Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 3, Article 79, 2004
RATE OF CONVERGENCE FOR A GENERAL SEQUENCE OF DURRMEYER
TYPE OPERATORS
NIRAJ KUMAR
S CHOOL OF A PPLIED S CIENCES
N ETAJI S UBHAS I NSTITUTE OF T ECHNOLOGY
S ECTOR 3, DWARKA
N EW D ELHI 110045, I NDIA .
neeraj@nsit.ac.in
Received 21 January, 2004; accepted 14 June, 2004
Communicated by C.P. Niculescu
A BSTRACT. In the present paper we give the rate of convergence for the linear combinations
of the generalized Durrmeyer type operators which includes the well known Szasz-Durrmeyer
operators and Baskakov-Durrmeyer operators as special cases.
Key words and phrases: Szasz-Durrmeyer operators, Baskakov-Durrmeyer operators, Rate of convergence, Order of approximation.
2000 Mathematics Subject Classification. 41A16, 41A25.
1. I NTRODUCTION
Durrmeyer [1] introduced the integral modification of Bernstein polynomials so as to approximate Lebesgue integrable functions on the interval [0, 1]. We now consider the general family
of Durrmeyer type operators, which is defined by
Z ∞
∞
X
(1.1)
Sn (f (t), x) = (n − c)
pn,v (x)
pn,v (t) f (t) dt, x ∈ I
v=0
0
v
(v)
where n ∈ N , n > max {0, −c} and pn,v (x) = (−1)v xv! ϕn (x). Also {φn } is a sequence
of real functions having the following properties on [0, a], where a > 0 and for all n ∈ N,
v ∈ N ∪ {0}, we have
(I) φn ∈ C ∞ [0, a], φn (0) = 1.
(II) φn is complete monotonic.
(v+1)
(v)
(III) There exist c ∈ N : φn
= −nφn+c , n > max {0, −c} .
Some special cases of the operators (1.1) are as follows:
(1) If c = 0,φn (x) = e−nx , we get the Szasz-Durrmeyer operator.
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
The author is extremely grateful to the referee for making valuable suggestions leading to the better presentation of the paper.
016-04
2
N IRAJ K UMAR
(2) If c = 1, φn (x) = (1 + x)−n , we obtain the Baskakov-Durrmeyer operator.
−n
(3) If c > 1, φn (x) = (1 + cx) c , we obtain a general Baskakov-Durrmeyer operator.
(4) If c = −1, φn (x) = (1 − x)n , we obtain the Bernstein-Durrmeyer operator.
Very recently Srivastava and Gupta [9] studied a similar type of operators and obtained the
rate of convergence for functions of bounded variation. It is easily verified that the operators
(1.1) are linear positive operators and these operators reproduce the constant ones, while the
operators studied in [9] reproduce every linear functional for the case c = 0. Several researchers
studied different approximation properties on the special cases of the operators (1.1), the pioneer
work on Durrmeyer type operators is due to S. Guo [3], Vijay Gupta (see e. g. [4], [5]), R. P.
Sinha et al. [8] and Wang and Guo [11], etc. It turns out that the order of approximation for
such type of Durrmeyer operators is at best O (n−1 ), how so ever smooth the function may be.
In order to improve the order of approximation, we have to slacken the positivity condition of
the operators, for this we consider the linear combinations of the operators (1.1). The technique
of linear combinations is described as follows:
1 d−1 d−2 ..... d−k −1 Sd n (f, x) d−1 d−2 ..... d−k 0
0
0
0
0
0
0
1 d−1 d−2 ..... d−k Sd n (f, x) d−1 d−2 ..... d−k 1
1
1
1
1
1
1
Sn,k (f, x) = ... ..
..
..
..... .. .
..
..... .. ...
... ..
..
..... .. ...
..
..
..... .. 1 d−1 d−2 ..... d−k Sd n (f, x) d−1 d−2 ..... d−k k
k
k
k
k
k
k
Such types of linear combinations were first considered by May [7] to improve the order of
approximation of exponential type operators. In the alternative form the above linear combinations can be defined as
k
X
Sn,k (f, x) =
c(j, k)Sdj n (f, x),
j=0
where
C(j, k) =
k
Y
i=0
i6=j
dj
, k 6= 0
dj − di
and
C(0, 0) = 1.
If f ∈ Lp [0, ∞), 1 ≤ p < ∞ and 0 < a1 < a3 < a2 < b2 < b3 < b1 < ∞ and Ii = [ai , bi ],
i = 1, 2, 3, the Steklov mean fη,m of mth order corresponding to f is defined as
Z η
Z η
m
Y
2
2
−m
Pm
fη,m (t) = η
f (t) + (−1)m−1 ∆m
f
(t)
dti ; t ∈ I1 .
···
i=1 ti
−η
2
−η
2
i=1
It can be verified [6, 10] that
(m−1)
(m)
(i) fη,m has derivative up to order m, fη,m ∈ AC(I1 ) and fη,m exist almost every where
and
to Lp (I1 );
belongs
(r) (ii) fη,m ≤ K1 η −r ωr (f, η, p, I1 ), r = 1, 2, 3, . . . , m;
Lp (I1 )
(iii) kf − fη,m kLp (I2 ) ≤ K2 ωm (f, η, p, I1 );
m (iv) fη,m
≤ K3 η m kf kLp (I1 )
Lp (I2 )
(m) (v) fη,m ≤ K4 η −r ωr (f, η, p, I1 )
Lp (I1 )
Ki0 s,
where
i = 1, 2, 3, 4 are certain constants independent of f and η.
In the present paper we establish the rate of convergence for the combinations of the generalized Durrmeyer type operators in Lp −norm.
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
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C ONVERGENCE F OR A S EQUENCE O F D URRMEYER T YPE O PERATORS
3
2. AUXILIARY R ESULTS
To prove the rate of convergence we need the following lemmas:
Lemma 2.1. Let m ∈ N ∪ {0}, also µn,m (x) is the mth order central moment defined by
Z ∞
∞
X
m
µn,m ≡ Sn ((t − x) , x) = (n − c)
pn,v (x)
pn,v (t)(t − x)m dt,
0
v=0
then
(i) µn,m (x) is a polynomial in x of degree m.
(ii) µn,m (x) is a rational function in n and for each 0 ≤ x < ∞
µn,m (x) = O n−[(m+1)/2] .
Remark 2.2. Using Hölder’s inequality, it can be easily verified that Sn (|t − x|r , x) = O(n−r/2 )
for each r > 0 and for every fixed 0 ≤ x < ∞.
Lemma 2.3. For sufficiently large n and q ∈ N, there holds
Sn,k ((t − x)q , x) = n−(k+1) {F (q, k, x) + o(1)},
where F (q, k, x) are certain polynomials in x of degree q and 0 ≤ x < ∞ is arbitrary but fixed.
Proof. For sufficiently large values of n we can write, from Lemma 2.1
Q[q/2] (x)
Q0 (x)
Q1 (x)
+ [(q+1)/2]+1 + · · · +
,
[(q+1)/2]
n
n
nq
where Qi (x),i = 0, 1, 2, . . . are certain polynomials in x of at most degree q.
Therefore Sn,k ((t − x)q , x) is given by
Q0 (x)
1 (x)
+ (d0 n)Q[(q+1)/2]+1
+ .... d−1
d−2
1 d−1 d−2 ..... d−k −1 (d0 n)[(q+1)/2]
0
0
0
0
0
Q1 (x)
Q0 (x)
−1
−2
1 d−1 d−2 ..... d−k 1
1
1
(d1 n)[(q+1)/2] + (d1 n)[(q+1)/2]+1 + .... d1 d1
... ..
..
..... .. ............................
..
..
... ..
............................
..
.....
..
..
..
1 d−1 d−2 ..... d−k Q0 (x)
Q1 (x)
−1
−2
+
+
....
d
d
k
k
k
k
k
(dk n)[(q+1)/2]
(dk n)[(q+1)/2]+1
Sn ((t − x)q , x) =
= n−(k+1) {F (q, k, x) + o(1)} ,
..... d−k
0
..... d−k
1
..... .. ..... .. ..... d−k
k
0 ≤ x < ∞.
Lemma 2.4 ([4]). Let 1 ≤ p < ∞, f ∈ Lp [a, b], f (m) ∈ AC [a, b] and f (m+1) ∈ Lp [a, b], then
n
o
(j) f (m+1) f ≤
C
+
kf
k
Lp [a,b] ,
Lp [a,b]
Lp [a,b]
j = 1, 2, 3, . . . , m and C is a constant depending on j, p, m, a, b.
Lemma 2.5. Let f ∈ Lp [0, ∞), p > 1. If f (2k+1) ∈ AC(I1 ) and f (2k+2) ∈ Lp (I1 ), then for all
n sufficiently large
n
o
−(k+1) (2k+2) (2.1)
kSn,k (f, ·) − f kLp (I2 ) ≤ C1 n
f
+ kf kLp [0,∞) .
Lp (I2 )
Also if f ∈ L1 [0, ∞), f (2k+1) ∈ L1 (I1 ) with f (2k) ∈ AC(I1 ) and f (2k+1) ∈ BV (I1 ), then for
all n sufficiently large
n
o
(2k+1) −(k+1) (2k+1) (2.2) kSn,k (f, ·) − f kLp (I2 ) ≤ C2 n
f
+ f
+ kf kL1 [0,∞) .
BV (I1 )
L1 (I2 )
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
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4
N IRAJ K UMAR
Proof. First let p > 1. By the hypothesis, for all t ∈ [0, ∞) and x ∈ I2 , we have
Sn,k (f, x) − f (x) =
2k+1
X
i=1
f (i) (x)
Sn,k ((t − x)i , x)
i!
Z t
1
+
Sn,k (φ(t)
(t − w)2k+1 f (2k+2) (w)dw, x)
(2k + 1)!
x
+ Sn,k (F (t, x)(1 − φ(t)), x)
= E1 + E2 + E3 , say,
where φ(t) denotes the characteristic function of I1 and
F (t, x) = f (t) −
2k+1
X
i=0
(t − x)i (i)
f (x).
i!
Applying Lemma 2.3 and Lemma 2.4, we have
kE1 kLp (I2 ) ≤ C3 n
−(k+1)
2k+1
X
n
o
(i) (2k+2) −(k+1)
f ≤
C
n
kf
k
+
f
.
4
Lp (I2 )
Lp (I2 )
Lp (I2 )
i=1
Next we estimate E2 . Let Hf be the Hardy Littlewood maximal function [10] of f (2k+2) on I1 ,
using Hölder’s inequality and Lemma 2.1, we have
Z t
1
2k+1 (2k+2)
|E2 | ≤
Sn,k φ(t) |t − w|
f
(w) dw , x
(2k + 2)!
x
Z t
(2k+2)
1
2k+1 f
Sn,k φ(t) |t − x|
≤
(w)dw , x
(2k + 2)!
x
1
Sn,k φ(t) |t − x|2k+2 |Hf (t)| , x
≤
(2k + 2)!
n
o 1q
1
1
q(2k+2)
≤
Sn,k φ(t) |t − x|
,x
{Sn,k (φ(t) |Hf (t)|p , x)} p
(2k + 2)!
! p1
Z b1
k
∞
X
X
.
≤ C5 n−(k+1)
C(j, k)
(dj n − c)
pdj n,v (x)pdj n,v (t) |Hf (t)|p dt
a1
j=0
v=0
Next applying Fubini’s theorem, we have
kE2 kpLp (I2 )
≤ C6 n
−p(k+1)
k
X
Z
≤ C6 n−p(k+1)
≤ C7 n
(dj n − c)
a2
Z
b1
a1
b2
(dj n − c)
a1
kHf kpLp (I1 )
∞
X
≤ C8 n
pdj n,v (x)pdj n,v (t) |Hf (t)|p dtdx
v=0
Z
C(j, k)
j=0
−p(k+1)
b1
Z
C(j, k)
j=0
k
X
b2
a2
−p(k+1)
∞
X
!
pdj n,v (x)pdj n,v (t)dx |Hf (t)|p dt
v=0
(2k+2) p
f
.
Lp (I1 )
Therefore
kE2 kLp (I2 ) ≤ C8 n−(k+1) f (2k+2) Lp (I1 ) .
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
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C ONVERGENCE F OR A S EQUENCE O F D URRMEYER T YPE O PERATORS
5
For t ∈ [0, ∞)\[a1 , b1 ], x ∈ I2 there exists a δ > 0 such that |t − x| ≥ δ. Thus
E3 ≡ Sn,k (F (t, x)(1 − φ(t)), x))
≤δ
−(2k+2)
k
X
C(j, k)Sdj n |F (t, x)| (t − x)2k+2 , x
j=0
≤ δ −(2k+2)
k
X
"
C(j, k) Sdj n |f (t)| (t − x)2k+2 , x
j=0
+
2k+1
X
(i) f (x)
i=0
i!
Sdj n |t − x|2k+2+i , x
#
= E31 + E32 , say.
Applying Hölder’s inequality and Lemma 2.1, we have
|E31 | ≤ δ −(2k+2)
k
X
o 1q
1 n
C(j, k) Sdj n (|f (t)|p , x) p Sdj n (|t − x|q(2k+2) , x)
j=0
≤ C9
k
X
1
C(j, k) Sdj n (|f (t)|p , x) p
j=0
1
.
(dj n)k+1
Finally by Fubini’s theorem, we obtain
Z b2
p
kE31 kLp (I2 ) =
|E31 |p dx
a2
≤ C9
k
X
−p(k+1)
Z
b2
∞
Z
(dj n − c)pdj n,v (x)pdj n,v (t) |f (t)|p dtdx
C(j, k)(dj n)
a2
j=0
0
≤ C10 n−p(k+1) kf kpLp [0,∞) .
Again by Lemma 2.4, we have
n
o
kE32 kLp (I2 ) ≤ C11 n−(k+1) kf kLp (I2 ) + f (2k+2) Lp (I2 ) .
Thus
n
o
kE3 kLp (I2 ) ≤ C12 n−(k+1) kf kLp [0,∞) + f (2k+2) Lp (I2 ) .
Combining the estimates of E1 , E2 , E3 , we get (2.1).
Next suppose p = 1. By the hypothesis, for almost all x ∈ I2 and for all t ∈ [0, ∞), we have
2k+1
X
f (i) (x)
Sn,k ((t − x)i , x)
i!
i=1
Z t
1
+
Sn,k (φ(t)
(t − w)2k+1 f (2k+1) (w)dw, x)
(2k + 1)!
x
+ Sn,k (F (t, x)(1 − φ(t)), x)
= M1 + M2 + M3 ,
say,
Sn,k (f, x) − f (x) =
whereφ(t) denotes the characteristic function of I1 and
F (t, x) = f (t) −
2k+1
X
i=0
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
(t − x)i (i)
f (x),
i!
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6
N IRAJ K UMAR
for almost all x ∈ I2 and t ∈ [0, ∞).
Applying Lemma 2.3 and Lemma 2.4, we have
n
o
kM1 kL1 (I2 ) ≤ C13 n−(k+1) kf kL1 (I2 ) + f (2k+1) L1 (I2 ) .
Next, we have
kM2 kL1 (I2 )
k
X
1
≤
|C(j, k)|
(2k + 1)! j=0
Z
b2
Z
b1
(dj n − c)
a2
a1
∞
X
pdj n,v (x)pdj n,v (t) |t − x|2k+1
v=0
Z t
(2k+1)
× df
(w) dtdx.
x
For each dj n there exists a non negative integer r = r(dj n) satisfying
r(dj n)−1/2 < max(b1 − a2 , b2 − a1 ) ≤ (r + 1)(dj n)−1/2 .
Thus
(Z
k
r Z b2
x+(l+1)(dj n)−1/2
X
X
1
kM2 kL1 (I2 ) ≤
|C(j, k)|
φ(t)(dj n − c)
(2k + 1)! j=0
−1/2
a
x+(l)(d
n)
2
j
l=0
) Z
!
∞
x+(l+1)(dj n)−1/2
X
(2k+1)
2k+1
.
pdj n,v (x)pdj n,v (t) |t − x|
φ(w) df
(w) dtdx
x
v=0
1
+
(2k + 1)!
×
∞
X
k
X
|C(j, k)|
j=0
r Z b2
X
l=0
pdj n,v (x)pdj n,v (t) |t − x|
x−(l)(dj n)−1/2
φ(t)(dj n − c)
x+(l+1)(dj n)−1/2
a2
)
2k+1
(Z
Z
x
·
x−(l+1)(dj n)−1/2
v=0
!
(2k+1)
φ(w) df
(w) dtdx.
Suppose φx,c,s (w)
denotes
the
characteristic
function
of
the
interval
x − c(dj n)−1/2 ,
−1/2
x + s(dj n)
, where c, s are nonnegative integers. Then we have
(Z
k
r Z b2
x+(l+1)(dj n)−1/2
X
X
1
kM2 kL1 (I2 ) ≤
|C(j, k)|
φ(t)(dj n)2 (dj n − c)
(2k + 1)! j=0
x+(l)(dj n)−1/2
l=1 a2
)
∞
X
×
pdj n,v (x)pdj n,v (t)l−4 |t − x|2k+5
v=0
x+(l+1)(dj n)−1/2
Z
×
!
(2k+1)
φ(w)φx,0,l+1 (w) df
(w) dtdx
x
k
r
X
X
1
+
|C(j, k)|
(2k + 1)! j=0
l=1
×
∞
X
Z
b2
a2
(Z
x−(l)(dj n)−1/2
φ(t)(dj n)2 (dj n − c).
x+(l+1)(dj n)−1/2
)
bdj n,v (x)pdj n,v (t)l−4 |t − x|2k+5
v=0
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
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C ONVERGENCE F OR A S EQUENCE O F D URRMEYER T YPE O PERATORS
Z
x
×
x−(l+1)(dj n)−1/2
!
(2k+1)
φ(w)φx,l+1,0 (w) df
(w) dtdx
k
X
1
×
|C(j, k)|
(2k + 1)! j=0
×
∞
X
7
Z
(Z
b2
a1 +(dj n)−1/2
φ(t)(dj n − c)
(dj n)−1/2
a2
)
2k+1
pdj n,v (x)pdj n,v (t) |t − x|
v=0
Z
x+(dj n)−1/2
×
x−(dj n)−1/2
!
(2k+1)
φ(w)φx,1,1 (w) df
(w) dtdx
k
r
×
∞
X
(Z
b2
Z
X
X
1
|C(j, k)|
l−4
≤
(2k + 1)! j=0
l=1
x+(l+1)(dj n)−1/2
φ(t)(dj n − c)
x+(l)(dj n)−1/2
a2
)
pdj n,v (x)pdj n,v (t) |t − x|2k+5
v=0
Z
b1
(2k+1)
×
φx,0,l+1 (w) df
(w) dtdx
a1
Z b2 (Z x−(l)(dj n)−1/2
k
r
X
X
1
+
|C(j, k)|
l−4
φ(t)(dj n − c)
(2k + 1)! j=0
a2
x+(l+1)(dj n)−1/2
l=1
)
∞
X
×
pdj n,v (x)pdj n,v (t) |t − x|2k+5
v=0
b1
(2k+1)
×
φx,l+1,0 (w) df
(w) dtdx
a1
Z b2 (Z a1 +(dj n)−1/2
k
X
1
+
|C(j, k)|
φ(t)(dj n − c)
(2k + 1)! j=0
a2
(dj n)−1/2
)
∞
X
×
pdj n,v (x)pdj n,v (t) |t − x|2k+1
Z
v=0
Z
b1
(2k+1)
φ(w)φx,1,1 (w) df
(w) dtdx.
×
a1
Applying Lemma 2.1 and using Fubini’s theorem we get
kM2 kL1 (I2 )
≤ C14
k
X
−(2k+1)/2
|C(j, k)|(dj n)
j=0
+ C14
r
X
k
X
|C(j, k)|(dj n)
φx,1,1 (w) df (2k+1) (w) dx
b2
Z
a2
−(2k+1)/2
b1
+
Z
l=1
j=0
Z
l
−4
r
X
l=1
l
−4
b1
φx,0,l+1 (w) df (2k+1) (w) dx
a1
Z
b2
a2
Z
b1
φx,l+1,0 (w) df (2k+1) (w) dx
a1
a1
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
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8
N IRAJ K UMAR
≤ C14
k
X
−(2k+1)/2
|C(j, k)|(dj n)
r
X
j=0
l
k
X
−(2k+1)/2
|C(j, k)|(dj n)
r
X
l
−4
φx,1,1 (w) df (2k+1) (w) dx
φx,0,l+1 (w)dx df (2k+1) (w)
a2
Z
b1
Z
b2
φx,l+1,0 (w) df (2k+1) (w) dx
a2
a1
l=1
b2
b2
Z
a1
j=0
+
b1
Z
l=1
+ C14
Z
−4
a2
≤ C15
k
X
|C(j, k)|(dj n)−(2k+1)/2
r
X
j=0
l−4
k
X
|C(j, k)|(dj n)−(2k+1)/2
j=0
b1
r
X
l−4
w+(dj
+
w−(dj n)−1/2
!
w
w−(l+1)(dj n)−1/2
b1
a1
!
n)−1/2
Z
Z
l=1
Z
a1
b1
a1
l=1
+ C15
Z
(Z
( Z
)
dx df (2k+1) (w)
w+(l+1)(dj n)−1/2
!
dx df (2k+1) (w)
w
)
dx df (2k+1) (w)
≤ C16 n−(k+1) f (2k+1) B.V.(I1 ) .
Finally we estimate M3 . It is sufficient to choose the expression without the linear combinations.
For all t ∈ [0, ∞)\[a1 , b1 ] and all x ∈ I2 , we choose a δ > 0 such that |t − x| ≥ δ. Thus
kSn (F (t, x)(1 − φ(t)), x)kL1 (I2 )
Z b2 Z ∞
∞
X
≤
pn,v (x)pn,v (t) |f (t)| (1 − φ(t))dtdx
(n − c)
a2
≤
2k+1
X
i=0
0
1
i!
v=0
Z
b2
Z
∞
(n − c)
a2
0
= M4 + M5 ,
∞
X
pn,v (x)pn,v (t) f (i) (t) . |t − x|i (1 − φ(t))dtdx
v=0
say.
2k+2
For sufficiently large t there exist positive constants C17 , C18 such that (t−x)
> C17 for all
t2k+2 +1
t ≥ C18 , t ∈ I2 . Applying Fubini’s theorem and Lemma 2.1, we obtain
Z C18 Z b2 Z ∞ Z b2 X
∞
M4 =
+
(n − c)pn,v (x)pn,v (t) |f (t)| (1 − φ(t))dtdx
0
a2
C18
a2
v=0
Z ∞ Z b2 X
∞
1
(t − x)2k+2
≤ C19 n
|f (t)| dt +
pn,v (x)pn,v (t) 2k+2
|f (t)| dxdt
C17 C20 a2 v=0
(t
+ 1)
0
Z C20
Z ∞
−(k+1)
≤ C20 n
|f (t)| dt +
|f (t)| dt
≤ C21 n−(k+1) kf kL1 [0,∞) .
−(k+1)
Z
C20
0
C20
Finally by the Lemma 2.4, we have
M5 ≤ C22 n
−(k+1)
n
o
(2k+1) kf kL1 (I2 ) + f
.
L1 (I2 )
Combining M4 and M5 , we obtain
n
o
M3 ≤ C23 n−(k+1) kf kL1 [0,∞) + f (2k+1) L1 (I2 ) .
This completes the proof of (2.2) of the lemma.
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
http://jipam.vu.edu.au/
C ONVERGENCE F OR A S EQUENCE O F D URRMEYER T YPE O PERATORS
9
3. R ATE O F C ONVERGENCE
Theorem 3.1. Let f ∈ Lp [0, ∞), p ≥ 1. Then for n sufficiently large
n
o
kSn,k (f, ·) − f kLp (I2 ) ≤ C24 ω2k+2 (f, n−1/2 , p, I1 ) + n−(k+1) kf kLp [0,∞) ,
where C24 is a constant independent of f and n.
Proof. We can write
kSn,k (f, ∗) − f kLp (I2 ) ≤ kSn,k (f − fη,2k+2 , ∗)kLp (I2 )
+ kSn,k (fη,2k+2 , ∗) − fη,2k+2 )kLp (I2 ) + k(fη,2k+2 − f )kL
= E1 + E2 + E3 ,
p (I2 )
say.
It is well known that
(2k+1) fη,2k+2 B.V.(I3 )
(2k+2) = fη,2k+2 .
L1 (I3 )
Therefore from Lemma 2.5 (p > 1) and (p = 1) we have
−(k+1) (2k+2) E2 ≤ C25 n
+ kfη,2k+2 kLp [0,∞)
fη,2k+2 Lp (I3 )
≤ C26 n−(k+1) n−(k+2) ω2k+2 (f, η, p, I1 ) + kf kLp [0,∞) ,
which follows from the properties of Steklov means.
Let φ(t) be the characteristic function of I3 , we have
Sn ((f − fη,2k+2 )(t), x) = Sn (φ(t)(f − fη,2k+2 )(t), x) + Sn ((1 − φ(t))(f − fη,2k+2 )(t), x)
= E4 + E5 ,
say.
By Hölder’s inequality
Z b2
Z
p
|E4 | dx ≤
a2
b2
a2
Z
b1
(n − c)
a1
∞
X
pn,v (x)pn,v (t) |(f − fη,2k+2 )(t)|p dtdx.
v=0
Applying Fubini’s theorem, we have
Z b2
|E4 |p dx ≤ kf − fη,2k+2 kLp(I2 ) .
a2
Similarly, for all p ≥ 1
kE5 kLp(I ) ≤ C27 η −(k+1) kf − fη,2k+2 kLp(0,∞) .
2
Consequently, via the property of Steklov means, we find that
n
o
kSn (f − f η, 2k + 2, ·)kLp(I2) ≤ C28 ω2k+2 (f, η, p, I1 ) + η −(k+1) kf kLp[0,∞) .
Hence
o
n
E1 ≤ C29 ω2k+2 (f, η, p, I1 ) + η −(k+1) kf kLp[0,∞) .
Thus, with η = n−1/2 , the result follows.
This completes the proof of the theorem.
J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
http://jipam.vu.edu.au/
10
N IRAJ K UMAR
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J. Inequal. Pure and Appl. Math., 5(3) Art. 79, 2004
http://jipam.vu.edu.au/