BIBECHANA
A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online)
Journal homepage: http://nepjol.info/index.php/BIBECHANA
Approximation of a generalized Lipschitz class function by Euler -
Cesàro means of Fourier series
Binod Prasad Dhakal
Central Department of Education Mathematics Tribhuvan University, Nepal
E-mail: [email protected]
Article history: Received 19 November, 2011; Accepted 27 November, 2012
Abstract
In this paper, I have taken product of two summability methods, Euler and Cesàro; and establish a new theorem on the degree of approximation of the function f belonging to W(Lp, ξ(t)) classes by Euler - Cesàro method.
Key words and phrases: Degree of approximation; (E,1) (C,1) Summability; Fourier series.
1. Definitions and Notations
A function f(x)∈ Lipα, if
( )
α =−
+t) f(x) O t x
(
f for 0 < α ≤ 1 and f ∈ Lip (α, p), if
( )
t
,
O
dx
)
x
(
f
)
t
x
(
f
p 1 2
0
p α
π
=
−
+
∫
0 < α ≤ 1, p ≥ 1.Given a positive increasing function ξ(t), p ≥ 1, f (x) ∈ Lip (ξ(t), p), if
))
t
(
(
O
dx
)
x
(
f
)
t
x
(
f
p 1 2
0
p
ξ
=
−
+
∫
π and f ∈ W (Lp, ξ(t)), if(
f
(
x
t
)
f
(
x
)
)
sin
x
dx
O
(
(
t
)),
(
0
).
p 1 2
0
p
≥
β
ξ
=
−
+
∫
π β[1]
It is noted that,
W
(
L
p,
ξ
(
t
))
β→
=0Lip
(
ξ
(
t
),
p
)
ξ →
(t)=
tαLip
(
α
,
p
)
→
p→
∞Lip
α
We define the norm
pby f f(x) dx , p 1 p
1 p 2
0
p ≥
=
∫
π
. The degree of approximation En(f) of function f: R→R is given by
p n n (f) Min t f
E = − .
Where tn is trigonometric polynomial of degree n [2].
Let f be 2π periodic, integrable over (-π,π) in the sense of Lebesgue and belonging to W
(
Lp,ξ(t))
class, then its “Fourier series” is given by)
nt
sin
b
nt
cos
a
(
a
2
1
)
t
(
f
n n1 n
o
+
+
=
∑
∞=
. (1)
Let
∑
∞
=0 n
n
u be the infinite series whose nth partial sum is given by
∑
=
= n
0 i
i
n u .
S
The Cesåro means (C, 1) of sequence {Sn} is S . 1 n
1 n
0 k
k
n
∑
= + = σ
If
σ
n→
S
,
as
n
→
∞
then sequence {Sn} or the infinite series∑
∞
=0 n
n
u is said to be summable by Cesåro means method (C,1) to S. It is denoted by
n S(C,1), as n [3].
σ → → ∞
The Euler means (E, 1) of sequence {Sn} is
∑
=
= n
0 k
k n
1
n S
k n
2 1
E
If E1n →Sas, n→∞then sequence {Sn} or infinite series
∑
∞
=0 n
n
u is said to be summable by Euler means method (E, 1) to S. It is denoted by
1 n
E →S(E,1), as n→ ∞ [4].
The E1ntransformation of {σn} is denoted by t 1 1 ,
C , E
n which is (E, 1) (C, 1) transformation of {Sn} and
defined as
∑
= σ
= n
0 k
k n
C , E n
k n
2 1
t 1 1
∑
∑
= + =
= n
0 k
r k
o r
n k 1 S .
1 k n
2 1
If tE1,C1 →S, as n →∞
n then sequence {Sn} or infinite series
∑
∞
=0 n
n
u is said to be summable by (E, 1) (C, 1) means method to S. It is denoted by
tE1,C1 S(E,1)(C,1), as n .
n → →∞
We use following notations.
) x ( f 2 ) t x ( f ) t x ( f ) t
( = + + − −
φ (2)
2 t 2
2 t 2
n 0 k 1 n C , E n
sin
)
1
k
(
)
1
k
(
sin
k
n
2
1
N
1 1+
+
π
=
∑
=
2. Main Theorem
In present paper, the degree of approximation of a function f∈W
(
Lp,ξ(t))
class by (E,1) (C,1) means of a Fourier series has been determined in the following form:Theorem: If f: R → R is 2π periodic, Lebesgue integrable function in (−π,π)and is W
(
Lp,ξ(t))
, then the degree of approximation of function f by (E,1)(C,1) means of Fourier series (1) satisfies,( )
+ ξ
=
− β+ n1+1
p C , E
n p
1 1
1
) 1 n ( O f
t , for n=1,2,3,4,...
Provided ξ(t) satisfy the following conditions;
ξ
t ) t (
is monotonic decreasing (4)
+
=
ξ
φ
β∫
+sin
t
dt
O
n
1
1
)
t
(
)
t
(
t
p1 p p
0
1 n
1
,
(5)
(
)
(
δ)
δ − π
+ =
ξ φ
∫
+
1 n O dt
) t (
) t (
t p
1 p
1 n
1
(6)
where δ is an arbitrary number such that q(1-δ)-1> 0, condition (5) and (6) hold uniformly in x.
3. Lemmas
We need the following Lemmas for the proof of our theorem.
Lemma 1: Let
∑
=
+
+
+
π
=
n0
k 2 2t
2 t 2
1 n C
, E
n
,
sin
)
1
k
(
)
1
k
(
sin
k
n
2
1
)
t
(
N
1 1 then) 1 n ( O ) t ( NE1,C1
n = + , for .
1 n
1 t 0
+ < <
Proof:
∑
=
+
+
+
π
=
n0
k 2 2t
2 t 2
1 n C
, E n
sin
)
1
k
(
)
1
k
(
sin
k
n
2
1
)
t
(
N
1 1
∑
=
+
+
+
π
≤
n0
k 2 2t
2 t 2 2 1
n
(
k
1
)
sin
sin
)
1
k
(
k
n
2
1
∑
=
+ +
π
= n
0 k 1
n k (k 1)
n
+
π
=
∑
∑
= =
+
n 0 k
n 0 k 1
n
k
n
k
n
k
2
1
[
n 1 n]
1n n2 2
2 1
+ π
= −
+
π + =
4 2 n
(
)
π
+
≤
2
1
n
=O(n+1) (7)
Lemma 2: Let E1,C1
n
N be given as Lemma I, then
+ =
2 C
, E n
t ) 1 n (
1 O
) t (
N 1 1 , for < < π
+1 t
n 1
Proof:
∑
=
+ +
+
π
≤ n
0
k 2 2t
2 t 2
1 n C
, E n
sin ) 1 k (
) 1 k ( sin
k n
2 1 )
t ( N 1 1
∑
=
+
+
+
−
π
=
n0
k 2 2t
1 n
sin
)
1
k
(
2
t
)
1
k
(
cos
1
k
n
2
1
∑
=
+
+
π
≤
n0
k 2 2t
1 n
sin
)
1
k
(
1
k
n
2
1
∑
=
+ +
π
= n
0 k 2 1
n (k 1)
1 k n
t 2
+
−
π
=
+ +1
n
1
2
t
2
1 n 2 1 n
− +
π
= +
1 n
2 2
1 1 t ) 1 n (
2
t
)
1
n
(
+
π
≤
. t ) 1 n (
1 O
2
+
= (8)
4. Proof of the Theorem
(
)
dt sin t n sin ) t ( 2 1 ) x ( f ) x ( S 2 t 2 t 0 n + φ π = −∫
π . (C,1) transform of Sn i.e. σn is given by(
)
sin
(
k
)
t
dt
sin
)
t
(
)
1
n
(
2
1
)
x
(
f
)
x
(
S
1
n
1
2 t n 0 k 2 t 0 k n 0 k+
φ
π
+
=
−
+
∑
∫
∑
= π =dt
sin
)
1
n
(
sin
)
t
(
)
1
n
(
2
1
)
x
(
f
)
x
(
2 t 2 2 t 2 0 n+
φ
π
+
=
−
σ
∫
π .Similarly, (E,1) transform of σn i.e. E1,C1
n t is
(
)
(
)
dt
sin
)
1
k
(
1
k
sin
k
n
)
t
(
2
1
)
x
(
f
)
x
(
k
n
2
1
2 t 2 2 t 2 n 0 k 0 1 n k n 0 k n+
+
φ
π
=
−
σ
∑
∫
∑
= π + =dt
sin
)
1
k
(
)
1
k
(
sin
k
n
2
1
)
t
(
)
x
(
f
)
x
(
t
2 t 2 2 t 2 n 0 k 1 n 0 C , E n 1 1+
+
π
φ
=
−
∫
∑
= + π dt ) t ( N ) t( E1,C1
n 0
φ =
∫
π
(
t
)
N
(
t
)
dt
(
t
)
N
1 1(
t
)
dt
1 n 1 1 1 1 n 1 C , E n C , E n 0
φ
+
φ
=
∫
∫
π + +
=
I
1+
I
2, say. (9)For I1, applying Holder inequality and fact thatφ(t)∈W
(
Lp,ξ(t))
, we have
p 1 p
0
1
sin
t
dt
)
t
(
)
t
(
t
I
1 n 1
ξ
φ
≤
β∫
+ q 1 q C , E n 0dt
)
t
(
N
t
sin
t
)
t
(
1 1 1 n 1
ξ
β∫
+=
( )
11+ n O q 1 1 n 1 dt t ) 1 n )( t ( 0 q 1 ξ +
∫
+ β+= O
(
( 11))
+ n
ξ
q 1 1 n 1 dt t q( 1)
∫
+ ε + β −, by the Mean Value Theorem, where
1 n 1 0 + < ε < .
(
)
q 1 1 n 1 1 ) 1 ( q t ) ( O 1 ) 1 ( q 1 n 1 + + β − ξ = + ε + + β − +
(
)
{
(
)
}
q1 1 q ) 1 ( 1 n
1 ) n 1 (
O ξ + + β+ −
+ ξ
=O (n 1)β+1−q (n1+1)
1 + ξ
=O (n 1)β+p (n1+1)
1
. (10)
For I2, applying Holder’s inequality and taking δ as an arbitrary number such that q (1- δ) -1 > 0, we have
p 1 p
2 sin t dt
) t ( ) t ( t I 1 n 1 ξ φ ≤ β δ − π
∫
+ q 1 q C , E n dt t sin t ) t ( N ) t ( 1 1 1 n 1 ξ β δ − π∫
+ p 1 p dt ) t ( ) t ( t 1 n 1 ξ φ = δ − π∫
+ q 1 q 2dt
)
1
n
(
t
)
t
(
1 n 1
+
ξ
+ β + δ − π∫
+=O
(
(n+1)δ−1)
q 1 2 q 2 y 1 1 n
y
dy
y
)
(
1
−
ξ
− β − δ +∫
π= O
+ ξ + δ 1 n 1 ) 1 n ( − δ − β + +
∫
π q 1 1 dy yO q(1 ) 2 1
n
Using condition ( 4)
= O
(
(n 1) (n11))
+ δξ +
−
δ
−
β
+
+ − δ − β + π q 1 1 1 n 1 ) 1 ( q1
)
1
(
q
y
O
=O
(
(n 1) (n11))
+ δξ +
(
)
− + − δ − β + − δ − β + π − δ − β + q 1 1 ) 1 ( q 1 1 ) 1 ( q ) ( ) 1 n ( 1 ) 1 ( q 1 O= O
(
(n 1) (n11))
+ δξ
+
(
(
)
q)
1
1
1
n
O
+
+β−δ−= O
+ ξ + − + β ) ( ) 1 n
( 1 q n11
1
= O
+ ξ + + β ) ( ) 1 n
( p n11
1
. (11)
By (9), (10) and (11), we have
+ ξ =
−f (n 1)β+ ( + ) tE,C n11
= O
+ ξ
+ + β
) ( ) 1 n
( p n11
1 p
1
dx 2
0
∫
π= O
+ ξ
+ + β
) ( ) 1 n
( p n11
1
. (12)
This completes the proof of theorem.
5. Corollaries
Corollary 1: If β=o andξ(t)= tα, 0<α≤1 then the degree of approximation of a 2
π
periodic function f belonging to class Lip(α,p)is given by
+ =
− α−
p 1 1
1
) 1 n (
1 O
f t
p C , E n
Proof:
( )
+ ξ
=
− β+ n1+1
p C , E
n p
1 1
1
) 1 n ( O f
t
= O
+ ξ
+)
( ) 1 n
( p n11
1
= O
+
α
+1) n (
1
p 1
) 1 n (
= O
+ α−p 1
) 1 n (
1
. (13)
Corollary 2: If p→∞ in corollary 1 then the degree of approximation of a 2
π
periodic function f belonging to class Lipα (0<α<1)is given by1 0
for , ) 1 n (
1 O f
t
1 1,C
E
n <α<
+ =
− α
∞
. (14)
Remarks: An independent proof of corollary can be developed along the same line as the theorem.
Example: Consider the infinite series,
∑
∞=
− − −
1 n
1 n ) 3 ( 4
1 . (15)
The (E,1) (C,1) means of the sequence {Sn} is given by
∑
= σ
= n
0 k
k n
C , E n
k n
2 1 t 1 1
(
1 ( 1)n 1)
)1 n ( 2
1 − − +
+
= . (16)
and (E,1). Consequently, (E,1) (C,1) means gives the better approximation than individual methods (C,1) and (E,1).
References
[1] B. E. Rhoades, Journal of Mathematics, 3(2003) 245-247.
[2] A .Zygmund: Trigonometric Series, Cambridge University Press (1959).
[3] G. H. Hardy: On the Summability of Fourier series, Proc. London Math. Soc.,12(1913) 365-372. [4] G. H. Hardy : Divergent Series, The University Press, Oxford ( 1949).
