J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
On Degree of Approximation of Conjugate Series of Fourier Series by Product Means (E , q )(N , p n ) B. P. PADHY1 , M. K. MUDULI2, U. K. MISRA3 and MAHENDRA MISRA4 1,2
Roland Institute of Technology, Golanthara-760008, Odisha, India. 3 Department of Mathematics, Berhampur University Berhampur – 760007, Orissa, India 4 Principal, Government Science College, Mlkangiri, Orissa. India ABSTRACT A theorem on degree of approximation of conjugate series of Fourier series by product means (E , q )( N , p n ) has been established. Keywords: Degree of Approximation, function,
(E, q )
mean,
(N , p n )
f ∈ Lip α mean,
class of
(E , q )(N , pn )
product mean, Fourier series, conjugate of Fourier series, Lebesgue integral . 2010-Mathematics subject classification: 42B05, 42B08.
1. INTRODUCTION Let
∑a
n
{t n } of the (N , p n ) mean of the sequence {s n } generated by the by sequence of coefficient {p n }. If defines the sequence
be a given infinite series
with the sequence of partial sums
{s n } .Let
{p n } be a sequence of positive real numbers
such that
(1.3)
t n → s , as n → ∞ ,
n
(1.1)
Pn = ∑ pυ → ∞ , as n → ∞ , υ =0
( P−i = p −i = 0 , i ≥ 0 ) The sequence –to-sequence transformation (1.2)
1 tn = Pn
then the series
∑a
n
is said to be ( N , p n )
summable to s .
n
∑p υ =0
n −υ
sυ
The conditions for regularity of Nörlund summability ( N , p n ) are easily seen to be1
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
pn (i ) P → 0 as n → ∞ n . n (ii ) p = O( P ) as n → ∞ k n ∑ k =0
(1.4)
f ( x) ~
(1.9)
(1.5)
Tn =
n n −υ q sυ ∑ υ = 0 υ
defines the sequence {Tn }
mean of the sequence {s n } .If
.
of the (E, q )
Tn → s , as n → ∞ ,
(1.6)
then the series
∑a
n
is said to be (E, q )
summable to s . Clearly
(E, q )
Further, the
method is regular1.
(E, q )
(1.7) If (1.8) then
=
1
(1 + q )n
is defined by
n n−k q Tk ∑ k =0 k n
n ∑ k q n−k k =0 n
n =0
(1.10)
∞
∞
n =1
n =1
∑ (an cos nx − bn sin nx ) ≡ ∑ Bn ( x)
(
)
Let sn f ; x be the n-th partial sum of (1.10). The L∞ -norm of a function f : R → R is defined by (1.11)
f
∞
1 Pk
∑ pk −υ sυ υ =0 k
τ n → s , as n → ∞ , ∑ 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
= sup{ f ( x) : x ∈ R
}
and the Lυ -norm is defined by
transform of the
(N , p n ) transform of {sn } 1 τn = (1 + q )n
∑ An ( x)
and the conjugate series of the Fourier series (1.8) is
n
(1 + q )n
∞
≡
The sequence –to-sequence transformation,1
1
a0 ∞ + ∑ (a n cos nx + bn sin nx ) 2 n =1
1
2π
(1.12)
f
υ
υ = ∫ f ( x ) 0
υ
, υ ≥1
The degree of approximation of a function f : R → R by a trigonometric polynomial
Pn (x) of degree n under norm
.
∞
is
4
defined by .
{
(1.13) Pn − f ∞ = sup Pn ( x) − f ( x) : x ∈ R
}
and the degree of approximation E n ( f ) of a function f ∈ Lυ is given by (1.14)
E n ( f ) = min Pn − f Pn
ν
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
This method of approximation is called trigonometric Fourier approximation.
f ∈ Lipα
A function (1.15)
if
( ) ,0 < α ≤ 1
f ( x + t ) − f ( x) = O t
α
We use the following notation throughout this paper:
ψ (t ) =
(1.16)
1 { f ( x + t ) − f ( x − t )}, 2
and K n (t ) =
π (1 + q )
(1.17) 1
k
1
n
cos
∑ pk −υ Pk υ =0
n n n −k ∑ q k =0 k
t 1 − cosυ + t . 2 2 t sin 2
Further, the method (E , q )(N , p n ) is assumed to be regular and this case is supposed through out the paper.
Dealing with the degree of approximation by the product (E , q ) (C ,1) mean of Fourier series, Nigam2 proved the following theorem.
,belonging to class
f ,2π - periodic
∞
∑ A (t ) is given by n =0
n
,0 < α < 1
(E, q ) transform of (C ,1) transform of s n ( f ; x ) . represents the
3. MAIN THEOREM In this paper, we have proved a theorem on degree of approximation by the product mean (E , q )(N , p n ) of conjugate series of Fourier series (1.10) .we prove: Theorem -3.1
f is a 2π − Periodic function
If
Lipα , then degree of of class approximation by the product (E, q )(N , pn ) summability means on its conjugate series of Fourier series (1.10) is given by ∞
1 = O α (n + 1)
,0 < α < 1 ,
where τ n as defined in (1.7) . 4. REQUIRED LEMMAS We require the following Lemmas to prove the theorem-3.1. Lemma -4.1
Lipα , then its degree
(E , q ) (C ,1) of approximation by summability mean on its Fourier series
∞
1 = O α (n + 1)
E nq C n1
,where
τn − f
2. KNOWN THEOREM
Theorem-2.1 If a function
Enq Cn1 − f
583
K n (t ) = O ( n)
,0 ≤ t ≤
1 . n +1
Proof of Lemma-4.1 For 0 ≤ t ≤
1 , we have sinnt ≤ nsint n +1
then
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
K n (t ) =
≤
t 1 cos − cosυ + t n n − k 1 k 2 2 ∑ q ∑ pk −υ t k P k =0 k υ =0 sin 2 n
1
π (1 + q )n
1 π (1 + q )n
n n−k 1 q ∑ k =0 k Pk n
k
∑p
υ =0
n−k 1 q Pk
1 ≤ π (1 + q )n
n ∑ k =0 k
1 ≤ π (1 + q )n
n n−k 1 q ∑ k =0 k Pk
n
n
k −υ
t t t cos − cosυ t. cos + sin υ t. sin 2 2 2 t sin 2
t t 2 cos 2 sin υ k 2 2 p k −υ + sin υ t ∑ t υ =0 sin 2
t t p O 2 sin υ sin υ + υ sin t ∑ k −υ 2 2 υ =0 k
n n−k 1 k q ∑ p k −υ (O (υ ) + O (υ )) n ∑ π (1 + q ) k = 0 k Pk υ = 0 n n n− k O(k ) k 1 ≤ ∑ q P υ∑=0 pk −υ π (1 + q )n k =0 k k 1
≤
n
= O (n). This proves the lemma-4.1. Lemma-4.2:
1 1 K n (t ) = O , for ≤t ≤π . n +1 t Proof of Lemma-4.2: For
t t 1 ≤ t ≤ π , we have by Jordan’s lemma , sin ≥ n +1 2 π
.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
K n (t ) =
Then
1 π (1 + q )n
n n − k 1 q ∑ k =0 k Pk n
k
p ∑ υ =0
k −υ
585
t 1 cos − cosυ + t 2 2 t sin 2
t t t t t υ υ cos − cos . cos + sin . sin k n n−k 1 1 2 2 2 2 2 q p = ∑ k −υ ∑ t π (1+ q)n k=0 k Pk υ=0 sin 2 n
1 ≤ π (1 + q )n
≤
n n −k 1 q ∑ k =0 k Pk n
π
π
t t t t 2 p + cos 2 sin υ sin υ . sin ∑ k −υ 2 2 2 2 υ =0 2t k
n n−k 1 q ∑ n 2π (1 + q ) t k =0 k Pk n
n n−k 1 q ∑ k =0 k Pk n n n−k 1 q = ∑ n 2 (1 + q ) t k =0 k
1 = n 2 (1 + q ) t
n
k
p ∑ υ =0
k
∑p υ =0
k −υ
k −υ
.
1 t
= O . This proves the lemma-4.2. 5. PROOF OF THEOREM- 3.1 Using Riemann –Lebesgue theorem, we have for the n-th partial sum s n ( f ; x ) of the conjugate Fourier series (1.10) of f (x ) , following Titechmarch3
s n ( f ; x ) − f ( x) =
2
π
π
∫ψ (t ) K
n
dt ,
0
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
586
B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
the (N , p n ) transform of sn ( f ; x ) using (1.2) is given by
t n − f ( x) =
denoting the
τn − f =
π
2 ψ (t )∑ p n − k π Pn ∫0 k =0
(E, q )(N , pn ) transform of sn ( f ; x ) 2 π (1 + q )n
n n− k 1 q ψ ( t ) ∫0 ∑ k =0 k Pk
π
1 t cos − sin n + t 2 2 dt , t 2 sin 2
n
n
by τ n , we have k
∑p
υ =0
k −υ
t 1 cos − sin υ + t 2 2 dt t 2 sin 2
π
=
∫ψ (t )
K n (t ) dt
0
n1+1 π = ∫ + ∫ ψ (t ) K n (t ) dt 1 0 n +1 (5.1)
= I 1 + I 2 , say
Now 1
I1 =
2 π (1 + q )n
n n−k 1 ψ (t )∑ q k =0 k Pk
n +1
∫ 0
≤ O ( n)
n
1 n +1
∫
ψ (t ) dt
k
∑p
υ =0
k −υ
t 1 cos − cosυ + t 2 2 dt t 2 sin 2
, using Lemma-4.1
0
= O ( n)
1 n +1
∫
t α dt
0 Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011) 1 n +1
α +1
t = O ( n) α + 1 0
1 = O( n) α +1 . (α + 1)(n + 1)
1 = O α (n + 1)
(5.2)
Next
I2 ≤
π
∫
ψ (t )
K n ( t ) dt
1 n +1
=
π
1 t
ψ (t ) O dt
∫
1 n +1
=
π
∫
1 n +1
π
=
∫
t
α −1
, using Lemma-4.2
1 t α O dt t
dt
1 n +1
(5.3)
α (n + 1)
= O
1
Then from (5.2) and (5.3) , we have
τ n − f (x ) = O α ( ) n + 1 1
, for 0 < α < 1 .
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
587
588
B. P. Padhy, et al., J. Comp. & Math. Sci. Vol.2 (4), 581-588 (2011)
Hence,
τ n − f (x ) ∞ = sup τ n − f (x ) = O
1 α (n + 1)
−π < x <π
,0 < α < 1
.
This completes the proof of the theorem-3.1. REFERENCES 3. 1. 2.
Hardy, G. H. Divergent Series, First edition, Oxford University Press 70 (19). Nigam, H. K. and Kusum Sharma. The degree of Approximation by Product Means, Ultra scientist, Vol. 22 (3) M,
4.
889-894, (2010). Titechmalch, E. C. The theory of functions, Oxford University Press, p.p 402-403 (1939). Ziggmund, A. Trigonometric Series, Second Edition, Vol. I, Cambridge University Press, Cambridge, (1959).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)