Abstract In the present paper we introduce the summation-integral type modified Lupasoperators with weights of Beta basis functions. We define the operators in terms of hyper-geometric series and, using such an approach, establish moments. Our main results are anasymptotic formula and an error estimate in terms of modulus of continuity and weightedapproximation and rate of convergence for functions having bounded derivatives.
Keywords Lupas operators · Rate of convergence · Asymptotic formula · Weightedapproximation
In 1995, Lupas [9] introduced the operator Ln(f, x) for f : [0,∞) → R and x ≥ 0 as
Ln(f, x) =∞∑
k=0
ln,k(x)f
(k
n
), (1)
where
ln,k(x) = 2−nx
(nx + k − 1
k
)2−k.
In order to approximate the integrable functions, the most common integral modificationsof the discrete operators are the Kantorovich- and Durrmeyer-type modifications. Agra-
V. Gupta (B) · R. YadavDepartment of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka,New Delhi 110078, Indiae-mail: [email protected]
tini [1] obtained some approximation properties and an asymptotic formula for the Lupas–Kantorovich operators. The Durrmeyer variant of the operators (1) introduced in [1] is de-fined by
Mn(f,x) =∞∑
k=0
cn,kln,k(x)
∫ ∞
0ln,k(u)f (u)du, (2)
where
cn,k = 1∫ ∞0 ln,k(t) dt
.
For er(t) = t r , operators (2) satisfy the following relation:
Mn(er , x) =∞∑
k=0
ln,k(x)
nr(log r)r
∑k
i=0(−1)isk,i(i+r)!(log 2)i∑k
i=0(−1)isk,i(i)!
(log 2)i
,
where sk,i are the Stirling numbers of first kind, which are related to the Pochhammer symbol(x)k as
(x)k = x(x + 1)(x + 2) · · · (x + k − 1) =k∑
i=0
(−1)k−i sk,ixi .
In the recent years, some Durrmeyer-type operators were discussed in [2–4, 7, 8]. Forf ∈ L1[0,∞), we now propose a new integral modification of the Lupas operators havingweights of Beta basis function as
Dn(f, x) =∞∑
k=1
ln,k(x)
∫ ∞
0bn,k−1(t)f (t) dt + ln,0(x)f (0), x ∈ [0,∞), (3)
where ln,k(x) is given by (1), and
bn,k(t) = 1
B(n, k + 1)
tk
(1 + t)n+k+1
are the Beta basis functions.The hypergeometric function is defined as
2F1(a, b; c;x) =∞∑
k=0
(a)k(b)k
(c)kk! xk.
The operators (3) can be written in terms of hypergeometric form:
Dn(f, x)
=∞∑
k=1
ln,k(x)
∫ ∞
0bn,k−1(t)f (t) dt + ln,0(x)f (0)
= 2−nx
∞∑
k=1
(nx + k − 1
k
)2−k
∫ ∞
0
1
B(n, k)
tk−1
(1 + t)n+kf (t) dt + 2−nxf (0)
ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS 195
= 2−nx
∫ ∞
0
f (t)
(1 + t)n
∞∑
k=1
(nx + k − 1)!2−k
k!(nx − 1)!(n + k − 1)!
(n − 1)!(k − 1)!tk−1
(1 + t)kdt + 2−nxf (0)
= 2−nx
∫ ∞
0
f (t)
(1 + t)n
∞∑
k=1
(nx)(nx + 1) · · · (nx + k − 1)
k!2k
× n(n + 1) · · · (n + k − 1)
(k − 1)!tk−1
(1 + t)kdt + 2−nxf (0)
= 2−nx
∫ ∞
0
f (t)
(1 + t)n
∞∑
k=0
(nx)(nx + 1)k
(2)k2k+1
n(n + 1)k
k!tk
(1 + t)k+1dt + 2−nxf (0)
= n2x
2nx+1
∫ ∞
0
f (t)
(1 + t)n+1 2F1
(nx + 1, n + 1;2; t
2(1 + t)
)dt + 2−nxf (0).
The present article deals with the approximation properties of the operators Dn(f, x). Westudy an asymptotic formula and an error estimate in terms of modulus of continuity,weighted approximation, and also the rate of convergence for the operators (3).
2 Auxiliary results
In order to prove the main results, we need the following two lemmas.
Lemma 1 For r > 0, Dn(er , x) satisfies the relation
Dn(er , x) = r!(n − r − 1)!nx
(n − 1)! 2F1(nx + 1,1 − r;2;−1).
Proof By the definition (3) we have
Dn(er , x) = 2−nx
∞∑
k=1
(nx + k − 1
k
)2−k
∫ ∞
0
1
B(n, k)
tk+r−1
(1 + t)n+kdt
= 2−nx
∞∑
k=1
(nx + k − 1)!k!(nx − 1)! 2−k 1
B(n, k)B(k + r, n − r)
= 2−nx (n − r − 1)!(n − 1)!(nx − 1)!
∞∑
k=0
(nx + k)!(k + r)!(k + 1)!2k+1k!
= 2−nx (n − r − 1)!(n − 1)!
∞∑
k=0
(nx + 1)k(nx)(r + 1)kΓ (r + 1)
(2)k2k+1k!
= 2−nx r!(nx)(n − r − 1)!2(n − 1)! 2F1
(nx + 1, r + 1;2; 1
2
).
Using 2F1(a, b; c;x) = (1 − x)−a2F1(a, c − b; c; x
Remark 1 By a simple computation using Lemma 1 we have
Dn(e0, x) = 1,
Dn(e1, x) = nx
n − 1,
Dn(e2, x) = nx(nx + 3)
(n − 1)(n − 2).
Remark 2 For ψx(t) = t − x and n ∈ N , from Remark 1 we have
Dn(ψx, x) = x
n − 1,
Dn
(ψ2
x , x) = x2(n + 2) + 3nx
(n − 1)(n − 2).
Further, for r = 0,1,2, . . . , we have
Dn
(ψr
x , x) = O
(n−[(r+1)/2]).
Applying the Schwarz inequality, we have
Dn
(∣∣ψrx
∣∣, x) ≤
√Dn
(ψ2r
x , x) = O
(n−r/2
).
For n ≥ 4, we can write
Dn
(|ψx |, x) ≤
√2x(3 + x)
n − 2.
Rewrite the operators (3) as
Dn(f, x) =∫ ∞
0Kn(x, t)f (t) dt,
where Kn(x, t) is the kernel function given by
Kn(x, t) =∞∑
k=1
ln,k(x)bn,k−1(t) + ln,0(x)δ(t),
and δ(t) is the Dirac delta function.
Lemma 2 For n ≥ 4 and for fixed x ∈ (0,∞), we have
λn(x, y) =∫ y
0Kn(x, t) dt ≤ 2x(3 + x)
(n − 2)(x − y)2, 0 ≤ y < x,
1 − λn(x, z) =∫ ∞
z
Kn(x, t) dt ≤ 2x(3 + x)
(n − 2)(z − x)2, x < z < ∞.
ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS 197
3 Main theorems
Theorem 1 Let f be a continuous function on [0,∞). Then the sequence {Dn(f, x)} con-verges uniformly to f (x) in [a, b] ⊂ [0,∞) as n → ∞.
By Remark 1, the proof of Theorem 1 follows immediately by Bohman–Korovkin Theo-rem.
By CB [0,∞) we denote the class of real-valued continuous bounded functions f (x)
defined on [0,∞) with the norm
‖f ‖ = supx∈[0,∞)
∣∣f (x)∣∣.
For f ∈ CB [0,∞) and δ > 0, the first- and second-order moduli of continuity are definedas
ω(f, δ) = sup0≤h≤δ
supx∈[0,∞)
∣∣f (x + h) − f (x)∣∣
and
ω2(f, δ) = sup0≤h≤δ
supx∈[0,∞)
∣∣f (x + 2h) − 2f (x + h) + f (x)∣∣,
respectively.The Peetre K-functional is defined as
K2(f, δ) = infg∈C2
B[0,∞)
{‖f − g‖ + δ∥∥g′′∥∥ : g ∈ C2
B[0,∞)},
where
C2B [0,∞) = {
g ∈ CB [0,∞) : g′, g′′ ∈ CB[0,∞)}.
According to [5], there exists a constant C > 0 such that for δ > 0, we have
K2(f, δ) ≤ Cω2(f,√
δ). (4)
Theorem 2 For x ∈ [0,∞) and f ∈ CB[0,∞), there exists a constant C > 0 such that
∣∣Dn(f, x) − f (x)∣∣ ≤ Cω2
(f,
√(2 + n)x2 + 3nx
(n − 1)(n − 2)+ x2
(n − 1)2
)+ ω
(f,
x
n − 1
).
Proof We modify the operators (3) as
Dn(f, x) = Dn(f, x) − f
(nx
n − 1
)+ f (x). (5)
Using Remark 1, we have Dn((t − x), x) = 0. Let g ∈ C2B [0,∞). By Taylor’s theorem we
have
g(t) = g(x) + g′(t − x) +∫ t
x
(t − u)g′′(u) du.
198 V. GUPTA, R. YADAV
Therefore,
∣∣Dn(g, x) − g(x)∣∣
=∣∣∣∣Dn
(∫ t
x
(t − u)g′′(u) du, x
)∣∣∣∣
≤∣∣∣∣Dn
(∫ t
x
(t − u)g′′(u) du, x
)∣∣∣∣ +∣∣∣∣
(∫ xn−1
x
(nx
n − 1− u
)g′′(u) du, x
)∣∣∣∣. (6)
Since∣∣∣∣∫ t
x
(t − u)g′′(u) du
∣∣∣∣ ≤ (t − x)2∥∥g′′∥∥ (7)
and∣∣∣∣∫ nx
n−1
x
(nx
n − 1− u
)g′′(u) du
∣∣∣∣ ≤(
x
n − 1
)2∥∥g′′∥∥, (8)
from (6), (7), (8), and (5) we have
∣∣Dn(g, x) − g(x)∣∣ ≤ Dn
((t − x)2, x
)∥∥g′′∥∥ +(
x
n − 1
)2∥∥g′′∥∥.
By Remark 2 we get
∣∣Dn(g, x) − g(x)∣∣ ≤ (2 + n)x2 + 3nx
(n − 1)(n − 2)
∥∥g′′∥∥ + x2
(n − 1)2
∥∥g′′∥∥. (9)
Now by Remark 1 we have
∣∣Dn(f, x)∣∣ ≤
∞∑
k=1
ln,k(x)
∫ ∞
0bn,k−1(t)
∣∣f (t)∣∣dt ≤ ‖f ‖.
Therefore,∣∣Dn(f, x)
∣∣ ≤ ∥∥Dn(f, x)∥∥ + 2‖f ‖ ≤ 3‖f ‖. (10)
Finally, combining (5), (6), (9), and (10), we get
∣∣Dn(f, x) − f (x)∣∣
≤ ∣∣Dn(f, x) − f (x)∣∣ +
∣∣∣∣f (x) − f
(nx
n − 1
)∣∣∣∣
= ∣∣Dn
((f − g), x
) − (f − g)(x)∣∣ + ∣∣Dn(g, x) − g(x)
∣∣ +∣∣∣∣f (x) − f
(nx
n − 1
)∣∣∣∣
≤ 4‖f − g‖ +(
(2 + n)x2 + 3nx
(n − 1)(n − 2)+ x2
(n − 1)2
)∥∥g′′∥∥ +∣∣∣∣f (x) − f
(nx
n − 1
)∣∣∣∣
≤ C
[4‖f − g‖ +
((2 + n)x2 + 3nx
(n − 1)(n − 2)+ x2
(n − 1)2
)∥∥g′′∥∥]
ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS 199
+ supt,t− x
n−1 ∈[0,∞)
∣∣∣∣f(
t − x
n − 1
)− f (t)
∣∣∣∣
≤ C
[4‖f − g‖ +
((2 + n)x2 + 3nx
(n − 1)(n − 2)+ x2
(n − 1)2
)∥∥g′′∥∥]
+ ω
(f,
x
n − 1
),
which, on using (4) and taking infimum over all C2B [0,∞), completes the proof of the theo-
rem. �
Theorem 3 Let f be a bounded and integrable function on the interval [0,∞) such that thesecond derivative of f exists at a fixed point x ∈ [0,∞). Then
limn→∞n
(Dn(f, x) − f (x)
) = xf ′(x) +(
x2 + 3x
2
)f ′′(x).
Proof By the Taylor formula we may write
f (t) = f (x) + f ′(x)(t − x) + 1
2f ′′(x)(t − x)2 + r(t, x)(t − x)2, (11)
where r(t, x) is the remainder term, and limt→x r(t, x) = 0. Acting Dn on both sides ofEq. (11), we obtain
Dn(f, x) − f (x) = Dn(t − x, x)f ′(x) + Dn
((t − x)2, x
)f ′′(x)
2+ Dn
(r(t, x)(t − x)2, x
).
Applying the Cauchy–Schwarz inequality, we have
Dn
(r(t, x)(t − x)2, x
) ≤√
Dn
(r2(t, x), x
)√Dn
((t − x)4, x
). (12)
As r2(x, x) = 0 and r2(·, x) ∈ C∗2 [0,∞), we have from Remark 2 that
limn→∞Dn
(r2(t, x), x
) = r2(x, x) = 0 (13)
uniformly with respect to x ∈ [0,A]. Now from (12), (13), and Remark 1, we get
limn→∞Dn
(r(t, x)(t − x)2, x
) = 0.
Then we obtain
limn→∞n
(Dn(f, x) − f (x)
) = limn→∞n
(f ′(x)Dn
((t − x), x
) + 1
2f ′′(x)Dn
((t − x)2, x
)
+ Dn
(r(t, x)(t − x)2;x))
= xf ′(x) +(
x2 + 3x
2
)f ′′(x). �
For weighted approximation, we give the following result. Let C∗x2 [0,∞) be the space of
all continuous functions with finite limit limx→∞ f (x)
1+x2 and belonging to the class Bx2 [0,∞),
200 V. GUPTA, R. YADAV
where Bx2 [0,∞) = {f : |f (x)| ≤ C(f )(1 + x2) for all x ∈ [0,∞),C(f ) being a constantdepending on f }. The norm on C∗
x2 [0, ∞) is defined by
‖f ‖x2 = supx∈[0, ∞)
|f (x)|1 + x2
.
Theorem 4 For each f ∈ C∗x2 [0, ∞), we have
limn→∞
∥∥Dn(f ) − f∥∥
x2 = 0.
Proof By a theorem in [6], in order to get the desired conclusion, it is sufficient to show that
limn→∞
∥∥Dn(eν, x) − eν
∥∥x2 = 0, ν = 0,1,2.
Since Dn(e0, x) = 1, the above condition holds for ν = 0. Next, we can write
∥∥Dn(e1, x) − x∥∥
x2 ≤ supx∈[0, ∞)
nx
(n − 1)(1 + x2)− x
1 + x2,
which gives
limn→∞
∥∥Dn(e1, x) − x∥∥
x2 = 0.
Finally, we can write
∥∥Dn(e2, x) − x2∥∥
x2 ≤ supx∈[0,∞)
x2
1 + x2
3n − 2
(n − 1)(n − 2)+ sup
x∈[0,∞)
x
1 + x2
3n
(n − 1)(n − 2),
which implies that
limn→∞
∥∥Dn(e2, x) − x2∥∥
x2 = 0.
Thus, the result holds for ν = 0,1,2. This completes the proof of the theorem. �
4 Rate of convergence
In this section we discuss the convergence of operators (3) for the functions having boundedderivatives.
Let us define the class ΦDB of functions as
ΦDB ={f : f (x) − f (0) =
∫ x
0φ(t) dt;f (t) = O
(t r
), t → ∞
},
where φ is bounded on every finite subinterval of [0,∞). For a fixed x ∈ [0,∞), λ ≥ 0, andf ∈ ΦDB , let us define the metric form as
Ω(f,λ) = supt∈[x−λ,x+λ]∩[0,∞)
∣∣f (t) − f (x)∣∣.
ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS 201
Theorem 5 Let f ∈ ΦDB and x ∈ (0,∞) be fixed. If φ(x−) and φ(x+) exist, then forn ≥ 4, we have
∣∣∣∣Dn(f, x) − f (x) − φ(x+) − φ(x−)
2
√2x(x + 3)
n − 2
∣∣∣∣
≤ 2(5x + 9)
n − 2
[√n]∑
k=1
Ωx
(φx,
x
k
)+ |φ(x+) + φ(x−)|
2
√x
n − 1+ O
(n−r
),
where
φx(t) =
⎧⎪⎨
⎪⎩
φ(t) − φ(x−), 0 ≤ t < x,
0, t = x,
φ(t) − φ(x+), x < t < ∞.
Proof By simple computation we have
Dn(f, x) − f (x) ≤ φ(x+) − φ(x−)
2Dn
(|t − x|, x) + φ(x+) + φ(x−)
2Dn(t − x, x)
− An,x(φx) + Bn,x(φx) + Cn,x(φx), (14)
where
An,x(φx) =∫ x
0
(∫ x
t
φx(u) du
)dt
(λn(x, t)
),
Bn,x(φx) =∫ 2x
x
(∫ t
x
φx(u) du
)dt
(λn(x, t)
),
Cn,x(φx) =∫ ∞
2x
(∫ t
x
φx(u) du
)dt
(λn(x, t)
).
Integrating by parts, we have
An,x(φx) =∫ x
t
φx(u) duλn(x, t)|x0 +∫ x
0λn(x, t)φx(t) dt
=(∫ x−x/
√n
0+
∫ x
x−x/√
n
)λn(x, t)φx(t) dt.
Since λn(x, t) ≤ 1, from the monotonicity of Ωx(φx,λ) and the definition of φx(t) we have
∣∣∣∣∫ x
x−x/√
n
λn(x, t)φx(t) dt
∣∣∣∣ ≤ x√n
Ωx
(φx,
x√n
)≤ 2x
n
[√n]∑
k=1
Ωx
(φx,
x
k
).
Taking t = xx−u
and using Lemma 2, we get∣∣∣∣∫ x−x/
√n
0λn(x, t)φx(t) dt
∣∣∣∣ ≤ 2x(3 + x)
(n − 2)
∫ x−x/√
n
0
Ωx(φx, x − t)
(x − t)2dt
≤ 2(x + 3)
(n − 2)
∫ √n
1Ωx
(φx,
x
u
)du
≤ 2(x + 3)
(n − 2)
√n∑
k=1
Ωx
(φx,
x
k
).
202 V. GUPTA, R. YADAV
Collecting the above results, we get
∣∣An,x(φx)∣∣ ≤ 2(2x + 3)
(n − 2)
√n∑
k=1
Ωx
(φx,
x
k
). (15)
Further, we have
Bn,x(φx) =∫ 2x
x
(∫ t
x
φx(u) du
)dt
(λn(x, t)
)
= −∫ 2x
x
(∫ t
x
φx(u) du
)dt
(1 − λn(x, t)
)
= −∫ 2x
x
φx(u) du(1 − λn(x,2x)
) +∫ 2x
x
φx(t)(1 − λn(x, t)
)dt.
From Lemma 2 we have
∣∣∣∣−∫ 2x
x
φx(u) du(1 − λn(x,2x)
)∣∣∣∣ ≤ xΩx(φx, x)2x(x + 3)
(n − 2)x2= 2(3 + x)
n − 2Ωx(φx, x).
Also, we have
∣∣∣∣∫ 2x
x
φx(t)(1 − λn(x, t)
)dt
∣∣∣∣ ≤ 2(2x + 3)
(n − 2)
√n∑
k=1
Ωx
(φx,
x
k
).
Hence, we get
∣∣Bn,x(φx)∣∣ ≤ 2(3 + x)
n − 2Ωx(φx, x) + 2(2x + 3)
(n − 2)
√n∑
k=1
Ωx
(φx,
x
k
). (16)
Assume that there exists an integer r such that f (t) = O(t2r ) as t → ∞. Finally, for acertain constant M > 0 depending only on f,x, r , we obtain
∣∣Cn,x(φx)∣∣ = M
∞∑
k=1
ln,k(x)
∫ ∞
2x
bn,k−1(t)t2r dt.
Using Remark 2 and the inequality t ≤ 2(t − x) for t ≥ 2x, with M ′ = 22rM , we get
∣∣Cn(f, x)∣∣ ≤ 22rM
∞∑
k=1
ln,k(x)
∫ ∞
0bn,k−1(t)(t − x)2r dt
= M ′[Dn
(ψ2r
x , x) − ln,0(−x)2r
] = O(n−r
). (17)
The proof of the theorem follows immediately by combining the estimates in (14), (15),(16), and (17). �
ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS 203
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