Internat. J. Math. & Math. Scl.
Vol. 8 No. 2 (1985) 367-372
367
ON THE STRONG MATRIX SUMMABILITY
OF DERIVED FOURIER SERIES
K. N. MISHRA and R. S. L. SRIVASTAVA
Department of Mathematics
Indian Institute of Technology
Kanpur 208016, India
(Received April 5, 1983, and in revised form August 8, 1983)
ABSTRACT.
Strong summability with respect to a triangular matrix has been defined
and applied to derived Fourier series yielding a result which extends some known
results under a general criterion.
Strong Summabiliy, Toeplitz matrix, Fourier Sees.
KEY WORDS AND PHRASES.
1880 MATHEMATICS SUBJECT CLASSIFICATION CODE.
1.
40FO.
INTRODUCTION.
The triangular matrix
A
[an,k],
n, k
0,1,...
and
an, k
0
for
k
>
n
is
regular if
lima
n/
n
k=O
n,k
=0
lan,kl
M, M is independent of n
and
n
lim
n+-
}:
k=O
a n ,k
1
k
I: u by s k, Fekete [1], defined that the series
r=l r
strongly suable to the sum s, provided
n
Denoting the sum
Is-sl
I: u is
r
=o(n)
k=l
This type is now known as strong Cesro summability of order unity with index 1 or
[C,I] sumabil ity.
The series 11 u r is said to be strongly summable by Cesro means, with index q,
or summable [C,q], or summable H q to the sum s if
n
S
sl q o(n)
k=l k
Is
A special point of interest in the method of summability H lies in the fact
q
that it is given neither by Toeplitz matrix nor by a sequence to function
transforma-
368
K. N. MISHRA AND R. S. L. SRIVASTAVA
The relationship between summability H
q and some regular methods of summation given by A- matrices has been investigated by Kuttner, [2], who proved that if
A is any regular Toeplitz method of summability then for any q (0< q < 1) there is
a series which is not summable A but summable H
q
In the present paper e shall define strong summability of series I: u with
k
the help of a matrix.
DEFINIIION. The series z u k is sid to be strongly summable by the regular
method A determined by the matrlx
with index q(q > o) to the sum s if
n
}1 a
sl q o(I), as n
k=o n
tion.
Jan,k]
klSk
For
an,k
n+--#-i-’ k
<_
n,
vie
(C,I)
get
matrix.
MAIN RESULTS.
Let f(x) be a periodic function with period
(-,). Let
2.
1/2 a
f(x)
be the Fourier series of
2
and integrable
(L)
(a n cos nx + b n sin nx)
/
over
(2 I)
f(x) and
n(b n cos n x a n sin nx)
1
be the first derived series of (2.1) ottained by term by term differentiation.
%
(2.2)
Write
g(u)
where
f’(x)
f(x+u) f(x-u)
f(x),
2uf’(x)
(2.3)
is the derivative of
t
f ldg(u)l
o
G(t)
Here we shall take
(2.4)
q
2,
1,2. Since the cse q
1 is included in the strong
we omit the same. Precisely we prove the following"
THEOREM. Let g(u), G(t) be defined as ir (2.3) and (2.4). If g(u) is a
continuous function of bounded variation over [0,] and for some B
1
summability for q
G(t)
where
B(t)
o [t
B(t)]
is a positive function of
B(t)
it is monotonic in
(n-l,a)
t
as t
such that
o as
t
o
(2.6)
o
(6 being small but fixed) and
n1
xzB{t)
t
dt
0(I)
then the derived series (2.2) is strongly summable to f’(x) by the matrix
index 2.
Note (2 7) is equivalent to
(2.5)
2(t)
t--
L(o,a).
(2 7)
(C,1) with
MATRIX SUMMABILITY OF DERIVED FOURIER SERIES
369
In order to prove the theorem we need the following lemma.
LEMMA. If G(t) o(t) as t o then for small but fixed
Idg(u)l
b
du
6
o(n)
and
t 2-
n 1
PROOF.
’’U’I du
u
n 1
o(n)
Since
Idg(u)
u
+
du
du
n -1
o(I) +
u
S_
o()
n I
du, in view of (2.4)
o(log n)
Therefore
fl IdgIt
dt
n
/_1 Idg(u)lu
du
o(lof# n) 2
o(n)
n
Again
-I
n
Idg(t)It2
S_
dt
Idg(U)lu
n 1
du
t
6
t
+
n
n
/-I
Idg(t)l
dg(t) log nt}
o(I)
t2
n I
ttGt- In-] -f
lgnt
0, +
1
+ o
(S(Iiu2)
1
n
_, ,o n< <
n6
n6
o(n)
dt
+ o(I) + o(log n t)} dt
t2
n
S GGu-du}
du) + o
ulu 2) du]
[S(Ig
I
o(n).
3.
PROOF OF TIIE THEOREM.
The kth partial sum
Ok(X)
of the series (2.2) is given by [3],
i
_
K. N. MISHRA AND R. S. L. SRIVASTAVA
370
k(X)"
f’
(x)=o Sinsin k21-+1/2)t
dg(t)
Further, simplifying certain steps as given by [3] and [4] we have
1j_1
f’(x)
Ok(X)
T
sin kt
dg(t) + o(1)
t
n
-E
n
Therefore
k=l
f
1
2
n
f’(x)}
{Ok(X
dg(t)
t
n -I
g()
+-
n
/I
dg(t)
t
nl
dg(t)_
t
(g(u)
n
I: sin kt sn ku
u
I
"I:
{COS k
o(n)
dg(u} + o(n)
u
cos k(u+t)}
(u-t)
__
+
_.in(n+l/2)(u-t)
Sn- i (u-t)
I dg(u)
si-nln-+.1!2)(u-t)
"I (u+t)
ul-dg(u)
2 sin
+
o(n)
n
On simplifying and using the first part of the lemma we obtain
n
r.
k= 1
n_l )/1
Jl 1
dg(t
1
{ok(x)_ f,(x))2_
t
d -d
(
1
t
n
o(n), say.
P1 + P2 +
NOW, since
and
fl
sin
n(u-t) dg(u)
u(u-t)-
t
sin
n(u-t) dg(u)
lu+l} dg(u)
sin n
(u+t)
dgu)
n
n
u
sin
z t
+ o(n
n{u-t)dg(t)
(u-t)
Therefore
t
P1
sin
1
nlu-t
u’(u-t
n -I
t
dg(u) +
t
dg(t)
-t
1
n
sin
n(u-t) dg(u)
u(u-t)
1
dg(t)S
-I t
/
t
sin
nlu-t
u{u-t
dg(u)
MATRIX SUMMABILITY OF DERIVED FOURIER SERIES
t
.
-
n(u-1) dg(u)
sin
n -I
t’
n 1
t
n dn
6
t)
1
,
(u-
-1
-1
t
=--Z
t2
n
(u
n 1
__
t
6
n-t)dg(u)+ o [ f-1
sin
_
37]
dg(t)l
t
u
n
n
by virtue of the second part of the emma.
Similarly it can be proved that
o(n). Thus we get
t
n
sin n(u-t)
1
.dg(t)
f’(x)}2
}:
dg(u) + o(n)
{k(X)
2
u(u-t)
k=l
t
-1
-1
=- n
Integration by parts gives
t
J_
n
dg(u)sin(u-t
n 1
P2
l.u-tl
I
nr
t
t
n -1
n
f dg(u)]-I
lu-tl
sin n
(u-’t)
t
.n
[{
Using (2.5) this is equal to
[
n(u-t)
(u--{-) o
sin
(u-t)
coS,{u_tnlu-t), sin(u_t)
2n
t
{t X B
(t)}] t -I
o [
n
+
0
o [ n t
Therefore
{Ok(X)
o[n
f’(x)}
o(n)
dg(t)
n 1
J-1
G_
t
o(n)
n(u-t_ du]
(u-t)
n(u
(u.t) t)
sin
LB(t)
o(n) [G(t)
+
cos
(t)}
t
f
[
tBx
{n
1
n -I
k=l
dg(u)] du
t
B-l(t)
+
o(n)
B (t)} du]
]
B(g)]
B(t)]
B
{t
n -I
+
+
o(n)
o(n)If-1 dg(t)B)dt]
n
’(t)} dt]
f
n -1
/
o(n) [
B
n -1
).B(t) }.B-l(t)
X’(t) dt]
K.N. MISHRA AND R. S. L. SRIVASTAVA
372
d
1
o(n) + o(n) [
n -1
g [[
:X2B( t)
dt]
o(n) by the hypothesis (2.7).
Since
B(t)
is monotonic, hence its differential coefficient is of constant sign.
Thus we get
n
k=l
f’(x)l 2
]Ok(X
o(n)
and therefore
n
7.
an, ]Ok(X
k
k=l
This completes the proof of the theorem.
4.
an,k
f’(x)}2
o(r,)
SPECIAL CASES.
By way of an application of our theorem, we take B
1
1, x(t)
I/log (l/t)
and
then the following result follows, [4]"
At a point for which f’(x) exists and
THEOREM (Sharma).
G(t)
1
as t
o[ t/log ]
o
then
n
Z
}Ok(X) f’(x){2
o(n loglog n)
k=l
Since the above theorem is an extension of the result from [C, I] summability to the
case of [C, 2] summability, (Prasad and Singh [3]), our theorem further extends that
result under a general type of criterion.
ACKNOWLEDGEMENT.
We are thankful to the referee for his valuable suggestions.
REFERENCES
1.
FEKETE, M. Vizsgalatok a Fourier-Sovokral, Mathematikai es Terniezs Ertesitok
3__4, (1916), 769-786.
2.
KUTTNER, B. Note on Strong Summability, J. Lond. Math. Soc. 21, (1946), 118-122.
3.
PRASAD, B. N., and SINGH, U. N. On Stronu Sumability of Derived Fourier Series
and its Conjugate Fourier Series. Math. z_. 56, 3(1952), 280-288.
4.
SHARMA, R. M. On
H
6__1, (1969), 75-81.
Summability of Derived Fourier Series, Bull. Cal. Math. Soc.