O n the absolute summability \ C , r + i \ k oî the conjugate series of the r-th derived series of Fourier series

(1)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X IX (1976)

Y.

^{K . S}^{i n g h}

(Aligarh, India)

O n the absolute summability \ C , r + i \ k oî the conjugate series of the r-th derived series of Fourier series

1. Let £ a n be a given infinite series with sn as its n -th partial sum.

We write

(L, ==

_{A a}

Vi.

n k = о

a

> — 1 ,

n ~ x ~ 5 ] A * - k1cak‘*

n = 0

The series £ a n is said to be mmmable |G, a|, a > —1 if the sequence {a“} is of bounded variation, th a t is the infinite series

[2] and [4].

The series ^ an is said to be summable \G,

^{а | л ,}

a >

^{— 1 ,}

Jc

> 1 ,

if

(i.i) [3].

For ~k — 1 the summability j<7, a\k is the same as summability \G, a |.

However, for Jc> 1 the summability \C,l\k and summability \G, 1| are independent to each other [3]

By virtue of the well-known identity

£ lb [4] and [5],

condition (1.1) can be w ritten as (1.2)

Let f ( t ) e L ( — n, n) be periodic with period 2 tv , and let its Fourier series is given by

oo

f(t) ~ |c 0 + £ {ancosnt + bnsinnt)*

7 1 — 1 O O

=

n= о

(2)

Then th e conjugate series of Fourier series is

oo

£ (bncosnt — ansinnt) = £ Bn(t).

n = 1 n = l

We shall make use of the following notations:

P r(t) = У —h—, where *s arbitrary

г —0

9

t

**ЯЛ*)** = ^— J (t — u)a~1gr(u)du (a > 0), 0

gQ{ t)= g { t).

2. Eecently B hatt [1] proved th at the summability \C, r + l | of the r-th derived series of a Fourier series is not a local property of generating function and only by imposing some restriction upon the Fourier coef

ficients, the summability |<7, r + l | becomes a local property. Mazhar has generalized the results of B hatt [1]. Proceeding on the same lines, we present in this paper certain result for the case of conjugate series of the r-th derived series of Fourier series.

We establish the following theorems.

T

h e o r em

1. I f f(t) is a periodic function with period 2- k and integrable (L) over (0, 2л:), then summability \C, r + 1!5* of the conjugate series of the r-th derived series of the Fourier series of f(t) is not necessarily a local property of the generating function.

T

h e o r em

2 Lei f{t) satisfy the conditions of Theorem 1. Then the summability (0, r -f 1 |fc of the conjugate series of the r-th derived series of a Fourier series depends only on the behaviour of the generating function f{t) in the immediate neighbourhood of the point t — x, when

(

2

.

1

)

and when

J n ^{< oo,} i f r is even ;

\ M ® ) \ k

(

2

.

2

)

n if r is odd.

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Conjugate series of the r-th derived series

149

It may be remarked th a t our Theorems include, as a special case 7c = 1, the results of У. Singh [9].

3. We require the following lemmas.

L

emma

1 [3]. I f a series ]Ian is summable \C, a\k, a > 0, Jc > 1, then

^ i n lc- ak~ l \an\k < oo.

LE3IMA 2 [6]. I f

Z- j n + 1 then

an is summable \C, a + 11.*

L

emma

3 [8]. Let x(u) be of bounded variation for 0 < и < тс , and continuous with a?(+0) = 0, x'(u) is bounded and x r,(u) is integrable L \ then

J W (t)x(t) cos (n + $ + fyr)tdt

and 2rt

J Ф{t)x(t) cos (n-\-% + %r)tdt 0

i f r is even ;

= О A v{x)

(n — v)2 + 0{\An{œ)\), i f r is odd, where у denotes the summation over — oo < v < —1, 1 < v < w — 1 and (w + l ) < v < oo, A v(x) = A _ v(x) and Bv(x) == — B_v{x).

L

emma

4 [6]. I f

< O O ,

1,

then

У I I Y ' _ i ± ! _ \ <* oo J w \A-J (n ~ v ) 2]

4. P r o o f of T h e o r e m 1. Suppose r is even. The theorem will be

established if we prove th a t there exists such a function summable over

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150 Y. K. S i n g h

(a, fi) and equal zero in the remainder of (0, 2 тс) that the conjugate series of whose the r-th derived series of Fourier series is not summable |C, r + l | fc over (a, fi). By virtue of Lemma 1, it is sufficient to show th at there exists a function x(t) summable over (a, ft) and such th at

where A, A ' — const, whence it follows the divergence of the investigated • series ([7], p. 39, formula 1.320,1), and by Lemma 2 of Mazhar [6]. The case when r is odd can be treated similarly.

4. P r o o f of T h e o r e m 2. We shall establish the theorem when r is even, the proof for the odd values of r runs similarly.

Let ârn(x) be the n -th Cesàro mean of order r of the conjugate series of the r-th derived series of the Fourier series. Then we have

where K rn^ denotes the n-th Cesàro mean of order r of the series

S i n £17 + sin2£)7 + sin3£)7 + . . .

I t is known [10] th a t Now,

(5.1)

П

where

Thus '

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Conjugate series of the r-th derived series

151

where

A r = /w+r\ = j r + 1) (Г + 2) ... (r + n)

n \ r ! n\

and thus we have

n }/{r + 1) ’

and J.® = 1.

Let us now write

A r — A r

-Я-п- 1 — -4*

— A T

Г > —1.

U ( p , n , t )

=

£ A!e~M,

v —O

where /? is a non-negative integer.

Summation by parts gives

U ( p , n , t ) = { - A f i e - (n+1)if + u ( P - l , n , t ) }

(1

—

е- й )- 1 ,

and so,

u { p , n , t ) = + Uj t) ( 1 - е ~ й) - ^

3=1

Hence as it is easily seen f 2

JKim = ---- E e l — --- У ---~ ^ H--- ^ --- \ nV 7 A rn (2sin*/2

Z l / ( l - e - ^ (1

-e"**)r (2sint/2)

j

= E e ^ - f JV'a), say.

We observe th a t for some fixed number <5, 0 < 6 < t < тс, drN 1

Also,

d f

drN 2

d f ft=1

r—1

_ ^(n + i)< /7^

d r

d-» \ (1 - e~lt)* d f ( 2sin£/2*

tf{ n + V )r ei(n+i)t

1 VI

= ---- > 0(пП ,

A'„ ( l - e - ‘)'(2sin«/2) [Л — 1

0 ( n ) +* ir{n + W

^6i(n+h)t

(1 — e ?')r (2sini/2)

for fixed ô, 0 <

(6)

So we have

(

6

.

2

) dr - , [ir{n + W ex(n+i)t

— = 0 ( w ~ 1) + Be - Z --- ц--- d f nK ’ \ A rn (1 — e *) ( 2sint/2)

0{n {n + ^-Y cos(n + ± + ±r)t A rn (2sint/2)r+-

Now following B hatt [1] and using (5.1) and (5.2), we can write 2 r [dr - )

■<> - ~ J о » (i)r

2 (n + l Y r cos {n + ± + lr )4 t ' ( } (2sin£/2)r+1

n

0 ( n - x/ |^(«)| dt}

A r [ dr _ I

= --- ¥(){— K rJ t ) \ dt + n J , ' [ d f пУ

a

+ cos(w + £ + £r)<

Г— {'П+} ) f W(t)2&mtl2 1 a ' --7' dt* l -к A rn ) 1 ( 2 s i n ^ / 2 r 2 ]

|~J ÿ/(ÿ) 2sin</2cos(^ + j- + |y)< ш

+

2 (^ + j) r _____

A h I j КЧ* (2sin(5/2)r+2 ï / (t)cos(^H -| + ^*)^

J ^(2sint/2)

^{r + l}

-1 + с [ и_1/ |!Р(<)1<**]

t71 + dr2“b^r3_b ^

w say.

We observe th at, for positive ô, however, small, but fixed, the con

vergence of the series

i

depends only the behaviour oe f(t) in the immediate neighbourhood of the point t = x. Thus, it follows from Lemma 2 that, to prove the theorem, it is sufficient to show th at

V i_ : <^_ oo.

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Conjugate series o f the r-th derived series

153-

Let us now define a function

m -

2sint/2(2sinô/2)~r~2 (0<<<<5), (2sint/2)~r~1 ( ( 5 < « <

tc

).

Then, for 0 < t < 7Г, |(tf) is of bounded variation and continuous,.

!( +0) = 0 , i'(t) is bounded and £"(t) integrable (L). Therefore,

by Lemma 3. Therefore by virtue of Lemma 4 and hypothesis (2.1) the- result follows. Similarly we can prove the result when r is odd.

This completes the proof of Theorem 2.

The author wishes to express his thanks to the referee for his help- in its preparation.

[1] S. N. B h a tt, A n aspect of local property of the absolute summability of the r-tk- derived series of Fourier series, Indian J. Math. 9 (1967), p. 17-24.

[2] M. F ek ete, Zur Théorie der divergenten Beihen, Mathematikai es termeszettudo- manyi ertesito (Budapest) 29 (1911), p. 719-726.

[3] T. M. F le tt, On an extension of absolute summability and some theorems of Little- wood and Paley, Proc. London Math. Soc. 7 (3) (1957), p. 113-141.

[4] E. K o g b e tlia n tz , Sur les séries absolutement sommables par la méthods des moyennes arithmétiques, Bull. Sci. Math. (2) 49 (1925), p. 234-256.

[5] — Sommation des séries et intégrais divergentes par les moyennes arithmétiques et typiques, Memorial Sci. Math. No. 51 (1931).

[6] S. M. Mazhar, On the local property of summability \C, r + 1|& of the r-th derived series of Fourier series (1), J. Math. Pure Appl. 49 (1970), p. 99-107.

[7] J. M. R y zik and J. S. G-radstein, Tables of integrals, sums, series and deriva

tives (in Russian), Gittl. Moskow-Leningrad 1951.

[8] A. S axen a, D. Phil Thesis (1966), University of Allahabad, p. 125.

[9] V. Singh, A n aspect of local property of the absolute summability of the conjugate series of the r-th derived series of Fourier series, Bull. Acad. Polon. Sci. Sér. Math.

Astron. Phys. 17 (1969), p. 739-744.

[10] A. Z ygm und, Trigonometrical series, Warszawa-Lwôw 1935.

References

43, BRAHMAN PURI ALIGARH, U. P.

INDIA

O n the absolute summability \ C , r + i \ k oî the conjugate series of the r-th derived series of Fourier series

Y.

(Aligarh, India)

O n the absolute summability \ C , r + i \ k oî the conjugate series of the r-th derived series of Fourier series

1. Let £ a n be a given infinite series with sn as its n -th partial sum.

We write

(L, ==

Vi.

a

*n ~ x ~ 5 ] A * - k1cak‘

n = 0

The series £ a n is said to be mmmable |G, a|, a > —1 if the sequence {a“} is of bounded variation, th a t is the infinite series

[2] and [4].

The series ^ an is said to be summable \G,

a >

Jc

if

(i.i) [3].

For ~k — 1 the summability j<7, a\k is the same as summability \G, a |.

However, for Jc> 1 the summability \C,l\k and summability \G, 1| are independent to each other [3]

By virtue of the well-known identity

£ lb [4] and [5],

condition (1.1) can be w ritten as (1.2)

Let f ( t ) e L ( — n, n) be periodic with period 2 tv , and let its Fourier series is given by

f(t) ~ |c * 0 + £ {ancosnt + bnsinnt)

=

Then th e conjugate series of Fourier series is

oo

£ (bncosnt — ansinnt) = £ Bn(t).

We shall make use of the following notations:

P r(t) = У —h—, where *s arbitrary

9

ЯЛ*) = — J (t — u)a~1gr(u)du (a > 0), 0

gQ{ t)= g { t).

2. Eecently B hatt [1] proved th at the summability \C, r + l | of the r-th derived series of a Fourier series is not a local property of generating function and only by imposing some restriction upon the Fourier coef­

ficients, the summability |<7, r + l | becomes a local property. Mazhar has generalized the results of B hatt [1]. Proceeding on the same lines, we present in this paper certain result for the case of conjugate series of the r-th derived series of Fourier series.

We establish the following theorems.

T

1. I f f(t) is a periodic function with period 2- k and integrable (L) over (0, 2л:), then summability \C, r + 1!*5 of the conjugate series of the r-th derived series of the Fourier series of f(t) is not necessarily a local property of the generating function.

T

2 Lei f{t) satisfy the conditions of Theorem 1. Then the summability (0, r -f 1 |fc of the conjugate series of the r-th derived series of a Fourier series depends only on the behaviour of the generating function f{t) in the immediate neighbourhood of the point t — x, when

2

1

and when

J n < oo, i f r is even ;

2

2

n if r is odd.

Conjugate series of the r-th derived series

It may be remarked th a t our Theorems include, as a special case 7c = 1, the results of У. Singh [9].

3. We require the following lemmas.

L

1 [3]. I f a series ]Ian is summable \C, a\k, a > 0, Jc > 1, then

^ i n lc- ak~ l \an\k < oo.

LE3IMA 2 [6]. I f

Z- j n + 1 then

an is summable \C, a + 11*.

L

3 [8]. Let x(u) be of bounded variation for 0 < и < тс , and continuous with a?(+0) = 0, x'(u) is bounded and x r,(u) is integrable L \ then

J W (t)x(t) cos (n + $ + fyr)tdt

and 2rt

J Ф{t)x(t) cos (n-\-% + %r)tdt 0

i f r is even ;

= О A v{x)

(n — v)2 + 0{\An{œ)\), i f r is odd, where у denotes the summation over — oo < v < —1, 1 < v < w — 1 and (w + l ) < v < oo, A v(x) = A _ v(x) and Bv(x) == — B_v{x).

L

4 [6]. I f

1,

then

У I I Y ' _ i ± ! _ \ * < oo J w \A-J (n ~ v ) 2]

4. P r o o f of T h e o r e m 1. Suppose r is even. The theorem will be

established if we prove th a t there exists such a function summable over

(a, fi) and equal zero in the remainder of (0, 2 тс) that the conjugate series of whose the r-th derived series of Fourier series is not summable |C, r + l | fc over (a, fi). By virtue of Lemma 1, it is sufficient to show th at there exists a function x(t) summable over (a, ft) and such th at

where A, A ' — const, whence it follows the divergence of the investigated • series ([7], p. 39, formula 1.320,1), and by Lemma 2 of Mazhar [6]. The case when r is odd can be treated similarly.

4. P r o o f of T h e o r e m 2. We shall establish the theorem when r is even, the proof for the odd values of r runs similarly.

Let ârn(x) be the n -th Cesàro mean of order r of the conjugate series of the r-th derived series of the Fourier series. Then we have

where K rn^ denotes the n-th Cesàro mean of order r of the series

I t is known [10] th a t Now,

(5.1)

П

n ~ x ~ 5 ] A * - k1cak‘*

f(t) ~ |c 0 + £ {ancosnt + bnsinnt)*

**ЯЛ*)** = ^— J (t — u)a~1gr(u)du (a > 0), 0

2. Eecently B hatt [1] proved th at the summability \C, r + l | of the r-th derived series of a Fourier series is not a local property of generating function and only by imposing some restriction upon the Fourier coef

1. I f f(t) is a periodic function with period 2- k and integrable (L) over (0, 2л:), then summability \C, r + 1!5* of the conjugate series of the r-th derived series of the Fourier series of f(t) is not necessarily a local property of the generating function.

J n ^{< oo,} i f r is even ;

an is summable \C, a + 11.*

У I I Y ' _ i ± ! _ \ <* oo J w \A-J (n ~ v ) 2]

-Я-п- 1 — -4*

d-» \ (1 - e~lt)* d f ( 2sin£/2*

0 ( n ) +* ir{n + W

■<> - ~ J о » (i)r

= --- ¥(){— K rJ t ) \ dt + n J , ' [ d f пУ

Г— {'П+} ) f W(t)2&mtl2 1 a ' --7' dt* l -к A rn ) 1 ( 2 s i n ^ / 2 r 2 ]

A h I j КЧ* (2sin(5/2)r+2 ï / (t)cos(^H -| + ^*)^

J ^(2sint/2)

We observe th at, for positive ô, however, small, but fixed, the con