# MATRIX OF DERIVED FOURIER SERIES

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```					Internat. J. Math. & Math. Scl.                                                                                 367
Vol. 8 No. 2 (1985) 367-372

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.

KEY WORDS AND PHRASES.        Strong Summabiliy, Toeplitz matrix, Fourier Sees.
1880 MATHEMATICS SUBJECT CLASSIFICATION CODE.                     40FO.

1.    INTRODUCTION.
The triangular matrix         A     [an,k],        n, k     0,1,...   and   an, k   0   for   k   >   n   is
regular if
lima            =0
n/
n,k
n
k=O
lan,kl         M, M is independent of n

and
n
lim        }:   a n ,k      1
n+-       k=O
k
Denoting the sumI: u by s k, Fekete [1], defined that the series                              I: u is
r=l r                                                                              r
strongly suable to the sum s, provided
n
k=l                Is-sl         =o(n)
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
k=l k
sl q o(n)
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

tion.  The relationship between summability H
q and some regular methods of summa-
tion 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                                    Jan,k]
with index q(q > o) to the sum s if
n
}1 a       sl q o(I), as nklSk
k=o n
For     an,k     n+--#-i-’ k
<_   n,     vie   get          (C,I)     matrix.

2.  MAIN RESULTS.
Let f(x) be a periodic function with period                                                 2   and integrable    (L)    over
(-,). Let

f(x)         1/2 a     /      (a n cos nx + b n sin nx)                            (2 I)
be the Fourier series of                    f(x) and

n(b n cos n x a n sin nx)
%                                                                         (2.2)
1
be the first derived series of (2.1) ottained by term by term differentiation.
Write
g(u)          f(x+u) f(x-u)               2uf’(x)                                (2.3)
where    f’(x)      is the derivative of                     f(x),
t
G(t)
o
f ldg(u)l                                          (2.4)
Here we shall take                   q
1,2. Since the cse q       1 is included in the strong
summability for q                   2,
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
G(t)             o [t     B(t)]        as t         o                           (2.5)
where    B(t)       is a positive function of                            t     such that
B(t)            o as      t   o                                      (2.6)
it is monotonic in              (n-l,a)          (6 being small but fixed) and

xzB{t)
t      dt       0(I)                               (2 7)
n1
then the derived series (2.2) is strongly summable to f’(x) by the matrix                                            (C,1) with
index 2.

Note (2 7) is equivalent to                       2(t)               L(o,a).
t--
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                                                              6

Idg(u)l             du        o(n)
b

and

’’U’I du                   o(n)
n 1            t 2-                                 u
n 1
PROOF.       Since

Idg(u)        du                                +                         du
u
n -1                        u

o(I) +                 S_
n I
o()        du, in view of (2.4)

o(log n)
Therefore

Again
fl IdgIt
n
dt
/_1 Idg(u)lu
n
du        o(lof# n) 2     o(n)

-I
n
Idg(t)It2         dt     S_
n 1
Idg(U)lu
t
du

6                                                        t

n                                         n
+
S GGu-du}                 dt

/-I
n
Idg(t)l
t2
+ o(I) + o(log n t)} dt

o(I)                   dg(t) log nt}
n I
t2

ttGt- In-] -f
lgnt
1
0, +
n
_, ,o n< <            i
n6                                          n6
o(n)      + o    (S(Iiu2)
1
du) + o               [S(Ig ulu 2) du]
I
o(n).

3.        PROOF OF TIIE THEOREM.
The kth partial sum                    Ok(X)      of the series (2.2) is given by [3],
370

Further, simplifying certain steps as given by [3] and [4] we have

Therefore

n
k=l
{Ok(X        f’(x)}
2
K. N. MISHRA AND R. S. L. SRIVASTAVA

k(X)"

Ok(X)

1
f’

f’(x)

f
n -I
-E

dg(t)

g()
t

+-
T

_
(x)=o Sinsin k21-+1/2)t
1j_1
n

n
n
sin kt

"
I:
n

I
t

{COS k
dg(t)

dg(t) + o(1)

I: sin kt sn ku

(u-t)
(g(u)
u

cos k(u+t)}
+     o(n)

dg(u} + o(n)
u

__
I dg(u)
/I       dg(t)
t
nl
_.in(n+l/2)(u-t)
Sn- i (u-t)

n
dg(t)_t
si-nln-+.1!2)(u-t)
2 sin  "I (u+t)                ul-dg(u)   +     o(n)

On simplifying and using the first part of the lemma we obtain
n
r.
k= 1
{ok(x)_ f,(x))2_                   1
n_l )/1   dg(t
t
sin          n(u-t) dg(u)

1

P1 + P2 +
Jl 1     (-d
d
t

o(n), say.
n
sin n
(u+t)
lu+l} dg(u)
u       + o(n

NOW, since

and

fl         t
sin    n(u-t) dg(u)
u(u-t)-
n
dgu)
n
sin
z t
n{u-t)dg(t)
(u-t)

Therefore
t
P1              1           t
n -I
u’(u-t
sin
nlu-t          dg(u) +             1
/  dg(t)S
-I t             t
sin
u{u-t  nlu-t   dg(u)

t
dg(t)
-t          1
n
sin
u(u-t)
n(u-t) dg(u)
.

-          n -I

n dn
6
MATRIX SUMMABILITY OF DERIVED FOURIER SERIES

t’
t

n 1
t
sin             n(u-1) dg(u)

6
_   t
37]

__
,
1
-1
t)
-1
sin
(u-  n-t)dg(u)+ o [ f-1                                                 dg(t)l
t
n
u
n
t
=--Z                       t2                       (u
n                       n 1
by virtue of the second part of the emma.
Similarly it can be proved that        o(n). Thus we get                P2

t
n
}:
k=l
{k(X) f’(x)}2
Integration by parts gives
1
-1
.dg(t)
t 2
t

-1      =- n
sin n(u-t)
u(u-t)    dg(u) + o(n)
nr  t                         t

J_
n 1
n
dg(u)sin(u-t                     l.u-tl         I   sin n
(u-’t)
lu-tl
f dg(u)]-I
n -1                       n
t
[{       .n    coS,{u_tnlu-t), sin(u_t)
2n(u-t)                           dg(u)] du

Using (2.5) this is equal to
t
[    sin    n(u-t)                 {t X B       (t)}] t -I             o [                             tBx                          cos     n(u-t_ du]
(u--{-) o                                     n                               1
{n                (t)}                     (u-t)

t
n(u
f               (u.t) t)
sin                                 B (t)} du]
+    0       [                                              {t

n -I
o [ n t                      LB(t)           ]
Therefore

dg(t)               B(g)]
k=l
{Ok(X)             f’(x)}              o[n                             t
+    o(n)
n 1

o(n) [G(t)                         B(t)]
n -I
+
o(n)If-1 dg(t)B)dt]
n
G_                         B-l(t)
+   o(n)
J-1         t              B                                ’(t)} dt]

o(n)       +        o(n)                  f
n -1

/    o(n) [                         B     ).B(t) }.B-l(t)   X’(t) dt]
n -1
372                      K.N. MISHRA AND R. S. L. SRIVASTAVA

o(n) + o(n) [            1
g [[
d      :X2B( t)   dt]
n -1
o(n) by the hypothesis (2.7).
Since     B(t)    is monotonic, hence its differential coefficient is of constant sign.
Thus we get
n
]Ok(X      f’(x)l 2     o(n)
k=l
and therefore
n
k
7.
an, ]Ok(X          f’(x)}2         o(r,)
k=l
This completes the proof of the theorem.

4.      SPECIAL CASES.
By way of an application of our theorem, we take B                          1, x(t)     I/log (l/t)   and

an,k      1   then the following result follows, [4]"

THEOREM (Sharma).       At a point for which f’(x) exists and
1
G(t)       o[ t/log ] as t                o
then
n
k=l
Z    }Ok(X) f’(x){2            o(n loglog n)

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       Summability of Derived Fourier Series, Bull. Cal. Math. Soc.
6__1, (1969), 75-81.

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