Internat. J. Math. & Math. Sci.
VOL. 12 NO.
(1989) 99-106
ON
99
(J,pn) SUMMABILITY OF FOURIER SERIES
S.M. MAZHAR
Department of Mathematics
Kuwait University
P.O. BOX 5969
13060, Safat, Kuwait
(Received December 13, 1986 and in revised form April 8, 1987)
In this note two theorems have been established.
ABSTRACT.
the summabillty
(J,pn)
The first one deals with
of a Fourier series while the second on concerns with the
These results include, as a special
summabillty of the first derived Fourier series.
case, certain results of Nanda [I].
KEY WORDS AND PHRASES.
(J,pn)
summsbility, Fourier series, Derived Fourier series.
1980 AMS SUBJECT CLASSIFICATION CODES.
42A24, 40GI0.
INTRODUCTION.
Let
{Pn
be a sequence of non-negatlve numbers such that
o
Pn
diverges and let
the radius of convergence of power series
be I.
Given any series
(1.1)
pnx
pCz)
I a
n
E
Ps(X)
with the sequence of partial sums
PnSnx
{s
n
we write
n
(1.2)
and
J (x)=
PS (x)
(I 3)
If the series in (1.2) is convergent in [0,I) and
lira J (x)
s
x+l
we say that the series
is a finite number.
For
A
k
Pn=l,
n
with
E a
n
s,
or the sequence
([2], [3], p.80).
Po=0, and A k > -I
k,n
method of summability respectively.
we
get
{s
n
is summable
summablllty
A,
(J,pn)
to s, where s
summabllity
(L)
and
i00
S.M. MAZHAR
Suppose f is a periodic function with period
(-,). Let
2 and integrable in the sense of
Lebesgue over
f(x)
~-
+ Z (a n cos nx + bn sin nx)
E
(1.4)
An(X).
Then the first derived series of (1.4) is
r. n(b
n
cos nx- a
z n B (x).
sin nx)
n
(1.5)
n
We write
@(t)
*(t)
=
=
{f(x
+ t) + f(x
0
{f(X0 + t)
(t)
g(t)
M(t)
2sin
t/2
Z
n
Pnx
o
t)}
0
s
sin n t
f *(u.):u
(t)
f(X
2s}
t)
0
du
and
g(u)
(t)
2.
MAIN RESULTS.
In
(J,pn)
If
du.
u
this
propose
we
note
establish
to
the
following
theorems
on
summabillty of (1.4) and (1.5).
THEOREM I.
Let
(a)
n
(b)
E
nO
{pn
be a positive sequence such that
0(I), r.
Pn
A2(n p.
Pk
xn)l
0(pv
and
0(l-x),
<
0
<
x
I.
o(p(l-t)), t +0. then the Fourier series (1.4) is summable
Let {pn satisfy the hypothesis (a) of Theorem I. If
#(t)
(J,pn)
to s.
THEOREM 2.
oZ n)A
(c)
and G(t)
(J,pn)
(n
Pn xn)l
o(p(l-t)), as
O(l-x), 0
<
x
<
t+ 0+, then the first derived series
(1.5)
is
summable
to s.
It may be remarked that for
Pn
’
n
)
I,
Po
0 our theorems include two known
theorems of Nanda [I] on L-summabillty of Fourier series and its derived series.
an earlier result on
(J,pn)
For
summabillty of (1.4) under more stringent conditions see
SUMMABILITY OF FOURIER SERIES
Khan
[4].
Very
recently
It may be observed that
Pn <
see that n
For if
{n
pn
(n+l)
n
(J,pn)
0(x)
p’(x)
0,1,2,... and p’(x)
n
llm n
is not bounded then
n+
that
n
In
n
7.
x/l
n
.
7.
lim (l-x)
n
n
Pn
x
,
n
o
x
we find for
.
n
Pn
3.
n
s
n
Pn
0(
Also it is easy to
imply that
e
I,
n
8n
0(I) as
/-I-
a
and
n
Pn
n
0(I).
Pn
>
0 with
that
x)p’(x)
which contradicts the
Thus conditions (2.1) and (2.4) of [5]
-
(x O) denote the n-th partial sum of (1.4) at
s __(x
n
O)
2
s
so that
_f0
nE--0 Pn xn(sn(X0)2
:f0
2
/0
(t)
+
I
sin nt
at
t
2
s)
n
f0
(t) E n
O.
Then
n=E0 Pn
x
n
sin nt dt
+ o(p(x))
pn x
n
cos nt dt + o(p(x))
o
12
=- f0
ok
x
+ o(i)
+ o(p(x))
(3.1)
+ o(p(x)), say.
Now
2
x
@’(t)M(t) dt + o(p(x))
2
l-x
=(f0
+f-x.)...
Ii
I-
as x
0(I).
PROOF OF THEOREM i.
Let
have
n
This means llm (I
ypothesis that (l-x)p’(x)
imply that
[5]
Now using the well known result
where radius of convergence of each power series is
o
Sahney
and
n
0
in
c
Mohapatra
summabillty with another set of conditions.
0(I)
Pn
Pn+l’
Prem Chandra,
1985
in
established a similar theorem on
I01
l-x
f0l-x
(t)
n=E0 n Pn x
o(p(l-t))
f -x
n
at]
cos nt dt
S.M. MAZHAR
102
Khan
[4].
Very recently
in
(J,pn)
established a similar theorem on
For if
Mohapatra and
Pn <
In pn
n
0(.
r.
in
x+l
n
o
,
Z a
o
n
x
I ax
n
o
Pn
x
we find for
.
3.
n
I, 6n
a
n
a
and
n
Pn
>
0 with
that
x)p’(x)
This means lim (I
which contradicts the
x+l-
0(I) as
hypothesis that (l-x)p’(x)
imply that
Pn
n
n
where radius of convergence of each power series is
n
have
summabillty with another set of conditions.
Pn
In
lim (l-x)
[5]
"
that
x/l-
Sahney
0(I)
p’(x)
0(T_x) as x I- Also it is easy to
0(I).
(n+l)
x) imply that n
Pn+l’ n 0,1,2,... and p’(x)
Now using the well known result
llm n
is not bounded then
n+ Pn
It may be observed that
see that n
Prem Chandra,
1985
Thus conditions (2.1) and (2.4) of [5]
I-
x
0(I).
Pn
PROOF OF THEOREM I.
Let
s (x
n
denote the n-th partial sum of (1.4) at
0)
s
n(x0)
2
s
fO
so that
nZo
Pn
-2_
f’n’0
x
xn(sn(X0)-s)
n
Then
+ o(1)
dt
(t)
f0
t
x cos nt dt
o
+
2
t
0.
nZO Pn
x
n
sin nt at
+ o(p(x))
@’(t)M(t) dt + o(p(x))
=--2 fo (t) Z pn
=2 f-x/ fx-x)’"
I
sin _nt
(t)
x
x
12 +
/
+ o(p(x))
o(pCx))
o(p(x)), say.
(3.)
Now
II
= fo
2
o(
o( l-x
1-x
(t)
n
nZO n pn x
cos nt dt
-x
(3.2)
fOl"x nffiEO
Pk
(l-t)
k
at
SUMMABILITY OF FOURIER SERIES
o(1)
Z0
o(I)
x
v
o()k.Z0 k+--T Z0
Pk
k,Zv
k
Pk
Pk _xk+l
kY (’
k-zO
103
v
0 px
o(I)
x
o(p(x)
Thus
(3.3)
o(p(x)).
I
Again
= fl-x
2
12
!
7
"
l-x
(t)
where
k
*(t)
k.0 k Pk x cos kt
k.Z0 z (k Pk xk) Fk(t
k
k
k
v-0 D(t) Z0 Dv(t) -Z0
Fk(t)
sln(v)t
2 sin t/2
Under the hypothesis of Theorem
12
fl-x p(1-t) (1t-J,x) dt
(l-x) dt
o(p(x)) -l--x
2
t
0(I)
I
(3.4)
o(p(x)
(3.1), (3.3) and (3.4)
Thus in view of
Z
o
Pn
x
n
(Sn(X0)
s)
o(p(x)) as x
I-
This proves Theorem 1.
4.
As shown in ([6], p. 54) we can assume that s
PROOF OF THEOREM 2.
Let
denote the n-th partial sum of (1.5) at
Tn(X0)
sin nt
dt--_f0
0 g(t)
sin t/2’
2n
Tn(X0)
=
2
T
f0g(t)
n,1
sin nt
dt
t
+ o(1) + T
+ o(I)
say
n, 2
so that
Z
O
Tn(X0) Pn
n
x
n
x
2n
Z
Tnl Pn
L
+ o(p(x)) + L 2
say.
w
Then
x
0.
cos (n +
+ o(p(x)) +
O
x
)
f0 cos
0.
t
(n
Tn2 Pn
g(t) dt
+-)
x
t
g(t) dt
n
(4.1)
104
S.M. MAZHAR
As shown in the proof of Theorem
z
H()
Let
L2
px
in view of
cos(n +
) t.
-- - f0
2
fo g(t)
nZ0 n Pn x
g(t)
n
cos(n
2
H(t) dt
Now since
n
f0
(4.2)
)t g(t) dt
+-)
fo
_2, {JiG(t) li(t)] 0 0 G(t)
_2
o(p(x)).
Then
_2 Eo n pn x /0 cos(n +
2
L
(4.5),
d
t dt
t G’ (t)H(t) dt
(tH(t))}dt
R(t) + tH’(t)} dt
G(t)
2
w
fol-x+ /I-x
-x
o(p(l-t) dt
-l-:x
L21 + L22
say.
0(I)
Pn
L21
+
f-x o(p(l-t))
dt
o
J,-
+ o(
o(p(x)
E n
t
1-x
dt
o(p(x)
(4.3)
dt
(4.4)
as shown in (3.2).
In view of the hypothesis of Theorem 2.
L22
Since
n
Pn
2
a
d
(t
fl-x o(p(1-t) [-
0(I), we have by using Abel’s transformation
d
d-
t H(t)
g n
Pn
x
nd
-
t cos(n
Z a(n p x n) d
n
o
E
o
Thus
H(t))l
A2(n
Pn
n
x
d
+-)
t
t
(sin (n+l)t)
2 sin t/2
{
t
4 sln 2 t/2
(cos
t/2
cos(n +
3/2)t)}.
SUMMABILITY OF FOURIER SERIES
t2
105
t2
t2
where C is a positive constant not necessarily the same at each occurrence, and in
view of the fact that
implies that
o
nlA3(n Pn xn)
)82(n Pn xn))
0
(4.5)
O(l-x)
O(l"X)
Hence from (4.4)
o(1)
Thus from (4.1)
l-x p(1-t)
(l-x)
dt
o(p(x))
as shown in (3.3).
(4.6)
t2
(4.3) and (4.6) the proof of Theorem 2 follows.
REFERENCES
I.
NANDA, M.
The summability (L) of the Fourier series and the first
differentiated Fourier series,
2.
3.
4.
5.
6.
BORWEIN, D.
Qu_a. rt. J. Math. Oxford (2) 13 (1962), 229-234.
On methods of summability based on power series, Proc. Royal Soc.
Edinburgh 64 (1957), 342-349.
HARDY, G.H. Divergent Series (Oxford, 1949).
KAN, F.M. On (J,pn) summabillty of Fourier series, Proc. Edinburgh Math. Soc.
18 (II) (i972), 13-17.
CHANDRA, P., MOHAPATRA, R.N. and SAHNEY, B.N. Tamkang J. Math. 16 (1985)
No. 2, 37-41.
MOHANTY, R. and NANDA, M. The summabtltty by logarithmic means of the derived
Fourier series, _Quart. J. Math. Oxford (2) 6 (1955), 53-58.