

International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 1 (2014), 58-67
http://www.etamaths.com

APPROXIMATING DERIVATIVES BY A CLASS OF POSITIVE

LINEAR OPERATORS

BRAMHA DUTTA PANDEY1,∗ AND B. KUNWAR2

Abstract. Some Direct Theorems for the linear combinations of a new class

of positive linear operators have been obtained for both, pointwise and uni-

form simultaneous approximations. a number of well known positive linear
operators such as Gamma Operators of Muller, Post-Widder and Modified

Post-Widder Operators are special cases of this class of operators.

1. Introduction

During past few decades a number of sequences of positive linear operators (
henceforth written as operator) both, of the summation and those defined by inte-
grals have been introduced and studied by a number of authors. Some of wellknown
operators of latter type are the Gamma operators of Müller [7], Post-Widder and
Modified Post-Widder operators [6], Kunwar [4], Sikkema and Rathore [11].

Now we define our linear operator Ln [4] as
(1) Ln(f; x) = D(m,n,α)x

mn+α−1
∫∞

0
u−mn−αe−n(

x
u

)mf(u)du

where D(m,n,α) =
|m|nn+

α−1
m

Γn+ α−1
m

,m ∈ IR−{0},n > 0,α ∈ IR.
The equation (1) defines a linear positive approximation methods, which con-

tains as particular cases, a number of well known linear positive operators; e.g.
Post-Widder and Modified Post-Widder operators [6], and the Gamma-operators
of Muller [7] .

In the present paper we study the following problems:
(i) Is it possible to approximate the derivatives of f by the derivatives of Ln(f)?
(ii) Can we use certain linear combinations of Ln to obtain a better order of

approximation?
We introduce notations and definitions used in this paper.
Throughout the paper IR+ denotes the interval (0,∞),< a,b > open interval

containing [a,b] ⊆ IR+,χδ,x(χcδ,x) the characteristic function of the interval (x −
δ,x+δ) {IR+−(x−δ,x+δ)}. The spaces M(IR+),Mb(IR+),Loc(IR+),L1(IR+)
respectively denote the sets of complex valued measurable, bounded and measur-
able, locally integrable and Lebesgue integrable functions on IR+.

Let Ω(> 1) be a continuous function defined on IR+.We call Ω a bounding
function if for each K ⊆ IR+ there exist positive numbers nK and MK such that

LnK (Ω; x) < MK, x ∈ K.

2000 Mathematics Subject Classification. Primary 41A35, 41A38; Secondary 41A25, 41A60.
Key words and phrases. Positive linear operators, Simultaneous approximation, Linear

combinations.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

58


APPROXIMATING DERIVATIVES BY A CLASS OF POSITIVE LINEAR OPERATORS 59

here Ω(u) = u−a + ebu
m

+ uc, where a,b,c > 0.
For this bounding function

DΩ = {f : f is locally integrable on IR+ and is such that
lim supu→0

f(u)
Ω(u)

and lim supu→∞
f(u)
Ω(u)

exist}
D

(k)
Ω = {f : f ∈ DΩ and f is k- times cotinuously differentiable on

IR+ and f(i) ∈ DΩ, i = 1, 2, ...,k}
Cmb (IR

+) = {f : f is m-times continuously differentiable and is such that
f(k), k = 0, 1, 2, ...m are bounded on IR+}.

2. SIMULTANEOUS APPROXIMATION FOR CONTINUOUS
DERIVATIVES

We consider the elementary case of simultaneous approximation by the operators
Ln wherein the derivatives of f are assumed to be continuous. We have termed this
case elementary, for it is possible here to deduce the results on the simultaneous
approximation: (Lnf)

(k) → f(k)(k ∈ IN) from the corresponding results on the
ordinary approximation: Lnf → f.

Theorem 1. : If f ∈ D(k)Ω ,then L
(k)
n (f; x) for x ∈< a,b > exists for all sufficiently

large n and

(2) limn→∞L
(k)
n (f; x) = f

(k)(x), uniformly for x ∈ [a,b].

Proof. We have
Ln(f; x) = D(m,n,α)x

mn+α−1
∫∞

0
u−mn−αe−n(

x
u

)mf(u)du
A formal k-times differentiation within the integral sign and replacing α by α−k,

let the new operator be denoted by L∗n and the corresponding D(m,n,α) be denoted
by D∗(m,n,α).Then

(3) L
(k)
n (f; x) =

D(m,n,α)
D∗(m,n,α)

L∗n(f
(k)(x))

Applying the known approximation Lnf → f to (3), we find that
L

(k)
n (f; x) =

D(m,n,α)
D∗(m,n,α)

L∗n(f
(k)(x)) → f(k)(x) as n →∞.

This completes the proof of the theorem. �

Theorem 2. :If f ∈ D(k)Ω . then at each x ∈ IR
+ where f(k+2) exists

(4) L
(k)
n (f; x) −f(k)(x) = 12nm2 [(m + k − 2α + 2)kf

(k)(x)+

+(m+2k−2α+3)xf(k+1)(x)+x2f(k+2)(x)]+o( 1
n

),n →
∞.

Further if f(k+2) exists and is continuous on < a,b >, then (4) holds uniformly
in x ∈ [a,b].

Proof. Using Voronovskaya formula [1], [6], [10],[11], [12] for L∗n and (3), the result
follows. �

In a similer manner one can prove the following results:

Theorem 3. : If f is such that f(k) exists and is continuous on IR+, then

(5)
∣∣∣L(k)n (f; x) −f(k)(x)∣∣∣ ≤ ωf(k) (n−12 )[1+
+ min{x2( 1

m2
+ o(1)),x( 1

m2
+ o(1))

1
2 ] + o( 1

n
),

(n →∞,x ∈ IR+).
where ωf(k) is the modulus of continuity of f

(k) [13] [2] [3].


60 BRAMHA DUTTA PANDEY1,∗ AND B. KUNWAR2

Theorem 4. : Let f be such that f(k+1) exists on IR+.Then for x ∈ IR+

(6)
∣∣∣L(k)n (f; x) −f(k)(x)∣∣∣

≤
k|f(k)(x)|

2nm2
{|m + k − 2α + 2|} + x|

f(k+1)(x)|
2nm2

{|m + k − 2α + 3|}+
+ωf(k+1) (n

−1
2 )[xn−

1
2{ 1

m2(m−3) + o(1)} +
x2

2n
1
2
{ 1
m2(m−3) + o(1)}],

(n →∞,x ∈ IR+).

3. Pointwise Simultaneous Approximation

In the present section we consider the “non-elementary” case of simultaneous
approximation wherein assuming only that f(k)(x) exist at some point x, we solve
the problem of pointwise approximation. Before proving this result we establish:

Lemma 1. :Let n > p ∈ IN (set of natural numbers). Then
(7) ∂

p

∂xp
{xα+mn−1u−mne−n(

x
u

)m} = xmn+α−p−1up−mne−(n−p)(
x
u

)m

×
∑p
k=0

∑[ p−k
2

]
ν=0 (

m
u

)knν+k( x
u

)k(m−1)e−k(
x
u

)m[1−
( x
u

)m]kgν,k,p(x,u)
where [x] denotes the integral part of x ∈ IR+ and the function gν,k,p(x,u) are

certain linear combinations of products of the powers of u−1,x−1 and ∂
k

∂xk
{( x
u

)me−(
x
u

)m},k =
0, 1, 2, ...,p and are independent of n.

Proof. We proceed by induction on p. We note that
(8) ∂

∂x
{xmn+α−1u−mne−n(

x
u

)m}
= x(α−1)( x

u
)m(n−1)e−(n−1)(

x
u

)m[
(mn+α−1)

x
( x
u

)me−(
x
u

)m−mn
u

( x
u

)2m−1e−(
x
u

)m]

Putting g0,0,1(x,u) =
(α−1)
x

( x
u

)me−(
x
u

)m

g0,1,1(x,u) = u
−1

We observe that (8) is of the form (7). Hence the result is true for p = 1.
Next, let us assume that the lemma holds for a certain p. Then by the induction

hypothesis,

(9) ∂
p+1

∂xp+1
{xα+mn−1u−mne−n(

x
u

)m}
= xα−1( x

u
)m(n−p−1)e−(n−p−1)(

x
u

)m

×
∑p+1
k=0

∑[( p−k+1
2

)]
ν=0 (

m
u

)knν+k{( x
u

)m−1e−(
x
u

)m − ( x
u

)2m−1e−(
x
u

)m}kgν,k,p+1(x,u)
Wherewith gν,k,p ≡ 0 for k > p or k < 0,ν < 0 or ν > [p−k2 ], we have put

gν,k,p+1(x,u) =
mn+α−1

x
gν,k,p(x,u)(

x
u

)me−(
x
u

)m

−mn
u
{m
u

( x
u

)m−1e−(
x
u

)m−m
u

( x
u

)2m−1e−(
x
u

)m}gν,k,p(x,u)+
+ ∂
∂x
gν,k,p(x,u) +

1
u
gν,k−1,p(x,u)+

+(k+1
u

){m(m−1)
u2

( x
u

)m−2e−(
x
u

)m−(m
u

)2( x
u

)2(m−1)e−(
x
u

)m−
−m(2m−1)

u2
( x
u

)2(m−1)e−(
x
u

)m+(m
u

)2( x
u

)3m−2e−(
x
u

)m}gν−1,k+1,p(x,u).
For k = 0, 1, 2, ...,p + 1 and ν = 0, 1, 2, ...., [p+1−k

2
]

It is clear that gν,k,p+1(x,u) satisfies the other required properties and hence the
result is true for p + 1. Hence it follows that (8) holds for all p = 1, 2, .... This
completes the proof. �

Theorem 5. : Let m ∈ IN and f ∈ DΩ, then
(10) lim

n→∞
L

(k)
n (f; x) = f

(k)(x).

whenever x ∈ IR+ is such that f(k)(x) exists. Moreover if f(k) exists and is
continuous on < a,b >, (10) holds uniformly in x ∈ [a,b].


APPROXIMATING DERIVATIVES BY A CLASS OF POSITIVE LINEAR OPERATORS 61

Proof. If f(k)(x) exists at some x ∈ IR+, given an arbitrary � > 0 we can find a δ
satisfying x > δ > 0 s.t.

f(u) =
∑k
p=0

f(p)(x)
p!

(u−x)p + hx(u)(u−x)k; |u−x| ≤ δ,
where hx(u) is certain measurable function on [x − δ,x + δ] satisfying the in-

equality |hx(u)| ≤ �, |u−x| ≤ δ. Hence
(11) L

(k)
n (f; x) =

∑k
p=0

f(p)(x)
p!

∑p
j=0

(
p
j

)
(−1)jL(k)n (up−j; x)+

+L
(k)
n (hx(u)(u−x)kχδ,x(u); x) + L

(k)
n (fχ

c
δ,x; x)

=
∑

1 +
∑

2 +
∑

3, (say).
Using the fact that Ln maps polynomials to polynomials and the basic conver-

gence Theorem3, we obtain
(12)

∑
1 = f

(k)(x)Ln(u
k; 1) → f(k)(x),n →∞.

It follows from Lemma1 that

L
(k)
n (hx(u)(u−x)kχδ,x(u); x) = xmn+α−1D(m,n,α)

∑k
p=0

∑[ k−p
2

]
ν=0 n

ν+p

×
∫x+δ
x−δ u

−mn−αhx(u)(u−
x)m{( x

u
)me−(

x
u

)m}(n−k)
×[ ∂

∂x
{( x
u

)me−(
x
u

)m}]kgν,p,k(x,u)du
The δ above can be chosen so small that∣∣ ∂

∂x
{( x
u

)me−(
x
u

)m}
∣∣ ≤ A |u−x| , |u−x| < δ,

where A is some constant. Since the functions gν,p,k(x,u) are bounded on [x−
δ,x + δ], it is clear that there exists a constant M1 independent of n,� and δ s.t.
for all n sufficiently large,∣∣∣ L(k)n (hx(u)(u−x)kχδ,x(u); x)∣∣∣ ≤ �M1 ∑kp=0 ∑[ k−p2 ]ν=0 nν+p−k+p2

by (3) where M2 is another constant not depending on n,� and δ. Since ν ≤
[k−p

2
],ν + p− p+k

2
− [k−p

2
] − k−p

2
≤ 0 there exists a

constant M independent of n,� and δ s.t.
(13) |

∑
2| ≤ M for all sufficiently large n. To estimate

∑
3, first of all we

notice that there exist a positive integer p and a positive constant P such that∣∣[{(m
u

)e−(
x
u

)m( x
u

)m−1}{1 − ( x
u

)m−1}]kgν,p,k(x,u)
∣∣ ≤ P(1 + u−m),u ∈

IR+

and 0 ≤ p ≤ k, 0 ≤ ν ≤ [k−p
2

]. Hence by Lemma1, we have

|
∑

3| ≤ P
∑k
p=0

∑[ k−p
2

]
ν=0 n

ν+pxmn+α−1D(m,n,α)

×
∫∞

0
u−mn−α(1+u−m)( x

u
)mn−ke−(n−k)(

x
u

)mf(u)χcδ,x(u)du

= P
∑k
p=0

∑[ k−p
2

]
ν=0 n

ν+p D(m,n,α)
D(m,n−k,α)Ln−k(fχδ,x; x) +

D(m,n,α)
D∗∗(m,n−k,α)L

∗∗
n−k(fχ

c
δ,x; x)

where L∗∗n corresponds to to the operator (1) with α replaced by α + m and
D∗∗(m,n,α) refers to D(m,n,α) for L∗∗n . We observe that

limn→∞
D(m,n,α)

D(m,n−k,α) = limn→∞
D(m,n,α)

D∗∗(m,n−k,α)
Also, by the definition of the operator Ln, we have

limn→∞n
ν+pLn−k(fχ

c
δ,x; x) = limn→∞n

ν+pL∗∗n−k(fχ
c
δ,x; x)

= 0
It follows that

∑
3 → 0 as n → ∞. In view of this fact and (11) −−(13), it

follows that there exists an n0 s.t.∣∣∣L(k)n (f; x) −f(k)(x)∣∣∣ < (2 + M)�,n > n0.
Since M does not depend on � we have (10).


62 BRAMHA DUTTA PANDEY1,∗ AND B. KUNWAR2

The uniformity part is easy to derive from the above proof by noting that, to
begin with, δ can be chosen independent of x ∈ [a,b] so that |hx(u)| ≤ � for x ∈ [a,b]
whenever |u−x| ≤ δ. Then, it is clear that the various constants occuring in the
above proof can be chosen independent of x ∈ [a,b]. This completes the proof of
the theorem. �

Finally, we show that the asymptotic formula of Theorem2 remains valid in
the pointwise simultaneous approximation as well. We observe that the difference
between Theorem2 and the following one lies in the assumptions of f . We have

Theorem 6. : If f ∈ DΩ, then
(14) L

(k)
n (f; x) −f(k)(x) = − 12nm2 [f

(k)(x)k{(2α−k − 5)}+

+xf(k+1)(x){2(α−k−3) + (3−k)}+x2f(k+2)(x)]+
+o( 1

n
),n →∞.

whenever x ∈ IR+ is s.t. f(k+2)(x) exists. Also if f(k+2)(x) exists and is
continuous on < a,b >, (14) holds uniformly in x ∈ [a,b].

Proof. If f(k+2) exists, we have

f (u) =
∑k+2
p=0

f(p)(x)
p!

(u−x)p + h(u,x),
where h(u,x) ∈ DΩ and for any � > 0, there exist a δ > 0 s.t. |h(u,x)| ≤

� |u−x|k+2 for all sufficiently |u−x| ≤ δ. Thus,
(15) L

(k)
n (f; x) = L

(k)
n (Q; x) + L

(k)
n (h(u,x); x),

where Q =
∑k+2
p=0

f(p)(x)
p!

(u − x)p is a polynomial in u.Clearly
Q ∈ D(k)Ω . Also,

Q(p)(x) = f(p)(x), for p = k,k + 1,k + 2.
Hence, applying Theorem2, we have

(16) L
(k)
n (Q; x) −f(k)(x) = − 12nm2 [k(2α−k −m− 2)f

(k)(x)+

+(2α− 2k −m− 3)xf(k+1)(x) + x2f(k+2)(x)] + o( 1
n

),
n →∞.

To establish (14), it remains to show that

(17)
∣∣∣L(k)n (h(u,x); x)∣∣∣ ≤ D(m,n,α)xα−1 ∑kp=0 ∑[ k−p2 ]ν=0 mnν+p ∫∞0 xmnu−mn−α−1e−n( xu )m

×
∣∣( x
u

)m−1e−(
x
u

)m{1 − ( x
u

)}m−1
∣∣gν,p,k(x,u){h(u,x)χcδ,x(u) +

� |u−x|k+2}du
Proceeding as in the proof of Theorem5, we find that the term corresponding

to � in the above is bounded by �M
n

for some M independent of � and n and χcδ,x−
term contributes only a o( 1

n
) quantity (in fact o( 1

ns
) for an arbitrary s > 0). Then

in view of arbitraryness of � > 0, (17) follows.

The uniformity part follows as a remark similar to that made for the proof of
the uniformity part of Theorem5. This completes the proof of the theorem. �

In the rest of the paper, we study the second problem.

4. Some Direct Theorems for Linear Combinations

In this section we give some direct theorems for the the linear combinations of the
operators Ln. First, we give some definitions. The k

th-moment µn,k(x),k ∈ IN0
(set of non-negative integers) of the operators Ln [5] is defined by


APPROXIMATING DERIVATIVES BY A CLASS OF POSITIVE LINEAR OPERATORS 63

(18) µn,k(x) = Ln((u−x)k; x) = xkτn,k (say).

Clearly, τn,k is independent of x.Now we first prove the lemma on the moments
µn,k.

Lemma 2. : If k ∈ IN0.Then there exist constants γk,ν,ν ≥ [k+12 ] s.t. the follow-
ing asymptotic expansion is valid:

(19) τν,k =
∑∞
ν=[ k+1

2
] γk,νn ν, n →∞.

Proof. Let 1
3
< γ < 1

2
. Then

τn,k =
∫ 1+n−γ

1−n−γ s
α−k−2(1 −s)k exp[n log{e−1 −m2 (s−1)

2

2!
e−1+

... +
(s−1)2p

2p!
( d

2p

dx2p
{( x
u

)me−(
x
u

)m}) x
u

=1 + o((s− 1)2p)}]ds,
(p ≥ 2)

= e−n
∫ 1+n−γ

1−n−γ s
α−k−2(1 −s)k exp[−nm2 (s−1)

2

2
]

×exp[{C3(s−1)3 +C4(s−1)4 +...+C2p(s−1)2p +o((s−1)2p)}]ds
C′is being constants.

= e−n
∫ 1+n−γ

1−n−γ s
α−k−2(1−s)k exp[−nm2 (s−1)

2

2
]{1+

∑
3≤3i≤j≤[2p+ i−1

γ
] bijn

i(s−
1)j + o(n1−2pγ)}ds
b′ijs depending on C

′
is.

= e−n
∫ 1+n−γ

1−n−γ exp[−nm
2 (s−1)2

2
][{
∑2p−1

γ

l=0 al(s− 1)
k+l}

×{1 +
∑

3≤3i≤j≤[2p+ i−1
γ

] bijn
i(s−1)j}+

o(n1−(2p+k)γ)]ds

= e−n
∫ 1+n−γ

1−n−γ exp[−nm
2 (s−1)2

2
][
∑

3≤3i≤j≤[2p+ i−1
γ

]

0≤l≤[2p−1
γ

]

dijln
i(s − 1)j+k+l +

o(n1−(2p+k)γ)]ds
where d′ijls are certain constants depending on a

′
ls and b

′
ijs and vanish if j +k+l

is odd.
Using substitutions we get

= 2
1
2
e−n

mn
1
2

∫−n
0

t
[
j+k+l+1

2
]− 1

2

et
[1 + γm2

∑
(0≤3i≤j≤[2p+ i−1γ ])

d∗ijln
i−[ j+k+l−1

2
] +

o(n1−(2p+k)γ+1−2γ)]dt

where d∗ijl = dijl{
2
m2
}[
j+k+l−1

2
].

= 2
1
2
e−n

mn
1
2

[
∑

0≤3i≤j≤[2p+ i−1
γ

]

0≤l≤[2p−1
γ

]

d∗∗ijln
i−[ j+k+l−1

2
]

+ o(n2−(2p+2+k)γ)]

where d∗∗ijl = d
∗
ijlΓ(([

j+k+l−1
2

])γ + 1
2
) and we have made use of the fact that by

enlarging the integral in the above from 0 to ∞, we are only adding the terms in n
which decay exponentially and therefore can be absorbed in the o-term.

Next, we analyse the expression∫
(0,∞)−(1−n−γ,1+n−γ) s

mn+α−k−2(1 −s)ke−ns
m

ds = E(n) (say).

We have for any positive integer q,

|E(n)| ≤ nγqD∗∗(m,n,α)L∗∗n (|u− 1|
k+q

; 1),
where D∗∗(m,n,α) and L∗∗n are the same as considered in the proof of Theorem5.

By making use of an estimate for the operators L∗∗n , we have

|E(n)| ≤ Anγq−
k+q
2 D∗∗(m,n,α),


64 BRAMHA DUTTA PANDEY1,∗ AND B. KUNWAR2

where A is certain constant not depending upon n. Again making use of the
same estimate as above for D∗∗(m,n,α), we have

en |E(n)| = o(nγq−
k+q+1

2 ).
Thus, choosing q s.t.

p ≥ 2(2p+2+k)
1−2γ , we have∫∞

0
smn+α−k−2(1 −s)ke−ns

m

ds

= 2
1
2
e−n

mn
1
2

[
∑

(0≤3i≤j≤[2p+ i−1γ ])
0≤l≤[2p−1

γ
]

d∗∗ijln
i−[ j+k+l−1

2
] + o(n2−(2p+k+2)γ)].

Now, for all indices under consideration we have
[j+k+l+1

2
] − i = [j−2i+k+l+1

2
] ≥ [k+1

2
],

and since p could be chosen arbitrarily large, there exist constants Ck,ν,ν ≥ [k+12 ]
s.t. we have the following asymptotic expansion∫∞

0
smn+α−k−2(1 −s)ke−ns

m

ds

= 2
1
2
e−n

mn
1
2

∑∞
ν=[ k+1

2
]
Ck,ν
nν

Noting that C0,0 = 1, it follows that there exist constants γk,ν,ν ≥ [k+12 ]
s.t. (19) holds. This completes the proof of Lemma2. �

For any fixed set of positive constants αi, i = 0, 1, 2, ...,k following [9] the linear

combination Ln,k of the operators Lαi,n, i = 0, 1, 2, ...k is defined by

(20) Ln,k(f; x) =
1
4

∣∣∣∣∣∣∣∣∣∣
Lα0n(f; x) α

−1
0 α

−2
0 ... ... α

−k
0

Lα1n(f; x) α
−1
1 α

−2
1 ... ... α

−k
1

.... ... ... ... ... ...

.... ... ... ... ... ...

Lαkn(f; x) α
−1
k α

−2
k ... ... α

−k
k

∣∣∣∣∣∣∣∣∣∣
where 4 is the determinant obtained by replacing the operator column by the

entries ′1′. Clearly

(21) Ln,k =
∑k
j=0 C(j,k)Lαjn,

for constants C(j,k),j = 0, 1, 2, ...,k which satisfy
∑k
j=1 C(j,k) = 1.

Ln,k is called a linear combination of order k. Ln,0 denotes the operator Ln

itself.

Theorem 7. : If f ∈ DΩ. If at a point x ∈ IR+,f(2k+2)exists, then

(22) |Ln,k(f; x) −f(x)| = O(n−(k+1)),
(23) |Ln,k+1(f; x) −f(x)| = o(n−(k+1)),
where k = 0, 1, 2, ... . Also, if f(2k+2) exists and is continuous on < a,b >⊂ IR+,

(22) −−(23) hold uniformly on [a,b].

Proof. First we show that

(24) Ln(f; x) −f(x) =
∑2k+2
j=1

xjf(j)(x)
j!

τn,j + o(n
−(k+1)),

if x ∈ IR+ is such that f(2k+2) exists and f ∈ DΩ.To prove (24) with the
assumption on f, we have

f(u) −f(x) =
∑2k+2
j=1

f(j)(x)
j!

(u−x)j + Rx(u); u → x,


APPROXIMATING DERIVATIVES BY A CLASS OF POSITIVE LINEAR OPERATORS 65

where Rx(u) = o((u−x)2k+2),u → x. It is clear from the definition of τn,j that
we only have to show that

(25) Ln(Rx(u); x) = o(n
−(k+1)).

Obviously, Rx(u) ∈ DΩ. Now, given an arbitrary � > 0, we can choose a δ > 0
s.t.

|Rx(u)| ≤ �(u−x)2k+2, |u−x| ≤ δ.
Hence, by using the basic properties of Ln[1], we note that the result follows.In

this case the uniformity part is obvious. Now, using Lemma2 and (24) we get

(26) Ln(f; x) −f(x) =
∑2k+2
j=1

xjf(j)(x)
j!

∑k+1
ν=[

j+1
2

]

γj,ν
nν

+ o(n−(k+1)),

which, in the uniformity case holds uniformly in x ∈ [a,b].Since the coefficients
C(j,k) in (21) obviously satisfy the relation

(27)
∑k
j=0 C(j,k)α

−p
j = 0,p = 1, 2, 3, ...,k.

In view of (26), (22) −−(23) are immediate and so is the uniformity part. This
completes the proof of Theorem7. �

In the same spirit we have,

Theorem 8. :.Let f ∈ DΩ. If 0 ≤ p ≤ 2k + 2 and f(p) exists and is con-
tinuous on < a,b >⊂ IR+, for each x ∈ [a,b] and sufficiently large n then
(28) |Ln,k(f; x) −f(x)| ≤ max[Cn−

p
2 ω(f(p); n−

1
2 ),C′n−(k+1)]

where C = C(k) and C′ = C′(k,f) are constants and ω(f(p); δ) denotes the local
modulus of continuity of f(p) on < a,b > .

Proof. :There exists a δ > 0 s.t. [a − δ,b + δ] ⊂< a,b > . It is clear that if
u ∈< a,b >, there exists an η lying between x ∈ [a,b] and u s.t.

(29)
∣∣∣f(u) −f(x) −∑pj=1 f(j)(x)j! (u−x)j∣∣∣ ≤ |u−x|pp! (1+|u−x|n12 )ω(f(p); n−12 ),

using a well known result on modulus of continuity [13]. If the expression occur-
ing within the modulus sign on L.H.S. of the above inequality is denoted by Fx(u),
by a well known property of Ln, it follows that

Lαjn(Fx(u)χ
c
δ,x(u); x) = o(n

−(k+1)),

uniformly in x ∈ [a,b]. By (29), we have
(30)

∣∣∣Lαjn(Fx(u)χcδ,x(u); x)∣∣∣ ≤ bpp! (Ap + Ap−1)(αjn)−p2 ω(f(p); n−12 )
for all n sufficiently large and x ∈ [a,b]. Here Ap,Ap−1 are constants depending

on p. Hence, for a constant Cp independent of f such that for all x ∈ [a,b],
(31)

∣∣∣Ln,k(Fx(u)χcδ,x(u); x)∣∣∣ ≤ Cpn−p2 ω(f(p); n−12 ).
Applying the result (22) for the functions 1,u,u2,u3, ...,up, we find that there

exists a constant C′′ depending on max{|f′(x)| , ...,
∣∣f(p)(x)∣∣ ; x ∈ [a,b]} and p such

that for all x ∈ [a,b],
(32)

∣∣∣Ln,k(∑pj=1 f(j)(x)j! (u−x)j; x)∣∣∣ ≤ C′′n−(k+1)
Now, (28) is clear from (30) −−(32). This completes the proof of the Theorem.

�

Theorem 9. :Let f ∈ DΩ. If at a point x ∈ IR+,f(2k+p+2) exists then
(33)

∣∣∣L(p)n,k(f; x) −f(p)(x)∣∣∣ = O(n−(k+1)),and
(34)

∣∣∣L(p)n,k+1(f; x) −f(p)(x)∣∣∣ = o(n−(k+1)),
where k = 0, 1, 2, ... . Also, if f(2k+p+2) exists and is continuous on < a,b >⊂

IR+, (33) −−− (34) hold uniformly in x ∈ [a,b].


66 BRAMHA DUTTA PANDEY1,∗ AND B. KUNWAR2

Proof. If f(2k+p+2) exists , we can find a neighbourhood (a′,b′) of x s.t. f(p) exists
and is continuous on (a′,b′). Let g(u) be an infinitely differentiable function with
supp g ⊆ (a′,b′) s.t. g(u) = 1 for u ∈ [x − δ,x + δ] for some δ > 0. Then an
application of Lemma1 shows that

(35) L
(p)
n,k(f(u) −f(u)g(u); x) = o(n

−(k+1)).

In the uniformity case, we consider a g(u) with supp g ⊂< a,b > with g(u) = 1,

for u ∈ [a − δ,b + δ] ⊆< a,b > and then (34) holds uniformly in x ∈ [a,b]. since
f(u)g(u) ∈ C(p)b IR

+ we have

(36) L
(p)
n (fg; x) = x

−pLn(u
p{f(u)g(u)}(p); x).

Now, since up{f(u)g(u)}(p) is (2k+2)−times differentiable at x (and continuously
differentiable on (a−δ,b + δ) in the uniformity case), applying Theorem7 we have

(37)
∣∣∣L(p)n,k(fg; x) −f(p)(x)∣∣∣ = O(n−(k+1)),and

(38)
∣∣∣L(p)n,k+1(fg; x) −f(p)(x)∣∣∣ = o(n−(k+1)),

where, in the uniformity case these holds in x ∈ [a,b]. Thus, combining (35) −
−− (38), we get (33) −−− (34). This completes the proof . �

Theorem 10. : Let m ∈ IN, and f ∈ DΩ. If 0≤ q ≤ 2k + 2 and f(p+q) exists and
is continuous on < a,b >⊆ IR+ for each x ∈ [a,b],then for all sufficiently large n,

(39)
∣∣∣L(p)n,k(f; x) −f(p)(x)∣∣∣ ≤ max{Cpn−( k2 )ω(f(p+q); n−12 ),C′pn−(k+1)}

where Cp = Cp(k),C
′
p = C

′
p(k,f) are constants and ω(f

(p+q); δ) denotes the local

modulus of continuity of f(p+q) on < a,b > .

Proof. : The proof of this Theorem follows from Lemma1 and Theorem5−−9. �

References

[1] BRAMHA DUTTA PANDEY AND B. KUNWAR, On a class of positive linear opera-

tors(communicated).
[2] R. A. DEVORE AND G. G. LORENTZ, Constructive Approximation. Springer-Verlag Berlin

Heidelberg, New York, (1993).

[3] P.P. KOROVKIN, Linear Operators and Approximation Theory (1960), Delhi. (Translated

from Russian edition of 1959).
[4] B. Kunwar, Approximation of analytic functions by a class of linear positive operators, J.

Approx. Theory 44. 173-182 (1985).

[5] G. G. LORENTZ, Bernstein Polynomials, Toronto (1953).
[6] C. P. MAY, Saturation and inverse theorems for combination of a class of exponential type

operators. Canad. J. Math. 28,(1976), 1224-50.
[7] M. W.MULLER, Die Folge der Gamma operatoren.Thesis,Technische

Hoschule,Stuttgort.(1967).

[8] Paula Anamaria Piţul, Evaluation of the approximation order by positive linear operators,
Ph. D. Thesis, Universitatea Babeş Bolyai(Romania) 2007.

[9] R. K. S. RATHORE, Linear combinations of linear positive operators and generating relations

in special functions, Dissertation, IIT Delhi (India) (1973).
[10] R. K. S. RATHORE, Approximation of unbounded functions with Linear Positive Operators,

Doctoral Dissertation, Technische Hogeschool Delft (1974).

[11] P. C. SIKKEMA AND R. K. S. RATHORE,Convolution with powers of bell shaped functions.
Report, Dept. of Math. Technische Hogeschool (1976).


APPROXIMATING DERIVATIVES BY A CLASS OF POSITIVE LINEAR OPERATORS 67

[12] P. C. SIKKEMA, Approximation formulae of Voronovskaya- type for certain convolution
operators, J. Approx. Theory 26(1979), 26-45.

[13] A. F. TIMAN, Theory of Approximation of Functions of a Real variable, Peargamon Press

(1963).

1Department of Applied Sciences and Humanities, Institute of Engineering and Tech-
nology, Lucknow-21 (India)

2Department of Applied Sciences and Humanities, Institute of Engineering and Tech-
nology, Lucknow-21 (India)

∗Corresponding author

The author is thankful to Councel of Scientific and Industrial Research, INDIA for

providing financial assistance for this research work under grant no.09/827(0004).