 (x) + n
Lemma 2. If we define the central moment operator by ρ
ρ
Mn,m (x) = Sn
(ρ (t) − ρ (x))m ; x ,
then we have ρ
ρ
Mn,0 (x) = 1, ρ
Mn,2 (x) =
Mn,1 (x) = 0,
ρ (x) n
,
ρ
Mn,3 (x) =
(2.1)
ρ (x) n2
,
(2.2)
ρ
3ρ 2 (x) ρ (x) + 3 , n2 n
(2.3)
ρ
15ρ 3 (x) 25ρ 2 (x) ρ (x) + + 5 , n3 n4 n
(2.4)
Mn,4 (x) = Mn,6 (x) = for all n, m ∈ N.
Throughout the paper we consider the weight function ρ satisfying the following assumptions: (i) ρ is a continuously differentiable function on R+ and ρ (0) = 0. (ii) inf ρ (x) ≥ 1. x≥0
Let ϕ (x) = 1 + ρ 2 (x) and Bϕ R+ = f : |f (x)| ≤ Mf ϕ (x) , where Mf is constant which may depend only on f. Cϕ R+ denotes + + + ∗ the subspace of all continuous functions in Bϕ R . By Cϕ R , we denote the subspace of all functions f ∈ Cϕ R for which + f (x) limx→∞ ϕ( be the space of functions f ∈ Cϕ R+ such that f/ϕ is uniformly continuous. Bϕ R+ is the x) is finite. Also let Uϕ R f (x) linear normed space with the norm f ϕ = supx≥0 ϕ( x) . + For each f ∈ Cϕ R and for every δ > 0 the weighted modulus of continuity defined in [14] is given by
ωρ f, δ =
sup x,y≥0 |ρ( y)−ρ(x)|≤δ
|f ( y) − f (x)| ϕ ( y) + ϕ (x)
and it was shown that: Lemma 3. ([14]) For every f ∈ Uϕ R+ , limδ → 0 ωρ ( f, δ ) = 0 and
|f ( y) − f (x)| ≤ (ϕ ( y) + ϕ (x)) 2 +
|ρ ( y) − ρ (x)| ωρ f, δ . δ
Remark 1. If ρ (x) = x, then ωρ is equivalent with 2 given in [15]
2 f ; δ
f x + h − f (x) . = sup 2 1 + x2 x,y≥0 1 + h |h|≤δ
(2.5)
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
235
3. Main results We give quantitative form of Voronovskaya theorem and a new pointwise convergence result, called Grüss-type-Voronovskaya theorem in quantitative mean.
Theorem 1. If the function ρ satisfies the conditions (i) and (ii) and f /(ρ )2 , f .ρ /(ρ )3 ∈ Cϕ (R+ ), then we have for x ∈ R+ that
ρ (x) D2 f ◦ ρ −1 (ρ (x)) 2 f fρ ρ ρ 2 ≤ 6 ρ (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ 3 , δn (x)
ρ n Sn f ; x − f (x) −
ρ
where δn (x) =
ρ
√ 4
ρ
15ρ 3 (x)+25ρ 2 (x)+ρ(x) √ . n
Proof. By the Taylor expansion of f◦ρ −1 at the point x ∈ R+ we can write
f ◦ ρ −1
h (u, x) =
D2 f ◦ ρ −1 (ρ (x))(ρ (u) − ρ (x))2 + h (u, x)(ρ (u) − ρ (x))2 , 2
+ where
(ρ (u)) = f ◦ ρ −1 (ρ (x)) + D f ◦ ρ −1 (ρ (x))(ρ (u) − ρ (x))
(3.1)
D2 f ◦ ρ −1 ρ ξ − D2 f ◦ ρ −1 (ρ (x)) 2
and ξ is a number between x and u. If we apply the operator to both sides of equality (3.1), we immediately have
D2 f ◦ ρ −1 (ρ (x)) ρ ρ Mn,2 (x) Sn f ; x − f (x) − 2 ρ ≤ Sn |h (u, x)| (ρ (u) − ρ (x))2 ; x
and using Lemma 2 we write
ρ Sn f ; x − f (x) −
ρ (x) D2 f ◦ ρ −1 (ρ (x))
ρ
≤ Sn
2n
|h (u, x)| (ρ (u) − ρ (x))2 ; x .
ρ In order to complete the proof, we estimate the Sn |h (u, x)| (ρ (u) − ρ (x))2 ; x . Since
f ◦ ρ −1
(ρ (t)) =
ρ (t) (t) 2 − f (t) 3 ρ (t) ρ (t) f
and by (2.5) we can write
f ◦ ρ −1 1 = 2 =
1 2
ρ ξ − f ◦ ρ −1 (ρ (x))
2
ξ ρ ξ f (x) ρ (x) 2 − f ξ 3 − 2 + f (x) ρ ξ ρ ξ ρ (x) ρ (x) 3
f
ξ ρ ξ f (x) ρ (x) 2 − 2 + f (x) 3 − f ξ 3 ρ ξ ρ (x) ρ (x) ρ ξ
f
|ρ (u) − ρ (x)| ≤ (ϕ (u) + ϕ (x)) 2 + δ
ωρ
f 2 ;
ρ
δ + ωρ
f .ρ 3 ;
δ
ρ
.
On the other hand, in the case of |ρ (u) − ρ (x)| < δ since ϕ (t) + ϕ (x) δ 2 + 2ρ 2 (x) + 2ρ (x)δ + 2 we have
|h (u, x)| ≤ 3 δ + 2ρ (x) + 2ρ (x)δ + 2 2
2
f
ωρ 2 , δ + ωρ ρ
fρ 3 , δ
(3.2)
ρ
ρ(x) 2 2 δ + 2ρ 2 (x) + 2ρ (x)δ + 2 we get and in the case of |ρ (u) − ρ (x)| < δ since ϕ (u) + ϕ (x) ≤ ρ(u)− δ
ρ (u) − ρ (x) |h (u, x)| ≤ 3 δ
4
δ + 2ρ (x) + 2ρ (x)δ + 2 2
2
f
ωρ 2 , δ + ωρ ρ
fρ 3 , δ
ρ
.
(3.3)
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T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
Combining (3.2) and (3.3) and choosing δ < 1 we deduce
|h (u, x)| ≤ 6 ρ (x) + ρ (x) + 2 2
ρ (u) − ρ (x) 4
δ
Hence, using (2.2) and (2.4) we deduce
ρ n Sn f ; x − f (x) − ≤ 6n
ρ (x) + ρ (x) + 2 1
ρ
× Mn,2 (x) +
f
ωρ 2 , δ + ωρ ρ
+1
fρ 3 , δ
ρ
.
ρ (x) D2 f ◦ ρ −1 (ρ (x))
2
2
f
ωρ 2 , δ + ωρ ρ
ρ
fρ 3 , δ
ρ
Mn,6 (x) f fρ = 6 ρ 2 (x) + ρ (x) + 2 ωρ 2 , δ + ωρ 3 , δ
δ4
ρ 25ρ 2 (x)
ρ
15ρ (x) ρ (x) + + 4 n2 n3 n f fρ ≤ 6 ρ 2 (x) + ρ (x) + 2 ωρ 2 , δ + ωρ 3 , δ ×
×
1
ρ (x) +
δ4
ρ
1
ρ (x) +
If we choose δn (x) =
3
√ 4
δ4
ρ ρ 2 15ρ (x) + 25ρ (x) + ρ (x) 3
n2
15ρ 3 (x)+25ρ 2 (x)+ρ(x) √ , n
then we have
ρ (x) D2 f ◦ ρ −1 (ρ (x)) 2 f fρ ρ ρ 2 ≤ 6 ρ (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ 3 , δn (x)
ρ n Sn f ; x − f (x) −
ρ
ρ
which completes the proof.
Corollary 1. The followings hold: i): Let f ∈ Cϕ R+ . If we choose ρ (x) = x in Theorem 1, we have quantitative Voronovskaya theorem for classical Szász– Mirakyan operators as:
n Sn f ; x − f (x) − xf (x) ≤ 12 1 + x 3 2 f ; 2
δn (x) ,
√ 4 where δn (x) = 15x3 + 25x2 + x/ n. 2 3 , f .ρ / ρ ∈ Uϕ (R+ ). If we take limit with n → in Theorem 1, we have the Voronovskaya theorem ii): Let f / ρ obtained in [6]:
ρ lim n Sn f ; x − f (x) =
2
n→∞
iii): Let f / ρ
2
, f .ρ / ρ
3
ρ (x) D2 f ◦ ρ −1 (ρ (x))
∈ Uϕ R
+
. If we take the limit with n → with the selection of ρ (x) = x in Theorem 1, we have
the Voronovskaya theorem for classical Szász–Mirakyan operators as:
xf (x)
. lim n Sn f ; x − f (x) = 2
n→∞
Theorem 2. If f, g,
f .ρ , g .ρ , f , g ∈ Cϕ (R+ ) such that ( fg) .ρ3 , ( fg )2 ∈ Cϕ (R+ ), then we have at any point x ∈ R+ that (ρ )3 (ρ )3 (ρ )2 (ρ )2 (ρ ) (ρ )
ρ ρ (x) fg (x) ρ (x) ρ ρ n Sn fg; x − Sn f ; x Sn (g; x) − x f x − g ( ) ( ) (ρ (x)) n (ρ (x))2 fg fg .ρ ρ ρ 2 ≤ 6 ρ (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ 3 , δn (x)
ρ
ρ
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
+ 6 f ϕ
+ 6 g ϕ
237
g g .ρ ρ 2 (x) + ρ (x) + 2 1 + ρ (x) 2 ωρ 2 , δnρ (x) + ωρ 3 , δnρ (x) ρ ρ
ρ
2
(x) + ρ (x) + 2 1 + ρ (x) 2
ωρ
f ◦ρ −1
2
ϕ
f f .ρ ρ ρ 2 , δn (x) + ωρ 3 , δn (x)
ρ
+ nIn f In (g) , ϕ(x) where In f =
ρ
ρ ρ ρ(x) ρ 1 2Mn,2 (x) + 2ϕ( x) Mn,3 (x) + ϕ(x) Mn,4 (x) and In (g) is the analogues one.
Proof. For x ∈ R+ and n ∈ N, we can write
ρ g (x) f (x) ρ ρ ρ Sn fg; x − Sn f ; x Sn (g; x) − Mn,2 (x) (ρ (x))2 g (x) f (x)ρ (x) g (x) f (x)ρ (x) ρ ρ − Mn,2 (x) − Mn,2 (x) 3 (ρ (x)) (ρ (x))3 ρ Mn,2 (x) ρ = Sn fg; x − f (x) g (x) − fg ◦ ρ −1 (ρ (x)) 2 ρ Mn,2 (x) ρ −1 g◦ρ − f (x) Sn (g; x) − g (x) − (ρ (x)) 2 ρ Mn,2 (x) ρ −1 f ◦ρ − g (x) Sn f ; x − f (x) − (ρ (x)) 2 ρ ρ + g (x) − Sn (g; x) . Sn f ; x − f (x) so using (2.2) we can write
ρ ρ (x) ρ ρ g (x) f (x) − Sn fg; x − Sn f ; x Sn (g; x) − n (ρ (x))2
ρ (x) fg (x) (ρ (x))
≤ |I1 | + |I2 | + |I3 | + |I4 | . By Theorem 1, we have the estimates
|I1 | ≤ 6 ρ
2
(x) + ρ (x) + 2 1 + ρ (x)
|I2 | ≤ 6 f ϕ
fg
ρ
ωρ 2 , δn (x) + ωρ ρ
fg .ρ ρ 3 , δn (x)
ρ
g g .ρ ρ 2 (x) + ρ (x) + 2 1 + ρ (x) 2 ωρ 2 , δnρ (x) + ωρ 3 , δnρ (x) ρ ρ
|I3 | ≤ 6 g ϕ ρ
2
(x) + ρ (x) + 2 1 + ρ (x) 2
ωρ
f f .ρ ρ ρ 2 , δn (x) + ωρ 3 , δn (x) .
ρ
ρ
On the other hand, by the assumptions of theorem we can write
1 ρ ρ ρ Sn f ; x − f (x) = f ◦ ρ −1 (ρ (x)) Mn,1 (x) + Sn ρ ξ (ρ (t) − ρ (x))2 ; x f ◦ ρ −1 2 hence using (2.1) we have
ρ Sn f ; x − f (x) 1 ρ ξ (ρ (t) − ρ (x))2 ; x ≤ Sn f ◦ ρ −1 2 −1 1 ρ 2 2 ρ ≤ f ◦ 2 Sn 1 + ρ ξ (ρ (t) − ρ (x)) ; x , ϕ
where ξ is a number between t and x. If t < ξ < x, then 1 + ρ 2 (ξ ) 1 + ρ 2 (x). In this case we get
−1 ϕ (x) ρ f ◦ρ ϕ ρ Sn f ; x − f (x) ≤ Mn,2 (x) 2
,
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T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
or if x < ξ < t, then 1 + ρ 2 (ξ ) 1 + ρ 2 (t). In this case we get
ρ Sn f ; x − f (x) f ◦ ρ −1 ϕ ρ Sn 1 + ρ (t)2 (ρ (t) − ρ (x))2 ; x ≤ 2 f ◦ ρ −1 ϕ ρ ρ ρ = 1 + ρ 2 (x) Mn,2 (x) + 2ρ (x) Mn,3 (x) + Mn,4 (x) . 2
Therefore, for two cases of ρ (ξ ) we obtain
ρ Sn f ; x − f (x) f ◦ ρ −1 ϕ (x) ϕ ≤ 2 2ρ (x) ρ ρ × 2Mn,2 (x) + M (x) + ϕ (x) n,3 := In f
1 ρ Mn,4 (x) . ϕ (x)
Hence, we get
ρ ρ (x) fg (x) ρ (x) ρ ρ n Sn fg; x − Sn f ; x Sn (g; x) − g (x) f (x) − (ρ (x)) n (ρ (x))2 fg fg .ρ ρ ρ , δ x ≤ 6 ρ 2 (x) + ρ (x) + 2 1 + ρ (x) ωρ 2 , δn (x) + ωρ ( ) n 3
ρ
2 + 6 f ϕ ρ 2 (x) + ρ (x) + 2 1 + ρ (x) 2 + 6 g ϕ ρ 2 (x) + ρ (x) + 2 1 + ρ (x)
+ nIn f In (g) ,
ωρ
ρ
g g .ρ ρ ρ 2 , δn (x) + ωρ 3 , δn (x)
ρ f
ρ
ωρ 2 , δn (x) + ωρ ρ
ρ
f .ρ ρ 3 , δn (x)
ρ
which is desired. Corollary 2. The followings hold: i): Let f, g, f , g ∈ Cϕ R+ such that fg ∈ Cϕ R+ . If we choose ρ (x) = x in Theorem 2, we have quantitative Grüss-type Voronovskaya theorem for classical Szász–Mirakyan operators as:
x n Sn fg; x − Sn f ; x Sn (g; x) − g (x) f (x) n ≤ 6 x2 + x + 2 1 + x 2 fg ; δn (x) 2 + 6 f ϕ x2 + x + 2 1 + x 2 g , δn (x) 2 + 6 g ϕ x2 + x + 2 1 + x 2 f , δn (x) + nIn f In (g) ,
where δn (x) =
In f =
√ 4 15x3 + 25x2 + x/ n and In (f) reduces to
f 2 1 + x 2 2
2x 2x2 1 + + n 1 + x2 1 + x2 n 2
x 3x2 + 3 n2 n
.
ii): Let f, g, f , g ∈ Uϕ R+ such that fg ∈ Uϕ R+ . If we take the limit with n → in Theorem 2, we have Grüss type ρ Voronovskaya theorem for Sn as:
ρ
ρ ρ lim n Sn fg; x − Sn f ; x Sn (g; x) =
n→∞
ρ (x) fg (x) ρ (x) g . x f x − ( ) ( ) (ρ (x)) (ρ (x))2
iii): Let f, g, f , g ∈ Uϕ R+ such that fg ∈ Uϕ R+ . If we take the limit with n → with the selection of ρ (x) = x in Theorem 2, we have Grüss type Voronovskaya theorem for classical Szász–Mirakyan operators as:
lim n Sn fg; x − Sn f ; x Sn (g; x) = xg (x) f (x) .
n→∞
T. Acar / Applied Mathematics and Computation 263 (2015) 233–239
239
Acknowledgment The authors are thankful to referees for making valuable suggestions leading to the better presentation of the paper. This paper is dedicated to my wife Özlem Acar and our newborn son Ya˘gız Acar. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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