Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 3 (2012.), Pages 12-22
APPROXIMATION OF FUNCTIONS BY MATRIX-EULER
SUMMABILITY MEANS OF FOURIER SERIES IN
GENERALIZED HÖLDER METRIC
(COMMUNICATED BY HÜ SEIN BOR)
SHYAM LAL AND SHIREEN
Abstract. In this paper, a new estimate for the degree of trignometric ap(w)
proximation of a function f ∈ Hr , (r ≥ 1) class by Matrix-Euler means
(∆.E1 ) of its Fourier Series has been determined.
1. Introduction
The degree of approximation of a function f belonging to Lipα class by Nörlund
summability method (N, pn ) has been determined by several investigators like
Khan[6], Qureshi[7, 8], Chandra[11], Leindler[9], Stepants[2] and Lal[16]. Working
in quite different direction, Totik[17, 18], Mazhar[14], Totik and Mazhar[15] and
Chandra[12] have studied the approximation of functions in Hölder space H (w) .
But till now no work seems to have done to obtain the degree of approximation of
(w)
functions f ∈ Hr , (r ≥ 1), by Matrix-Euler (∆.E1 ) product summability means.
In an attempt to make an advance study in this direction, in this paper, a new
(w)
estimate for degree of trignometric approximation of a function f ∈ Hr , (r ≥ 1)
(w)
space has been determined. It is important to note that Hr , (r ≥ 1), is a gen(w)
eralization of H , H(α),r and H(α) spaces. Some important applications of main
theorem has been investigated.
2. Definition and Notations
Let f (x) be a 2π periodic function, integrable in the Lebesgue sense over [o, 2π]
(w)
and belonging to Hr class. Let the Fourier series of f (x) is given by
∞
X
1
f (x) = a0 +
(an cos nx + bn sin nx)
2
n=1
(1)
2010 Mathematics Subject Classification. 42B05, 42B08.
Key words and phrases. Trigonometric approximation, Fourier series, (E, 1) means, matrix
summability means and (∆.E1 )- summability means, generalized Minkowski’s inequality, generalized Hölder metric.
c
2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted April 17, 2012. Published date June 10, 2012.
12
APPROXIMATION OF FUNCTIONS BY MATRIX-EULER SUMMABILITY MEANS ......
13
withP
nth partial sums sn (f ; x).
Pn
∞
Let n=0 un be an infinite series having nth partial sum sn = ν=0 uν .
Let T = (an,k ) be an infinite triangular matrix satisfying the condition of regularity
(Silverman-Töeplitz [10]) i.e.
Pn
(i).
k=0 an,k = 1 as n → ∞ ,
(ii). a
n,k
Pn = 0 for k > n ,
(iii).
k=0 |an,k | ≤ M , a finite constant.
The sequence-to-sequence transformation
t∆
n =
n
X
an,k sk =
k=0
n
X
an,n−k sn−k
k=0
defines the sequence t∆
n of triangular matrix means of the sequence {sn }, generated
by the sequence of coefficients (an,k ).
P∞
If t∆
n → s as n → ∞ then the series
n=0 un is summable to s by triangular
matrix ∆- method (Zygmund[1], p.74).
n P∞
1X n
(1)
(1)
sk . If En → s as n → ∞, then n=0 un is said to be
Let En = n
2
k
k=0
summable to s by the Euler’s method, E1 (Hardy[5]).
The triangular matrix ∆-transform of EP
1 transform defines the (∆.E1 ) transform
∞
t∆E
of
the
partial
sums
s
of
the
series
n
n
n=0 un by
t∆E
=
n
n
X
an,k Ek1 =
k=0
If t∆E
→ s as n → ∞,
n
sn → s ⇒
En(1)
P∞
n=0
n
X
k=0
an,k
k 1X k
sν .
2k ν=0 ν
un is said to be summable (∆.E1 ) to s.
n 1X n
= n
sν → s as n → ∞, E1 method is regular,
2 ν=0 ν
(1)
∆E
⇒ t∆
→ s as n → ∞, ∆ method is regular,
n (En ) = tn
⇒ (∆.E1 ) method is regular.
Some important particular cases of triangular matrix-Euler means (∆.E1 ) are
1
1
).(E1 ) means, when an,k = (n−k+1)
(i). (H, n+1
log n .
Pn
pn−k
(ii). (N, pn ).E1 means, when an,k = Pn , where Pn = k=0 pk 6= 0.
qk
(iii). (N, pn , qn ).E1 means, when an,k = pn−k
Rn ,
Pn
where Rn = k=0 pk qn−k 6= 0.
Let C2π denote the Banach space of all 2π-periodic and continuous functions
defined on [0, 2π] under the supremum norm.
For 0 < α ≤ 1, let
α
H(α) = {f ∈ C2π : |f (x + t) − f (x)| = O(|t| )}
14
SHYAM LAL AND SHIREEN
The space H(α) is a Banach space (Prössdorfff [13]) under the norm
kf k(α)
=
sup |f (x)| + sup
x,t
t6=0
0≤x≤2π
|f (x + t) − f (x)|
, 0<α≤1
α
|t|
|f (x + t) − f (x)|
, 0 < α ≤ 1.
α
|t|
= kf k∞ + sup
x,t
t6=0
The metric induced by the norm kk(α) on H(α) is called the Hölder metric. Clearly
(H(α) , kf k(α) ) is a Banach space which decreases as α increases, i.e.,
H(α) ⊆ H(β) ⊆ C2π , for 0 ≤ β < α ≤ 1,
and
kf k(β) ≤ (2π)α−β kf k(α) .
In general,
|f (x + t) − f (x)|
, 0 < α ≤ 1.
α
|t|
sup |f (x)| =
6 sup
0≤x≤2π
x,t
t6=0
We define the norm kkr by
 n
o r1
R 2π
r

1
for 1 ≤ r < ∞
2π 0 |f (x)| dx
kf kr =
 ess sup |f (x)| for r = ∞.
0<x<2π
r
Let L [0, 2π] =
2π
Z
f : [0, 2π] → R :
r
|f (x)| dx < ∞ , r ≥ 1 , be the space of all
0
2π-periodic, integrable functions and for all t
(
)
Z 2π
r1
r
α
H(α),r = f ∈ Lr [0, 2π] :
|f (x + t) − f (x)| dx
= O (|t| ) .
0
The space H(α),r , r ≥ 1, 0 < α ≤ 1 is a Banach space under the norm kk(α),r :
kf k(α),r = kf kr + sup
t6=0
kf (. + t) − f (.)kr
.
α
(|t| )
kf k(0),r = kf kr .
The metric induced by the norm kk(α),r on H(α),r is called Hölder continuous with
degree r.
Easily, it can be obtained by
kf k(β),r ≤ (2π)α−β kf k(α),r , 0 ≤ β < α ≤ 1, r ≥ 1.
Since f ∈ H(α),r if and only if kf k(α),r < ∞, we have
H(α),r ⊆ H(β),r ⊆ Lr [0, 2π], 0 ≤ β < α ≤ 1, r ≥ 1.
For f ∈ Lr [0, 2π], r ≥ 1, the integral modulus of continuity is defined by
r1
Z 2π
1
r
wr (f, δ) = sup
|f (x + t) − f (x)| dx
, for f ∈ Lr [0, 2π] where
2π 0
0<t≤δ
APPROXIMATION OF FUNCTIONS BY MATRIX-EULER SUMMABILITY MEANS ......
15
1 ≤ r < ∞ and if r = ∞, then
w(f, δ) = w∞ (f, δ) = sup max |f (x + t) − f ((x)| for f ∈ C2π .
o<t≤δ
x
It is known (Zygmund [1], p.45) that wr (f, δ) → 0 as δ → 0.
Let w : [0, 2π] → R be an arbitrary function with w(t) > 0 for 0 < t ≤ 2π and
lim w(t) = w(0) = 0.
t→0+
The class of function H (w) has been defind by Leindler [9] as
H (w) = {f ∈ C2π : |f (x + t) − f (x)| = O(w (t))}
where w is a modulus of continuity, that is, w is a postive non-decreasing continuous
function with the property:
w(0) = 0, w(t1 + t2 ) ≤ w(t1 ) + w(t2 ).
)
(
kf
(.
+
t)
−
f
(.)k
r
<∞
We define Hr(w) = f ∈ Lr [0, 2π] : 1 ≤ r < ∞, sup
w (t)
t6=0
kf (. + t) − f (.)kr
(w)
(w)
(w)
and kf kr = kf kr +sup
, r ≥ 1. Clearly kkr is a norm on Hr .
w
(t)
t6=0
(w)
The completeness of the space Hr can be discussed considering the completeness
of Lr (r ≥ 1).
kf (. + t) − f (.)kr
(v)
kf kr = kf kr + sup
, r ≥ 1. If w(t)
tends to zero as t → o+ then
t
v
(|t|)
t6=0
f 0 (x) exists
and is zero everywhere and f is constant.
w(t)
v(t)
be positive non decreasing.
(v)
(w)
Then kf kr ≤ max 1, w(2π)
< ∞. Thus,
v(2π) kf kr
Let
Hr(w) ⊆ Hr(v) ⊆ Lr , r ≥ 1
Remarks.
(i). If we take w(t) = tα then H (w) reduces to H(α) class.
(w)
(ii). By taking w(t) = tα in Hr , it reduces to H(α),r .
(w)
(iii). If we take r → ∞ then Hr
class reduces to H (w) .
The degree of approximation
of a function f : R → R by a trignometric polynoP∞
mial tn (x) = 21 a0 + ν=1 (aν cos νx + bν sin νx) of order n is defined by (Zygmund
[1], p.114-115)
En (f ) = min ktn − f kr .
We write,
φ(x, t) = f (x + t) + f (x − t) − 2f (x), ∆an,k = an,k − an,k+1 , 0 ≤ k ≤ n − 1.
n
Kn∆E =
sin (n − k + 1)( 2t )cosn−k ( 2t )
1 X
an,k
.
2π
sin ( 2t )
k=0
16
SHYAM LAL AND SHIREEN
3. Theorem
In this paper, we prove the following theorem:
Theorem 3.1. Let A = (an,k ) be a regular lower triangular infinite matrix such
that
n−1
X
|∆an,k | = O
k=0
1
n+1
, (n + 1) |an,n | = O (1) .
(2)
If f : [0, 2π] → R be a 2π-periodic, Lebesgue integrable and belonging to the gen(w)
eralized class Hr ,r ≥ 1; w,v be modulus of continuity and w(t)
v(t) be positive, nondecreasing then the degree ofapproximation of f by triangular matrix-Euler means
Pk
Pn
t∆E
= k=0 an,k 21k ν=0 νk sν of its Fourier series (1) is given by
n
kt∆E
n
−
f k(v)
r
1
(n + 1)
=O
Z
π
1
n+1
!
w(t)
dt .
t2 v(t)
4. Lemmas
Following Lemmas are required to prove the theorems:
Lemma 4.1. For 0 < t ≤ (n + 1)−1 , Kn∆E (t) = O(n + 1).
Proof. For 0 < t ≤ (n + 1)−1 , sin 2t ≥
∆E Kn (t)
t
π,
sin nt ≤ nt, |cos t| ≤ 1. We have
n
1 X
sin (n − k + 1)( 2t )cosn−k ( 2t ) = an,k
2π
sin ( 2t )
k=0
n
(n − k + 1)( 2t ) cosn−k ( 2t )
1 X
|an,k |
≤
2π
( πt )
k=0
n
≤
X
1
(n + 1)
|an,k |
4
k=0
M
≤
(n + 1)
4
= O(n + 1).
Lemma 4.2. For (n + 1)−1 < t < π, Kn∆E (t) = O
1
(n+1)t2
.
(3)
APPROXIMATION OF FUNCTIONS BY MATRIX-EULER SUMMABILITY MEANS ......
Proof. For (n + 1)−1 < t < π, sin 2t ≥
∆E Kn (t) =
≤
≤
≤
=
=
=
t
π,
17
using Abel’s lemma, we get
n
1 X
sin (n − k + 1)( 2t )cosn−k ( 2t ) an,k
2π
sin ( 2t )
k=0
n−1
k
X
1 X
t
t
(an,k − an,k+1 )
sin (n − ν + 1)
cosn−ν
2t 2
2
ν=0
k=0
n
X
t
t n−k
+an,n
cos
sin (n − k + 1)
2
2 k=0
"n−1
t
sin (2n − k + 2)( 4t ) sin (n + 1)( 4t ) 1 X
+ |an,n | sin (n + 2)( 4 ) sin(n + 1)
|∆an,k | t
2t
sin ( 4 )
sin 4t
k=0
"n−1
#
π X
t
t |∆a
|
+
|a
|
max
sin
(2n
−
k
+
2)
sin(n
+
1)
n,k
n,n
2
0≤k≤n
t
2
2 k=o
"n−1
#
π X
|∆an,k | + |an,n |
t2
k=o
π
1
1
O
+O
by (2)
t2
n+1
n+1
1
.
O
(n + 1)t2
5. Proof of the Theorem3.1
Following Titchmarsh [4], sk (f ; x) of Fourier series (1) is given by
1
sk (f ; x) − f (x) =
2π
Z
0
π
sin k + 12 t
φ(x, t)
dt, k = 0, 1, 2... .
sin 2t
Then
n 1 X n
(sk (f ; x) − f (x))
2n
k
=
1
2π
En1 (x) − f (x)
=
1
2π
=
1
2π
=
1
2π
=
1
2π
k=0
or
n 1 X n sin k + 21 t
φ(x, t) n
dt
2
k
sin 2t
0
k=0
(
)
Z π
n X
1
n i(k+ 1 )t
2
φ(x, t) n
Im
e
dt
k
2 sin ( 2t )
0
k=0
(
)
Z π
n X
1
n ikt i t
2
φ(x, t) n
Im
e e
dt
k
2 sin ( 2t )
0
k=0
Z π
n
n t o
1
φ(x, t) n
Im 1 + eit .ei 2 dt
t
2 sin ( 2 )
0
Z π
sin (n + 1) ( 2t ) cosn ( 2t )
φ(x, t)
dt.
sin ( 2t )
0
Z
π
t
4
#
18
SHYAM LAL AND SHIREEN
Now
t∆E
n (x) − f (x)
=
=
n
1
1 X
an,k En−k
(x) − f (x)
Pn
k=0
Z π
n
X
sin (n − k + 1)( 2t ) cosn−k ( 2t )
1
φ(x, t)
an,k
dt.
2π 0
sin ( 2t )
k=0
Let
ln (x) = t∆E
n (x) − f (x)
Z
=
π
φ(x, t)Kn∆E (t)dt.
0
Then
Z
ln (x + y) − ln (x)
π
=
(φ(x + y, t) − φ(x, t))Kn∆E (t)dt.
0
By generalized Minkowski’s inequality (Chui[3], p.37), we get
Z π
kln (. + y) − ln (.)kr ≤
kφ(. + y, t) − φ(., t)kr Kn∆E (t) dt
0
Z
=
0
=
1
n+1
kφ(. + y, t) − φ(., t)kr Kn∆E (t) dt +
Z
π
1
n+1
kφ(. + y, t) − φ(., t)kr Kn∆E (t) dt
I1 + I2 .
(4)
Clearly
|φ(x + y, t) − φ(x, t)|
≤
|f (x + y + t) − f (x + y)| + |f (x + y − t) − f (x + y)|
+|f (x + t) − f (x)| + |f (x − t) − f (x)|.
Applying Minkowski’s inequality, we have
kφ(. + y, t) − φ(., t)kr
≤
kf (. + y + t) − f (. + y)kr + kf (. + y − t) − f (. + y)kr
+ kf (. + t) − f (.)kr + kf (. − t) − f (.)kr
= O (w(t)) .
(5)
Also
kφ(. + y, t) − φ(., t)kr
≤
kf (. + y + t) − f (. + t)kr + kf (. + y − t) − f (. − t)kr
+2 kf (. + y) − f (.)kr
=
O (w(|y|)) .
(6)
For v is positive, non decreasing, t ≤ |y|, we obtained
kφ(. + y, t) − φ(., t)kr
= O (w(t))
w(t)
= O v(t)
v(t)
w(t)
= O v(|y|)
.
v(t)
APPROXIMATION OF FUNCTIONS BY MATRIX-EULER SUMMABILITY MEANS ......
Since
w(t)
v(t)
w(t)
v(t)
is positive, non-decreasing, if t ≥ |y|, then
kφ(. + y, t) − φ(., t)kr
≥
w(|y|)
v(|y|) ,
19
so that
= O (w(|y|))
w(t)
= O v(|y|)
.
v(t)
Using lemma (4.1) and (7) we obtain
1
Z n+1
kφ(. + y, t) − φ(., t)kr Kn∆E (t) dt
I1 =
0
!
1
Z n+1
w(t)
= O
v(|y|)
(n + 1)dt
v(t)
0
!
1
Z n+1
w(t)
= O (n + 1)v(|y|)
dt
v(t)
0


1
1
Z n+1
w n+1
= O (n + 1)v(|y|) dt
1
0
v n+1


1
w n+1
.
= O v(|y|) 1
v n+1
(7)
(8)
Also,using Lemma (4.2) and (7) we get
Z π
I2 =
kφ(. + y, t) − φ(., t)kr Kn∆E (t) dt
1
n+1
!
1
w(t)
dt
v(|y|)
1
v(t) (n + 1)t2
n+1
!
Z π
1
w(t)
v(|y|) 2
dt .
1
n + 1 n+1
t v(t)
Z
=
O
=
O
π
(9)
By (4), (8) and (9), we have

kln (. + y) − ln (.)kr
=
w
1
n+1


O v(|y|) 1
v n+1
!
Z π
1
w(t)
+O
v(|y|) 2
dt .
1
n + 1 n+1
t v(t)
Thus,
sup
y6=0
kln (. + y) − ln (.)kr
v(|y|)

=
w
1
n+1


O 1
v n+1
!
Z π
1
w(t)
+O
dt .
1
n + 1 n+1
t2 v(t)
(10)
20
SHYAM LAL AND SHIREEN
Clearly
|φ(x, t)| =
≤
|f (x + t) + f (x − t) − 2f (x)|
|f (x + t) − f (x)| + |f (x − t) − f (x)|
Applying Minkowski’s inequality, we have
kφ(., t)kr
≤
kf (. + t) − f (.)kr + kf (. − t) − f (.)kr
=
O (w(t)) .
(11)
Using(11), Lemma (4.1),Lemma (4.2) we obtain
1
Z n+1
Z π !
∆E
+
kφ(., t)kr Kn∆E (t) dt
kln (.)kr = tn − f r ≤
1
n+1
0
1
n+1
Z
=
0
kφ(., t)kr (Kn∆E (t) dt +
π
Z
1
n+1
kφ(., t)kr Kn∆E (t) dt
!
Z π
1
w(t)
= O (n + 1)
w(t)dt + O
dt
1
(n + 1) n+1
t2
0
!
Z π
1
1
w(t)
= O w
+O
dt .
1
(n + 1)
(n + 1) n+1
t2
Z
!
1
n+1
(12)
Now, By (10) and (12)
kln (. + y) − ln (.)kr
v(|y|)
y6=0
!
Z π
w(t)
1
1
= O w
dt
+O
1
n+1
(n + 1) n+1
t2

 !
1
Z π
w n+1
1
w(t)
+O
dt .
+O  1
(n + 1) n+1
v(t)t2
v 1
(v)
kln (.)kr
= kln (.)kr + sup
n+1
w(t)
v(t) .v(t)
≤ v(π) w(t)
v(t) , 0 < t ≤ π, we get

 !
1
Z π
w n+1
1
w(t)
+O
= O dt .
1
(n + 1) n+1
v(t)t2
v 1
Using the fact that w(t) =
(v)
kln (.)kr
(13)
n+1
Since w and v are modulus of continuity such that
therefore
1
n+1
Z
π
1
n+1
w
1
n+1
w(t)
v(t)
w(t)
dt ≥ 1
v(t)t2
v n+1
1
n+1
π
Z
1
n+1
is positive, non decreasing,
w
1
n+1
dt
.
≥
1
t2
2v n+1
Then
w
v
1
n+1
1
n+1
= O
1
(n + 1)
Z
π
1
n+1
!
w(t)
dt .
t2 v(t)
(14)
APPROXIMATION OF FUNCTIONS BY MATRIX-EULER SUMMABILITY MEANS ......
21
By (13) and (14), we have
∆E
tn − f (v)
r
1
(n + 1)
= O
Z
!
w(t)
dt .
t2 v(t)
π
1
n+1
This completes the proof of theorem 3.1.
6. Applications
We obtain the following corollaries from the main Theorems.
Corollary 6.1. Let f ∈ H(α),r , r ≥ 1, 0 < α ≤ 1 then
 1
 O
, 0 ≤ β < α < 1,
∆E
α−β
tn − f (n+1)
=
(β),r
log(n+1)π
 O
, β = 0, α = 1.
n+1
Proof. If we take w(t) = tα , v(t) = tβ in theorem 3.1.
1
(n−k+1) log n , in theorem
(w)
1
).E1 means
Hr by (H, n+1
Corollary 6.2. If we take an,k =
approximation of a function f ∈
tHE
n
3.1, then degree of
n
k 1
1 X
1 X k
sν
log n
n − k + 1 2k ν=0 ν
=
k=0
of the fourier series (1) is given by
HE
tn − f (v)
r
1
(n + 1)
= O
Corollary 6.3. If we take an,k =
pn−k
Pn ,
E
tN
n
1
Pn
=
pn−k
1
n+1
k=0
(w)
Hr
1
2k
!
w(t)
dt .
t2 v(t)
π
Pn
where Pn =
then degree of approximation of a function f ∈
n
X
Z
k=0
pk 6= 0 in theorem 3.1,
by (N, pn ).E1 means
k
sν
ν
k X
ν=0
of the fourier series (1) is given by
NE
tn − f (v)
r
= O
Corollary 6.4. If we take an,k =
pn−k qk
Rn ,
1
(n + 1)
Z
π
1
n+1
E
tN
n
=
Pn
k=0 pk qn−k 6= 0 in theorem
(w)
Hr by (N, p, q).E1 means
where Rn =
3.1, then degree of approximation of a function f ∈
!
w(t)
dt .
t2 v(t)
n
k 1 X
1 X k
pn−k qk k
sν
Rn
2 ν=0 ν
k=0
of the fourier series (1) is given by
NE
tn − f (v)
r
= O
1
(n + 1)
Z
π
1
n+1
!
w(t)
dt .
t2 v(t)
22
SHYAM LAL AND SHIREEN
Acknowledgements. Authors are grateful to the referee for his valuable suggestions and comments for improvement of this paper. Authors mention their sense
of gratitude to Professor Huseyin Bor, Professor Faton Z. Merovci and Editorial
Board of BMAA for supporting this work.
Shyam Lal, one of the author, is thankful to DST-CIMS, Banaras Hindu University for encouragement to this work.
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Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi221005, India
E-mail address: shyam lal@rediffmail.com, shrn.maths.bhu@gmail.com