Journal of Computations & Modelling, vol.3, no.1, 2013, 145-152
ISSN: 1792-7625 (print), 1792-8850 (online)
Scienpress Ltd, 2013
Degree of Approximation of Fourier Series
by Hausdörff and Nörlund Product Means
Sunita Sarangi1 , S.K. Paikray2 ,
Minakshi Dash3 and U.K. Misra4
Abstract
In this paper a theorem on degree of approximation of a function
f ∈ Lip(α, r) by product summability (E, q)(N, pn ) of Fourier series
associated with f has been established.
Mathematics Subject Classification: 42B05, 42B08
Keywords: Degree of Approximation; Lipα class of function; (E, q) mean;
(N, pn ) mean; (E, q)(N, pn ) product mean; Fourier series, Lebesgue integral
1
Introduction
Let
1
2
3
4
P
an be a given infinite series with the sequence of partial sums {sn }.
P.G. Department of Mathematics, Ravenshaw University, Cuttack, Odisha, India
P.G. Department of Mathematics, Ravenshaw University, Cuttack, Odisha, India
P.G. Department of Mathematics, Ravenshaw University, Cuttack, Odisha, India
Department of Mathematics, National Institute of Science and Technology, Palur Hills,
Golanthara-761008, Brahmapur, Odisha, India, e-mail: umakanta misra@yahoo.com
Article Info: Received : December 30, 2012. Revised : January 28, 2013
Published online : March 31, 2013
146
Degree of Approximation of Fourier Series...
Let {pn } be a sequence of positive real numbers such that
Pn =
n
X
pυ −→ ∞, as n −→ ∞, (P−i = p−i = 0, i ≥ 0).
υ=0
The sequence-to-sequence transformation
n
1 X
pn−υ sυ ,
tn =
Pn υ=0
(1)
defines the sequence {tn } of the (N, pn )-mean of the sequence {sn } generated
by the sequence of coefficient {pn }. If
tn −→ s, as n −→ ∞,
(2)
P
then the series
an is said to be (N, pn ) summable to s.
The conditions for regularity of Nörlund summability (N, pn ) are easily
seen to be [1]
pn
(i)
→ 0 as n → ∞,
(3)
Pn
n
X
pk = 0(Pn )as n → ∞.
(4)
(ii)
k=0
The sequence-to-sequence transformation [1]
n µ ¶
X
1
n n−υ
Tn =
q sυ ,
n
(1 + q) υ=0 υ
(5)
defines the sequence {Tn } of the (E, q) mean of the sequence {sn }. If
Tn → s, as n → ∞,
(6)
then the series Σan is said to be (E, q) summable to s. Clearly (E, q) method
is regular [1]. Further, the (E, q) transformation of the (N, pn ) transform of
{sn } is defined by
µ ¶
n n−k
1
n
q Tk
Σk=0
τn =
n
k
(1 + q)
µ ¶
¾
½
1
n n−k 1 k
n
=
Σ
q
Σ pk−υ sυ
(7)
(1 + q)n k=0 k
Pk υ=0
If
τn → s, as n → ∞,
(8)
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Sunita Sarangi et al
P
then
an is said to be (E, q)(N, pn )-summable to s.
Let f (t) be a periodic function with period 2π, L-integrable over (−π, π).
The Fourier series associated with f at any point x is defined by
∞
∞
X
a0 X
f (x) ∼
+
(an cos nx + bn sin nx) ≡
An (x)
(9)
2
n=1
n=0
Let sn (f : x) be the n-th partial sum of (9). The L∞ -norm of a function
f : R → R is defined by
k f k∞ = sup{|f (x)| : x ∈ R}
and the Lυ -norm is defined by
µZ
2π
k f kυ =
|f (x)|
υ
¶ υ1
(10)
, υ ≥ 1.
(11)
0
The degree of approximation of a function f : R → R by a trigonometric
polynomial Pn (x) of degree n under norm k · k is defined by
k Pn − f k∞ = sup{|pn (x) − f (x)| : x²R}
(12)
and the degree of approximation En (f ) a function f ²Lυ is given by
En (f ) = min kPn − f kυ .
(13)
Pn
This method of approximation is called Trigonometric Fourier approximation.
A function f ²Lip α if
|f (x + 1) − f (x)| = O(|t|α ), 0 < α ≤ 1.
(14)
and f ²Lip(α, r), for 0 ≤ x ≤ 2π, if
¶ r1
µ Z 2π
r
|f (x + t) − f (x)| dx
= 0(|t|α ), 0 < α ≤ 1, r ≥ 1, t > 0.
(15)
0
We use the following notations throughout this paper:
φ(t) = f (x + t) + f (x − t) − 2f (x),
and
(16)
µ
½
sin υ +
n µ ¶
k
X
1
n n−k 1 X
Kn (t) =
q
pk−υ
2π(1 + q)n k=0 k
Pk υ=0
sin 2t
¶
1
2
t¾
.
Further, the method (E, q)(N, Pn ) is assumed to be regular and this case is
supposed through out the paper.
148
2
Degree of Approximation of Fourier Series...
Known Theorems
Dealing with the degree of approximation by the product (E, q)(C, 1)-mean
of Fourier series, Nigam et. al [3] proved the following theorem.
Theorem 2.1. If a function f is 2π-periodic and of class Lipα, then its degree of approximation by (E, q)(C, 1) summability mean on its Fourier series
P∞
n=0 An (t) is given by
µ
¶
1
q 1
kEn Cn − f k∞ = O
, 0 < α < 1,
(n + 1)α
where Enq Cn1 represents the (E, q) transform of (C, 1) transform of sn (f : x).
Subsequently Misra et. al. [2] have proved the following theorem on degree
of approximation by the product mean (E, q)(N, pn ) of Fourier series:
Theorem 2.2. If f is a 2π-Periodic function of class Lipα, then degree of
approximation by the product (E, q)(N, pn ) summability means on its Fourier
series (defined above) is given by
µ
¶
1
kτn − f k∞ = O
, 0 < α < 1,
(n + 1)α
where τn as defined in (7).
3
Main Theorem
In this paper, we have proved a theorem on degree of approximation by the
product mean (E, q)(N, pn ) of the Fourier series of a function of class Lip(α, r).
We prove:
Theorem 3.1. If f is a 2π- periodic function of the class Lip(α, r), then
degree of approximation by the product (E, q)(N, pn ) summability means on its
Fourier series (9) is given by
µ
¶
1
k τn − f k∞ = O
, 0 < α < 1, r ≤ 1,
1
(n + 1)α− r
where τn is as defined in (7).
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Sunita Sarangi et al
4
Required Lemmas
We require the following Lemmas for the proof the theorem.
Lemma 4.1.
|Kn (t)| = O(n), 0 ≤ t ≤
Proof For 0 ≤ t ≤
1
,
n+1
|Kn (t)| =
1
2π(1 + q)n
≤
1
2π(1 + q)n
≤
1
2π(1 + q)n
≤
(2n + 1)
2π(1 + q)n
1
.
n+1
we have sin nt ≤ n sin t then
µ
¶ ¯
¯
¯
¯
1
¯X
µ ¶
½
sin υ + 2 t ¾¯¯
k
¯ n n n−k 1 X
¯
¯
pk−υ
q
t
¯
¯
P
k
sin
k
¯ k=0
¯
2
υ=0
¯
¯
¯
¾¯
½
k
n µ ¶
¯X
(2υ + 1) sin 2t ¯¯
n n−k 1 X
¯
pk−υ
q
¯
¯
¯
¯
Pk υ=0
k
sin 2t
k=0
¯ n µ ¶
¯
½ X
¾¯
k
¯X n
1
¯
¯
n−k
q (2k + 1)
pk−υ ¯
¯
¯
¯
k
Pk υ=0
¯ k=0 µ ¶
¯
n
¯X
n n−k ¯¯
¯
q ¯
¯
¯
¯
k
k=0
= O(n).
This proves the lemma.
Lemma 4.2.
µ ¶
1
1
≤ t ≤ π.
|Kn (t)| = O
, f or
t
n+1
µ ¶
Proof
For
sin nt ≤ 1. Then
1
n+1
≤ t ≤ π, we have by Jordan’s lemma, sin
t
2
≥
t
,
π
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Degree of Approximation of Fourier Series...
|Kn (t)| =
≤
=
=
=
µ
¶ ¯
¯
¯
¯
1
¯ n µ ¶
½ X
¾¯
υ
+
sin
t
k
2
X
¯
¯
n n−k 1
1
¯
¯
q
pk−υ
t
¯
¯
n
2π(1 + q) ¯ k=0 k
Pk υ=0
sin 2
¯
¯
¯
¯
¯ n µ ¶
½ X
¾
k
¯X n
1
1
πpk−υ ¯¯
¯
n−k
q
¯
¯
¯
2π(1 + q)n ¯ k=0 k
Pk υ=0 t
¯
¯
½
¾¯
n µ ¶
k
¯X
1
n n−k 1 X
¯
¯
q
pk−υ ¯
¯
¯
2(1 + q)n t ¯ k=0 k
Pk υ=0
¯
¯ n µ ¶
¯
¯X n
1
¯
n−k ¯
q
¯
¯
¯
2(1 + q)n t ¯ k=0 k
µ ¶
1
O
.
t
This proves the lemma.
5
Proof of theorem 3.1
Using Riemann-Lebesgue theorem, for the n-th partial sum sn (f : x) of the
Fourier series (9) of f (x) and following Titchmarch [4], we have
µ
1
sn (f : x) − f (x) =
2π
Z
π
0
¶
1
2
sin n +
t
µ
¶
φ(t)
dt.
t
sin 2
Using (1), the (N, pn ) transform of sn (f : x) is given by
µ
1
tn − f (x) =
2πPn
Z
π
0
¶
1
2
sin n +
t
µ
¶
φ(t)
pn−k
dt.
t
k=0
sin 2
n
X
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Sunita Sarangi et al
Denoting the (E, q)(N, pn ) transform of sn (f : x) by τn , we have
¶
µ
1
½
Z π
sin υ + 2 t ¾
n µ ¶
k
X
1
n n−k 1 X
kτn − f k =
φ(t)
q
pk−υ
dt
2π(1 + q)n 0
k
Pk υ=0
sin 2t
k=0
½Z 1
Z π
Z π ¾
n+1
φ(t)Kn (t)dt
=
φ(t)Kn (t)dt =
+
0
1
n+1
0
= I1 + I2 .
(17)
Now
µ
¶
¯
¯
¯
¯
1
¯Z 1
½ X
sin υ + 2 t ¾ ¯¯
n µ ¶
k
X
¯
n+1
1
n n−k 1
¯
|I1 | =
φ(t)
pk−υ
dt¯¯
q
t
¯
n
2π(1 + q) ¯ 0
P
k
sin
k υ=0
¯
2
k=0
¯
¯
¯Z 1
¯
¯ n+1
¯
= ¯¯
φ(t)K(t)dt¯¯
0
¶ r1 µ Z 1
¶ 1s
µZ 1
n+1
n+1
r
s
(φ(t)) dt
(Kn (t)) dt , using Holder’s inequality
=
0
µ
= O
= O
Next
¶µ
0
¶ 1s
ns
n+1
¶
µ
=O
1
−1+α
1
(n + 1)α
µ
1
(n + 1) s
µZ
|I2 | ≤
µ
π
1
n+1
µ
¶r ¶ r1 µ Z
φ(t) dt
¶
1
1
(n + 1)α− r
π
1
n+1
s
(Kn (t)) dt
¶ 1s
,
using Holder’s inequality
¶µ Z π µ ¶s ¶ 1s
1
1
= O
dt , using Lemma 4.2
α
1
(n + 1)
t
n+1
µ
¶µ·
¸π ¶ 1s
1
−s+1
= O
t
,
(n + 1)α
1
n+1
µ
¶µ
µ
¶ 1−s
¶
µ
¶
s
1
1
1
1
= O
=O
=O
.
1
1
(n + 1)α
n+1
(n + 1)α−1+ s
(n + 1)α− r
Then from (17) and the above inequalities, we have
µ
¶
1
|τn − f (x)| = O
, for 0 < α < 1, r ≥ 1,
1
n + 1α− r
µ
¶
1
k τn − f (x) k∞ =
sup |τn − f (x)| = O
, 0 < α < 1, r ≥ 1.
1
−π<x<π
(n + 1)α− r
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Degree of Approximation of Fourier Series...
This completes the proof of the theorem.
References
[1] G.H. Hardy, Divergent Series, First Edition, Oxford University Press,
70,(19).
[2] U.K. Misra, M. Misra, B.P. Padhy and M.K. Muduli, On degree of approximation by product mean (E, q)(N, pn )of Fourier Series, Gen. Math. Notes,
6(2), (2011).
[3] H.K.Nigam and Ajay Sharma, On degree of Approximation by Product
Means, Ultra Scientist of Physical Science, 22(3), (2010), 88-89.
[4] E.C. Titchmarch, The Theory of Functions, Oxford University Press, pp.
402-403, 1939.
[5] A. Zygmund, Trigonometric Series, Second Edition, Cambridge University
Press, Cambridge, 1959.