Journal of Inequalities in Pure and
Applied Mathematics
DIRECT APPROXIMATION THEOREMS FOR DISCRETE
TYPE OPERATORS
volume 7, issue 5, article 163,
2006.
ZOLTÁN FINTA
Babeş-Bolyai University
Department of Mathematics and Computer Science
1, M. Kogălniceanu st.
400084 Cluj-Napoca, Romania.
Received 16 July, 2006;
accepted 10 October, 2006.
Communicated by: Z. Ditzian
EMail: fzoltan@math.ubbcluj.ro
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
189-06
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Abstract
In the present paper we prove direct approximation theorems for discrete type
operators
∞
X
(Ln f)(x) =
un,k (x)λn,k (f),
k=0
f ∈ C[0, ∞), x ∈ [0, ∞) using a modified K−functional. As applications we give
direct theorems for Baskakov type operators, Szász-Mirakjan type operators
and Lupaş operator.
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
2000 Mathematics Subject Classification: 41A36, 41A25.
Key words: Direct approximation theorem, K−functional, Ditzian-Totik modulus of
smoothness.
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Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
8
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1.
Introduction
We introduce the following discrete type operators Ln , n ∈ {1, 2, 3, . . .}, defined by
(Ln f )(x) ≡ Ln (f, x) =
(1.1)
∞
X
un,k (x)λn,k (f ),
k=0
where f ∈ C[0, ∞), x ≥ 0, un,k ∈ C[0, ∞) with un,k ≥ 0 on [0, ∞) and
λn,k : C[0, ∞) → R are linear positive functionals, k ∈ {0, 1, 2, . . .}.
The purpose of this paper is to establish sufficient conditions with the aim
of obtaining direct local and global approximation theorems for (1.1). In [3]
Ditzian gave the following interesting estimate:
1 1−λ
2
(1.2)
|Bn (f, x) − f (x)| ≤ Cωϕλ f, √ ϕ (x) ,
n
where
Bn (f, x) =
n X
n
k=0
k
k
n−k
x (1 − x)
k
f
,
n
f ∈ C[0, 1], x ∈ [0, 1]
is the Bernstein-polynomial, C > 0 is an absolute constant and ϕ(x) =
p
x(1 − x). This estimate unifies the classical estimate for λ = 0 and the norm
estimate for λ = 1. Guo et al. in [7] proved a similar estimate to (1.2) for the
Baskakov operator. For the more general operator (1.1) we shall give a result
similar to the estimate (1.2) and to the result established in [7].
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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To formulate the main results we need some notations: let CB [0, ∞) be
the space of all bounded continuous functions on [0, ∞) with the norm kf k =
supx≥0 |f (x)|. Furthermore, let
ωϕλ (f, t) = sup
0<h≤t
|f (x + hϕλ (x)) − 2f (x) + f (x − hϕλ (x))|
sup
x±hϕλ (x)∈[0,∞)
be the second order modulus of smoothness of Ditzian-Totik and let
K ϕλ (f, t) = inf kf − gk + tkϕ2λ g 00 k + t2/(2−λ) kg 00 k : g 00 , ϕ2λ g 00 ∈ CB [0, ∞)
be the corresponding modified weighted K−functional, where λ ∈ [0, 1] and
ϕ : [0, ∞) → R is an admissible weight function (cf. [4, Section 1.2]) such that
ϕ2 (x) ∼ xλ as x → 0+ and ϕ2 (x) ∼ xλ as x → ∞, respectively. Then, in view
of [4, p.24, Theorem 3.1.2] we have
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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(1.3)
2
K ϕλ (f, t ) ∼
ωϕ2 λ (f, t)
(x ∼ y means that there exists an absolute constant C > 0 such that C −1 y ≤
x ≤ Cy). Throughout this paper C1 , C2 , . . . , C6 denote positive constants and
C > 0 is an absolute constant which can be different at each occurrence.
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2.
Main Results
Our first theorem is the following:
Theorem 2.1. Let (Ln )n≥1 be defined as in (1.1) satisfying
(i) Ln (1, x) = 1,
x ≥ 0;
(ii) Ln (t, x) = x,
x ≥ 0;
(iii) Ln (t2 , x) ≤ x2 + C1 n−1 ϕ2 (x),
x ≥ 0;
(iv) kLn f k ≤ C2 kf k, f ∈ CB [0, ∞);
R
t
(v) Ln x |t − u| ϕ2λdu(u) , x ≤ C3 n−1 ϕ2(1−λ) (x),
2/(2−λ)
(vi) n−1 ϕ2 (x) ≤ C4 n−1 ϕ2(1−λ) (x)
,
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
x ∈ [1/n, ∞)
and
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x ∈ [0, 1/n) .
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Then for every f ∈ CB [0, ∞), n ∈ {1, 2, 3, . . .} and x ≥ 0 one has
−1
|(Ln f )(x) − f (x)| ≤ max{1 + C2 , C3 , C1 C4 } · K ϕλ f, n ϕ
2(1−λ)
Proof. From Taylor’s expansion
0
(x) .
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Z
g(t) = g(x) + g (x)(t − x) +
(t − u)g 00 (u)du,
x
and the assumptions (i), (ii), (iii), (v) and (vi) we obtain
Z t
00
(2.1)
|(Ln g)(x) − g(x)| ≤ Ln
(t − u)g (u)du, x x
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Z t
00
≤ Ln |t − u| · |g (u)|du , x
Zx t
du
≤ Ln |t − u| · 2λ , x · kϕ2λ g 00 k
ϕ (u)
x
C3
≤
· ϕ2(1−λ) (x) · kϕ2λ g 00 k,
n
where x ∈ [1/n, ∞), and
(2.2)
Z t
00
|(Ln g)(x) − g(x)| ≤ Ln |t − u| · |g (u)|du , x
x
≤ Ln (t − x)2 , x · kg 00 k
ϕ2 (x)
≤ C1
· kg 00 k
n
2/(2−λ)
1
2(1−λ)
≤ C1 C4
·ϕ
(x)
· kg 00 k,
n
where x ∈ [0, 1/n).
In conclusion, by (2.1) and (2.2),
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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(2.3) |(Ln g)(x) − g(x)| ≤ max{C3 , C1 C4 } ·
· ϕ2(1−λ) (x) · kϕ2λ g 00 k
n
)
2/(2−λ)
1
+
· ϕ2(1−λ) (x)
· kg 00 k
n
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for x ≥ 0. Using (iv) and (2.3) we get
|(Ln f )(x) − f (x)|
≤ |Ln (f − g, x) − (f − g)(x)| + |(Ln g)(x) − g(x)|
≤ (C2 + 1)kf − gk + max{C3 , C1 C4 }
(
)
2/(2−λ)
1
1
2(1−λ)
2λ 00
2(1−λ)
00
·ϕ
(x) · kϕ g k +
·ϕ
(x)
· kg k
·
n
n
≤ max{1 + C2 , C3 , C1 C4 } · {kf − gk
o
2
4/(2−λ)
+ n−1/2 · ϕ1−λ (x) · kϕ2λ g 00 k + n−1/2 · ϕ1−λ (x)
· kg 00 k .
Now taking the infimum on the right-hand side over g and using the definition of
K ϕλ (f, n−1 ϕ2(1−λ) (x)) we get the assertion of the theorem.
Corollary 2.2. Under the assumptions of Theorem 2.1 and for arbitrary f ∈
CB [0, ∞), n ∈ {1, 2, 3, . . .} and x ≥ 0 we have the estimate
|(Ln f )(x) − f (x)| ≤ Cωϕ2 λ f, n−1/2 ϕ1−λ (x) .
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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Proof. It is an immediate consequence of Theorem 2.1 and (1.3).
Remark 1. In Corollary 2.2 the case λ = 0 gives the local estimate and for
λ = 1 we obtain a global estimate.
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3.
Applications
The applications are in connection with Baskakov type operators, Szász - Mirakjan type operators and the Lupaş operator. To be more precise, we shall study
the following operators:
(Ln f )(x) =
∞
X
vn,k (x)λn,k (f ),
vn,k (x) =
sn,k (x)λn,k (f ),
sn,k (x) = e−nx ·
k=0
(Ln f )(x) =
∞
X
n+k−1
k
k=0
xk (1 + x)−(n+k) ;
(nx)k
,
k!
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
and their generalizations:
(L(α)
n f )(x)
=
∞
X
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(α)
vn,k (x)λn,k (f ),
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k=0
Qn
Qk−1
n+k−1
i=0 (x + iα)
j=1 (1 + jα)
=
,
Qn+k
k
r=1 (x + 1 + rα)
∞
X
(α)
(α)
(Ln f )(x) =
sn,k (x)λn,k (f ),
(α)
vn,k (x)
α ≥ 0;
(α)
nx(nx + nα) · · · (nx + n(k − 1)α)
,
k!(1 + nα)k
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k=0
sn,k (x) = (1 + nα)−x/α
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(the parameter α may depend only on the natural number n), and the Lupaş
operator [8] defined by
∞
X
nx(nx + 1) · · · (nx + k − 1)
k
−nx
(L̃n f )(x) = 2
f
.
k k!
2
n
k=0
For different values of λn,k we obtain the following explicit forms of the
above operators:
1) the Baskakov operator [2]
Direct Approximation Theorems
for Discrete Type Operators
∞
X
k
(Vn f )(x) =
vn,k (x)f
;
n
k=0
Zoltán Finta
Title Page
2) the generalized Baskakov operator [5]
(Vn(α) f )(x)
=
∞
X
(α)
vn,k (x)f
k=0
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k
;
n
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3) the modified Agrawal and Thamer operator [1]
∞
X
(L1,n f )(x) = vn,0 (x)f (0)+
vn,k (x)
k=1
1
B(k, n + 1)
∞
Z
0
tk−1
f (t)dt;
(1 + t)n+k+1
(α)
(α)
(L1,n f )(x) = vn,0 (x)f (0)+
k=1
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4) the generalized Agrawal and Thamer type operator
∞
X
II
I
(α)
vn,k (x)
1
B(k, n + 1)
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Z
0
∞
k−1
t
f (t)dt;
(1 + t)n+k+1
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5) Szász - Mirakjan operator [12]
∞
X
k
sn,k (x)f
(Sn f )(x) =
;
n
k=0
6) Mastroianni operator [9]
(Sn(α) f )(x)
=
∞
X
(α)
sn,k (x)f
k=0
k
;
n
7) Phillips operator [10], [11]
(L2,n f )(x) = sn,0 (x)f (0) + n
Direct Approximation Theorems
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Zoltán Finta
∞
X
∞
Z
sn,k (x)
sn,k−1 (t)f (t)dt;
0
k=1
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8) the generalized Phillips operator
(α)
(α)
(L2,n f )(x) = sn,0 (x)f (0) + n
∞
X
(α)
Z
sn,k (x)
k=1
∞
sn,k−1 (t)f (t)dt;
0
9) a new generalized Phillips type operator [6] defined as follows:
let I = {ki : 0 = k0 ≤ k1 ≤ k2 ≤ · · · } ⊆ {0, 1, 2, . . .}. Then we can
introduce the operators
X
Z ∞
∞
∞
X
k
+
sn,k (x)n
(L3,n f )(x) =
sn,k (x)f
sn,k−1 (t)f (t)dt
n
0
k=0
k=0
k∈I
k6∈I
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and its generalization
(α)
(L3,n f )(x)
=
∞
X
(α)
sn,k (x)f
k=0
X
Z ∞
∞
k
(α)
+
sn,k (x)n
sn,k−1 (t)f (t)dt.
n
0
k=0
k∈I
k6∈I
For the above enumerated operators we have the following theorem:
p
Theorem 3.1. If f ∈ CB [0, ∞), x ≥ 0, ϕ(x) = x(1 + x), λ ∈ [0, 1] then
a) |(Vn f )(x) − f (x)| ≤ Cωϕ2 λ f, n−1/2 ϕ1−λ (x) , n ≥ 1;
b) |(Vnα f )(x) − f (x)| ≤ Cωϕ2 λ f, n−1/2 ϕ1−λ (x) , n ≥ 1,
α = α(n) ≤ C5 /(4n), C5 < 1;
c) |(L1,n f )(x) − f (x)| ≤ Cωϕ2 λ
(α)
d) |(L1,n f )(x) − f (x)| ≤ Cωϕ2 λ
Zoltán Finta
Title Page
f, n−1/2 ϕ1−λ (x) ,
f, n−1/2 ϕ1−λ (x) ,
n ≥ 9;
n ≥ 9,
α = α(n) ≤ C6 /(4n), C6 < 1.
√
For f ∈ CB [0, ∞), x ≥ 0, ϕ(x) = x, λ ∈ [0, 1] and Ln ∈ {Sn , L2,n , L3,n , L̃n }
(α)
(α)
(α)
(α)
resp. Ln ∈ {Sn , L2,n , L3,n } we have
e) |(Ln f )(x) − f (x)| ≤ Cωϕ2 λ f, n−1/2 ϕ1−λ (x) , n ≥ 1;
(α)
f ) Ln f (x) − f (x) ≤ Cωϕ2 λ f, n−1/2 ϕ1−λ (x) , n ≥ 1,
α = α(n) ≤ 1/n;
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g) L̃n f (x) − f (x) ≤ Cωϕ2 λ f, n−1/2 ϕ1−λ (x) ,
n ≥ 1.
Proof. First of all let us observe that we have the integral representation
Z ∞
x
1
θ α −1
(α)
(3.1)
Ln f (x) =
(Ln f )(θ)dθ,
x
1
B αx , α1 + 1 0 (1 + θ) α + α +1
n
o
(α)
(α)
(α)
where 0 < α < 1 and Ln , Ln ∈
Vn , Vn
, L1,n , L1,n .
Analogously
x
1 α Z ∞
θ
x
α (3.2)
L(α)
e− α θ α −1 (Ln f )(θ)dθ,
n f (x) =
x
Γ α 0
n
o
(α)
(α)
(α)
(α)
where α > 0, and Ln , Ln ∈
Sn , Sn , L2,n , L2,n , L3,n , L3,n .
The relations (3.1) and (3.2) can be proved with the same idea. For example,
(α)
(α)
if Ln = Vn and Ln = Vn then
Z ∞
x
1
θ α −1
Vn (f, θ)dθ
1
x
B αx , α1 + 1 0 (1 + θ) α + α +1
Z ∞
x
∞ X
θk
k
θ α −1
n+k−1
1
dθf
=
1
x
x 1
n+k
+
+1
n
k
B α , α + 1 0 (1 + θ) α α (1 + θ)
k=0
∞
X
n + k − 1 B αx + k, α1 + n + 1
k
=
f
x 1
k
n
B α, α + 1
k=0
∞
X
k
(α)
=
vn,k (x)f
= Vn(α) (f, x).
n
k=0
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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The statements of our theorem follow from Corollary 2.2 if we verify the conditions (i) − (vi). It is easy to show that each operator preserves the linear
functions and
(3.3)
1
x(1 + x),
n
x(1 + x) αx(1 + x)
5
Vn(α) ((t − x)2 , x) =
+
≤
x(1 + x),
(1 − α)n
1−α
3n
2x(1 + x)
4
≤ x(1 + x),
L1,n ((t − x)2 , x) =
n−1
n
2x(1
+
x)
αx(1 + x)
17
(α)
L1,n ((t − x)2 , x) =
+
≤
x(1 + x),
(1 − α)(n − 1)
1−α
3n
1
Sn ((t − x)2 , x) = x,
n
3
1
(α)
2
Sn ((t − x) , x) = α +
x + αx ≤ x,
n
n
2
L2,n ((t − x)2 , x) = x,
n
2
3
(α)
L2,n ((t − x)2 , x) = x + αx ≤ x,
n
n
2
L3,n ((t − x)2 , x) ≤ x,
n
2
3
(α)
L3,n ((t − x)2 , x) ≤ x + αx ≤ x
(see [6, p. 179]),
n
n
2
L̃n ((t − x)2 , x) = x,
n
Vn ((t − x)2 , x) =
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which imply (i), (ii) and (iii). The condition (iv) can be obtained from the
integral representations (3.1) – (3.2) and the definition of L̃n .
For (v) we have in view of [4, p.140, Lemma 9.6.1] that
Z t
Z t
du
du
|t − u|
= |t − u|
(3.4)
ϕ2λ (u)
uλ (1 + u)λ x
x
1
1
(t − x)2
≤
·
+
xλ
(1 + x)λ (1 + t)λ
Direct Approximation Theorems
for Discrete Type Operators
or
(3.5)
Z t
Z t
2
|t − u| du = |t − u| du ≤ (t − x) .
ϕ2λ (u) x
uλ xλ
x
Because Ln is a linear positive operator, therefore either (3.4) and (3.3) or (3.5)
and (3.3) imply
Z t
17 x(1 + x)
1
du · λ
+ λ Ln (t − x)2 (1 + t)−λ , x ,
Ln |t − u| 2λ , x ≤
λ
ϕ (u)
3n x (1 + x) x
x
and
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Z t
3 x
3
du
Ln |t − u| 2λ , x ≤ · λ = x1−λ ,
ϕ (u)
n x
n
x
respectively. Thus we have to prove the estimation
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C x(1 + x)
Ln (t − x)2 (1 + t)−λ , x ≤ ·
,
n (1 + x)λ
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(3.6)
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x ∈ [1/n, ∞)
J. Ineq. Pure and Appl. Math. 7(5) Art. 163, 2006
for each Baskakov type operator defined in this section.
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1. By Hölder’s inequality and [4, p.128, Lemma 9.4.3 and p.141, Lemma
9.6.2] we have
1
λ
Vn ((t − x)2 (1 + t)−λ , x) ≤ {Vn ((t − x)4 , x)} 2 · {Vn ((1 + t)−4 , x)} 4
1
λ
≤ C(n−2 x2 (1 + x)2 ) 2 · ((1 + x)−4 ) 4
C x(1 + x)
= ·
,
n (1 + x)λ
where x ∈ [1/n, ∞);
Direct Approximation Theorems
for Discrete Type Operators
2. Using
3
Vn ((t − x) , x) = 2
n
4
2
1+
n
1
· x (1 + x) + 3 · x(1 + x),
n
2
(3.1) and [4, p.141, Lemma 9.6.2] we obtain
(3.7)
Zoltán Finta
2
Vn(α) ((t − x)4 , x)
3
2
x(x + α)(x + 1)(x + 1 − α)
= 2 1+
·
n
n
(1 − α)(1 − 2α)(1 − 3α)
1 x(x + 1)
+ 3·
n 1 − α
3
2
5
1
≤ 2 1+
· ·
· x2 (1 + x)2
4
n
n
4 (1 − C5 )
1
1
+ 2·
· x2 (1 + x)2
n (1 − C5 )2
2
≤ C n−1 x(1 + x) ,
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where x ∈ [1/n, ∞), and
(3.8)
Vn(α) ((1 + t)−4 , x)
Z ∞
x
θ α −1
C
dθ
≤
·
x
1
x 1
+
+1
(1 + θ)4
B α , α + 1 0 (1 + θ) α α
(1 + α)(1 + 2α)(1 + 3α)(1 + 4α)
=C
(1 + x + α)(1 + x + 2α)(1 + x + 3α)(1 + x + 4α)
≤ C(1 + x)−4 ,
where x ∈ [0, ∞), α = α(n) ≤ C5 /(4n), n ≥ 1, C5 < 1. Therefore the
(α)
Hölder inequality, (3.7) and (3.8) imply (3.6) for Ln = Vn ;
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
3. We have
(3.9) L1,n ((t − x)2 (1 + t)−λ , x)
1 λ
≤ L1,n ((t − x)4 , x) 2 · L1,n ((1 + t)−4 , x) 4 .
By direct computation we get
(3.10) L1,n ((t − x)4 , x)
= vn,0 (x)x4
∞
X
vn,k (x)
+
k=1
4
= vn,0 (x)x +
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1
B(k, n + 1)
∞
X
k=1
vn,k (x)
Z
0
∞
tk−1
(t − x)4 dt
(1 + t)n+k+1
1
{B(k + 4, n − 3)
B(k, n + 1)
− 4xB(k + 3, n − 2) + 6x2 B(k + 2, n − 1)
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− 4x3 B(k + 1, n) + x4 B(k, n + 1)
∞
X
k(k + 1)(k + 2)(k + 3)
4
vn,k (x)
= vn,0 (x)x +
n(n − 1)(n − 2)(n − 3)
k=1
k(k + 1)(k + 2)
k(k + 1)
k
− 4x ·
+ 6x2 ·
− 4x3 · + x4
n(n − 1)(n − 2)
n(n − 1)
n
4
3
2
(12n + 84)x + (24n + 168)x + (12n + 108)x + 11x
=
.
(n − 1)(n − 2)(n − 3)
Direct Approximation Theorems
for Discrete Type Operators
Hence, for x ∈ [1/n, ∞) and n ≥ 9 one has
(3.11)
L1,n ((t − x)4 , x)
(12n + 84)x4 + (24n + 168)x3 + (12n + 108)x2 + 11nx2
≤
(n − 1)(n − 2)(n − 3)
C
≤ 2 x2 (1 + x)2 .
n
Further,
(3.12)
L1,n ((1+t)−4 , x)
= vn,0 (x) +
∞
X
k=1
Z ∞
×
= vn,0 (x) +
0
∞
X
k=1
vn,k (x)
1
B(k, n + 1)
dt
tk−1
·
n+k+1
(1 + t)
(1 + t)4
vn,k (x)
B(k, n + 5)
B(k, n + 1)
Zoltán Finta
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= vn,0 (x) +
∞
X
k=1
vn,k (x)
(n + 1)(n + 2)(n + 3)(n + 4)
(n + k + 1)(n + k + 2)(n + k + 3)(n + k + 4)
∞
X vn−4,k (x)
1
= vn−4,0 (x) ·
+
(1 + x)4 k=1 (1 + x)4
(n + k − 4)(n + k − 3)(n + k − 2)(n + k − 1)
(n + k + 1)(n + k + 2)(n + k + 3)(n + k + 4)
(n + 1)(n + 2)(n + 3)(n + 4)
·
(n − 4)(n − 3)(n − 2)(n − 1)
≤ 16(1 + x)−4 ,
·
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
where n ≥ 9. Now (3.9), (3.11) and (3.12) imply (3.6) for Ln = L1,n ;
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4. Using (3.1), Hölder’s inequality, (3.10) and (3.12) we have
(α)
L1,n ((t
− x)2 (1 + t)−λ , x)
Z ∞
x
1
θ α −1
=
L1,n ((t − x)2 (1 + t)−λ , θ)dθ
x
1
x 1
+
+1
B α , α + 1 0 (1 + θ) α α
! 12
Z ∞
x
θ α −1
1
≤
L1,n ((t − x)4 , θ)dθ
1
x
B αx , α1 + 1 0 (1 + θ) α + α +1
! λ4
Z ∞
x
1
θ α −1
·
L1,n ((1 + t)−4 , θ)dθ
x
1
B αx , α1 + 1 0 (1 + θ) α + α +1
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≤C
1
Z
∞
! 12
x
θ α −1
L1,n ((t − θ)4 , θ) + (θ − x)4 dθ
x
1
+ 1 0 (1 + θ) α + α +1
!λ
B αx , α1 + 5 4
·
B αx , α1 + 1
Z ∞
x
θ α −1
1
1
·
≤C
(θ4 + θ3 + θ2 + n−1 θ)dθ
x
1
B αx , α1 + 1 n2 0 (1 + θ) α + α +1
! 12
Z ∞
x
−1
α
θ
+
(θ − x)4 dθ
1
x
+
+1
(1 + θ) α α
0
λ4
(1 + α)(1 + 2α)(1 + 3α)(1 + 4α)
·
(1 + x + α)(1 + x + 2α)(1 + x + 3α)(1 + x + 4α)
−2 4
≤ C n (x + x3 + x2 ) + n−2 (6α3 + 2α2 + α + n−1 )x + (18α3 + 3α2 )x4
1
+ (36α3 + 6α2 )x3 + (24α3 + 3α2 )x2 + 6α3 x 2 · (1 + x)−λ
C x(1 + x)
≤ ·
n (1 + x)λ
B
x 1
,
α α
for x ∈ [1/n, ∞), n ≥ 9, α = α(n) ≤ C6 /(4n), C6 < 1.
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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Condition (vi) follows by direct computation if ϕ2 (x) = x(1 + cx), c ∈
{0, 1} and x ∈ [0, 1/n). Thus the theorem is proved.
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References
[1] P.N. AGRAWAL AND K.J. THAMER, Approximation of unbounded functions by a new sequence of linear positive operators, J. Math. Anal. Appl.,
225 (1998), 660–672.
[2] V.A. BASKAKOV, An example of a sequence of linear positive operators
in the space of continuous functions, Dokl. Akad. Nauk SSSR., 113 (1957),
249–251.
[3] Z. DITZIAN, Direct estimate for Bernstein polynomials, J. Approx. Theory, 79 (1994), 165–166.
[4] Z. DITZIAN
Berlin, 1987.
AND
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
V. TOTIK, Moduli of Smoothness, Springer Verlag,
[5] Z. FINTA, Direct and converse theorems for integral-type operators,
Demonstratio Math., 36(1) (2003), 137–147.
[6] Z. FINTA, On converse approximation theorems, J. Math. Anal. Appl., 312
(2005), 159–180.
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[7] S.S. GUO, C.X. LI and G.S. ZHANG, Pointwise estimate for Baskakov
operators, Northeast Math. J., 17(2) (2001), 133–137.
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[8] A. LUPAŞ, The approximation by some positive linear operators, In: Proceedings of the International Dortmund Meeting on Approximation Theory
(Eds. M.W. Müller et al.), Akademie Verlag, Berlin, 1995, 201–229.
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[9] G. MASTROIANNI, Una generalizzazione dell’operatore di Mirakyan,
Rend. Accad. Sci. Mat. Fis. Napoli, Serie IV, 48 (1980/1981), 237–252.
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J. Ineq. Pure and Appl. Math. 7(5) Art. 163, 2006
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[10] R.S. PHILLIPS, An inversion formula for semi groups of linear operators,
Ann. Math., 59 (1954), 352–356.
[11] C.P. MAY, On Phillips operators, J. Approx. Theory, 20 (1977), 315–322.
[12] O. SZÁSZ, Generalization of S. Bernstein’s polynomials to the infinite
interval, J. Res. Nat. Bur. Standards, Sect. B, 45 (1950), 239–245.
Direct Approximation Theorems
for Discrete Type Operators
Zoltán Finta
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