Norlund summability oî the derived series oî Fourier seriesсо

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ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I : PRACE MATEMATYCZNE X V I (1972)

B

amesii

C

handra

(India)

Norlund summability oî the derived series oî Fourier series

со

, 1. Let У an be a given infinite series with the sequence of partial

n — 0

sums {

8

n}. Let {pn} be a sequence of real constants and P n = Р о У Р г У • • '-\~Vni P - i = P - i = 0.

In the sequel it is assumed that for n ^ 0, P n Ф 0 and ->■ oo as П -> oo.

Let p(t) be continuous in (0, oo), linear in each interval {n ,n -\-1) {n = 0 , 1 , 2 , ...) and such that p (n) = p n for n = 0 , 1 , 2 , . . . Also we

U

write P(u) - J p{x)dx so that P{n) ^ P n as n -> oo.

о

The transformation

tП 1

P~n

n

r=0

defines the n-th {N ,p n) mean or the n-th. Norlund mean of the sequence

8

n.

If

lim tn =

8

,

n—>oo

the series ]? a n is said to be summable (N , p n) to the sum

8

.

2. Let f(t) be a continuous function of bounded variation, periodic with period 27c and integrable in ( — 7

t

, 7t). Let the function f(t) have a derivative f {pc) at the point t = x and let the Fourier Series associated with f(t) be

OO

(2.1 ) £a0+ ^ {anco&nt+ bn&mnt), W —1

where the constants an and bn are given by the usual Euler-Fourier for

mulae.

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The derived series of Fourier series is

CO

(

2

.

2

)

n (bn cos nt — an sin nt)

n = 1

and its allied series is

00

(2.3) JT1 n {an

cosr

£+ bn&innt).

In this paper we adopt the following notations:

g (t) = f( x + t)—f { x —t) —

2

tf(x ) Ji{t) = f { x + t ) + f ( x - t ) -

2

f{x ),

t

d ( t ) = / » ( « ) ! ,

0

t

X{t) = J \dh(u)\, о

3. The Nôrlund summability of a Fourier series and its allied series has been studied by a number of workers such as Iyengar [4], Siddiqi [9]r Pati [7], Singh [10], Bajagopal [8], Hirokawa and Kayashima [2], Hiro- kawa [1], M. Izumi and S. Izumi [5] and others. In this paper we establish the following two theorems on the Norlund Summability of the derived series of Fourier Series and its allied series.

Th e o r e m 1 . I f

as n^~

^{o o ,}

then the derived series o f Fourier series of f{x ) is summable (N , p n) to the sum f'(x ) at the point x.

Th e o r e m

2. I f p n is a positive sequence satisfying condition (3.1) (3.1)

as t - 0 and p n is a positive sequence such that* П

(3.2)

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and

t

X{t) = J \dh(u)\ = о о

as t-+ 0, then series (2.3) is summable { N ,p n) to the sum 1

4tz

f h(t) cosec2 \tdt at every point x at which this integral exists.

It is interesting to note that these theorems generalize earlier results due to Tripathi ([11], [12]) and Joshi [6].

The following lemma is required in the proof of Theorem 1:

L

emma

[5]. F or t ^ l / n ,

n n

^ p ksm {n-Tc + ±)t = 0(P {llt)) + 0 {llt)\ p [llt) + p n+ f \p'{u)\du}.

fc=o ljt

4. P ro o f of T heorem 1. Before proceeding with the proof of the theorem we note that (see [5]) hypothesis (3.2) of the theorem implies that

(4-1) npn — 0 (P n)

as n-> oo. Clearly if (4.1) is satisfied, then (3.1) gives

(4.2) G{t) = o{t)

as t - 0.*

We have

on(x) ^ v{bvc,osvx— av sinr#)

r = l

d П

= --- Г V * ( ^ c o s r ^ + ^ s i n r a ? )

dx L ^ J

V = 1

_ _ 1 _ Г , \ d sin (n + j) {oo— u) 2к J

_о

_J _{\ dx} s in l(æ—u)

Orr

d [ sin(w + 1 ) (x— u) du

du

1 2ти

du [ sin|(æ— u) U

rK d [ sin(w-f |)w)

/ ж \ s i n i — \du ’

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so that on integration by parts

sin (n 4- %)u

1

r sin(%4-l)M

= - x — I --- ---

d { f { x + u ) - f { x - u ) } .

2тг J

sinpt It is known that

and therefore

n

f sin (w+ l)u 2 тс

^J

о sin

^Ш

2

du,

{ x ) - f ( x ) = - i - f sm^ + 2^ , [Д{/(ж4-ц )—/(ж —ад)} —2/(а?)]Дц 2 тс J s in k sin \u

1 r sin(% + I )u

J

2

tc

J о sinA'W 2

dg{u).

In order to establish the theorem we have to show that

(4.3) lim

ft->O0

1

П 1

= o .

- n k^o J

Now

n

S <»-—/4®)) l _{Г d9(t)}

₁

2

ъ Р п

^J

sin U

\ *

0 2 A

1

Г %(*) /

2 n P n

J sin U \

0 2

1

_c

_d9(t) , '

2

n P n

J

sink 1

n

1

In 2

k = 0

k—0

/ JE Pk&in^n~1cJ^ =7i+Za

In 2 A=0

say.

Let ns first consider I r We have

(4.6) =

t

A = 0

1 In

=

0

{ n ) f \dg(t)\

= 0 ( n ) - o ^ - j = o(

1 ):

as n — >

^{с о ,}

by the application of (4.2).

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Making use of the lemma, we have (4.6) I 2 = 0 ( - —“ 1 ( Ш ^ т И

' n 1 L 1/» ' ' 1 In1 In

\dg(t)\

\dg{t)\ 1 1 , ,

~ ё ~ р

T l +

, _ f WWI , г №№1 r , „ , , , 1

+

Pn

J ^ Ь J ——— J |p (ад)|йм|

1 / n 1 In ljt **

« (М Я З Ч т К я г +

^' n ' L Mn x ' 1In

remembering that p 1 = 0 |P

(

t

)).

1 In

+ j\p'(u)\d«\

1 In

lit J

as t -> 0.

By integration by parts and hypothesis (3.1) we have

i - / J S S ! L p ( i

И 1 In '

1 P G(t)

l/n

1 f PD ⁱ

t

3

T i î If

j ± P dt -)- о

- , 2 \Pn / J* t2

l /И P I I \ n / 1/w

= o(l) + o - p(u)du = o (l), as % -» oo. Also

(4.8) Pn

P .

7T

l/n /

!%(*)!

*2

P. [>M I

¹^In

^{+ 2 —}

^P

П 1 In

f

m t

3

dt

= «(l)+o(fj j 4 - o(1)+o(^|=0(1), _г2

as n~> oo, by the application of (4.1).

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(4.9)

Finally

TC n

n 1 In lit

n ^7Г ÏI

J + ~^r / ~~p~dt j \P'(u)\du-

n *- lit -* 1/n n l/n l/<

7t

- i f flffl b'(i/<)l J2 <2

TC _ n

= j \v'iv)\du + o ^ ^ f f № ^ du + 1/тг

+ 0 \v'W ) I

P - / V

_l/n

<8

n / 1 \ П ”

o (l) + o ( - ^ - ) J J ll>'(w)l<ü*4-

l/rt и

+ o ( - H J u\p'{u)\du

' n ' l/n

= 0(l) + o |2>'(w)|fÏM = 0(1 ),

l П

— J l^'(w)|^ = 0(1),

if we show that

(4.10)

as n oo. We have

n

1 1 1

(4.11) — f\ f{u )\ du = — — f «|

j

)'(

m

)|Æ» + — f — f v)p'{v)\dv

lTM J J Ÿ u Г

Л tt

=оШ+оШ / ^ <г“

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and since the series £ 1 /^2 s convergent by virtue of a known result* (see Hobson [3], example 3 on p. 8) it follows that

(4.12) 1

A=1

as n -> oo. Combining the estimates in (4.11) and (4.12) we find that (4.10) is established.

Again combining the estimates in (4.6) through (4.9) we find that

(4.13)

1

2 = 0(1 )

as % —

oo.

Finally from (4.4), (4.5) and (4.13) we get that

£ Pk(<n-k—f ’№) = o (P J,* k

—0

as n

^oo

and this establishes (4.3). Hence the theorem.

P ro o f of T heorem 2. If we denote by an{x) the sum of the first n terms of the series (2.3) at a point t = x, then proceeding as in [11]

we have

1 r cos n %)t 2 тс

₁

Щ J sin И J

Hence in order to establish the theorem we have to show that 1

2nPn

_{1 In} _{k = 0}

- , + « ' = 0 (1)

as

7

i

—

>■

oo.

But this can be proved by exactly similar arguments as in the case of the proof of I

2

— o(l) as n

oo,

in the proof of Theorem 1 . This completes the proof of the theorem.

The author is deeply indebted to Dr. S. H. Lai for his valuable guidance and suggestions.

R e fe re n ce s

[1] H iro s h i H iro k a w a , On the Nôrlund summability of Fourier series and its conjugate series, Proc. Japan Acad. 44 (1968), p. 449-451.

[2] — and Ik u k o K a y a s h i m a , On the Nôrlund summability of the conjugate series of Fourier series, ibidem 44 (1968), p. 301-306.

[3] E. W. H ob son , The theory o f functions of areal variable and the theory o f Fourier's series, New York 1957.

7 — Roczniki PTM — P race Matematyczne XVI

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[4] К. В. К. I y e n g a r , А Tauberian theorem and its application to convergence of Fourier series, Proc. Indian Acad. Sci. Sec. A, 18 (1943), p. 81-87.

[5] M. Iz u m i and S. Iz u m i, Nôrlund summability of Fourier series, Pacific J . of Math. 26 (1968), p. 289-301.

[6] H .P . J o s h i , On the Nôrlund summability of the derived series of a Fourier series, J . of Sci. Res. 2, 16 (1965/66), p. 183-189.

[7] T. P a t i , A generalization of a theorem of Iyenger on the harmonic summability of Fourier series, Indian J . Math. 3 (1961), p. 85-90.

[8] С. T. R a j a g o p a l, On the Nôrlund summability of Fourier series, Proc. Cambridge Phil. Soc. 59 (1963), p. 47-53.

[9] J . A. Siddiqi, On the harmonic summability of Fourier series, Proc. Indian Acad. Sci. Sec. A, 28 (1948), p. 527-531.

[10] T. Singh, Nôrlund summability of Fourier series and its conjugate series, Ann.

Mat. Рига Appl. Ser. IV, 64 (1964), p. 123-132.

[11] L. M. T r i p a t h i , On the harmonic summability of the derived Fourier series and its conjugate series, Proc. Nat. Acad. Sci. India Sec. A, 33 (1963), p. 443-454.

[12] — On the Nôrlund summability of the derived Fourier series, ibidem 30 (1964), p. 548-555.

DEPARTM ENT OF MATHEMATICS FACULTY OF SCIENCE

BANARAS HINDU U N IV ER SITY VARANASI, INDIA

Norlund summability oî the derived series oî Fourier seriesсо

B

C

(India)

Norlund summability oî the derived series oî Fourier series

, 1. Let У an be a given infinite series with the sequence of partial

sums {

n}. Let {pn} be a sequence of real constants and P n = Р о У Р г У • • '-\~Vni P - i = P - i = 0.

In the sequel it is assumed that for n ^ 0, P n Ф 0 and ->■ oo as П -> oo.

Let p(t) be continuous in (0, oo), linear in each interval {n ,n -\-1) {n = 0 , 1 , 2 , ...) and such that p (n) = p n for n = 0 , 1 , 2 , . . . Also we

write P(u) - J p{x)dx so that P{n) ^ P n as n -> oo.

о

The transformation

tП 1

P~n

r=0

defines the n-th {N ,p n) mean or the n-th. Norlund mean of the sequence

n.

If

lim tn =

,

the series ]? a n is said to be summable (N , p n) to the sum

.

2. Let f(t) be a continuous function of bounded variation, periodic with period 27c and integrable in ( — 7

, 7t). Let the function f(t) have a derivative f {pc) at the point t = x and let the Fourier Series associated with f(t) be

(2.1 ) £a0+ ^ {anco&nt+ bn&mnt), W —1

where the constants an and bn are given by the usual Euler-Fourier for­

mulae.

The derived series of Fourier series is

2

2

n (bn cos nt — an sin nt)

and its allied series is

(2.3) JT1 n {an

£+ bn&innt).

In this paper we adopt the following notations:

g (t) = f( x + t)—f { x —t) —

tf(x ) Ji{t) = f { x + t ) + f ( x - t ) -

f{x ),

t

0

X{t) = J \dh(u)\, о

as n^~

then the derived series o f Fourier series of f{x ) is summable (N , p n) to the sum f'(x ) at the point x.

2. I f p n is a positive sequence satisfying condition (3.1) (3.1)

as t -* 0 and p n is a positive sequence such that П

(3.2)

and

t

X{t) = J \dh(u)\ = о о

as t-+ 0, then series (2.3) is summable { N ,p n) to the sum 1

f h(t) cosec2 \tdt at every point x at which this integral exists.

It is interesting to note that these theorems generalize earlier results due to Tripathi ([11], [12]) and Joshi [6].

The following lemma is required in the proof of Theorem 1:

L

[5]. F or t ^ l / n ,

^ p ksm {n-Tc + ±)t = 0(P {llt)) + 0 {llt)\ p [llt) + p n+ f \p'{u)\du}.

4. P ro o f of T heorem 1. Before proceeding with the proof of the theorem we note that (see [5]) hypothesis (3.2) of the theorem implies that

(4-1) npn — 0 (P n)

as n-> oo. Clearly if (4.1) is satisfied, then (3.1) gives

(4.2) G{t) = o{t)

as t -* 0.

We have

on(x) ^ v{bvc,osvx— av sinr#)

d П

dx L ^ J

_ _ 1 _ Г , \ d sin (n + j) {oo— u) 2к J

J \ dx s in l(æ—u)

d [ sin(w + 1 ) (x— u) du

du

1 2ти

du [ sin|(æ— u) U

rK d [ sin(w-f |)w)

/ ж \ s i n i — \du ’

so that on integration by parts

sin (n 4- %)u

r sin(%4-l)M

= - x — I --- ---

sinpt It is known that

and therefore

where the constants an and bn are given by the usual Euler-Fourier for

as t - 0 and p n is a positive sequence such that* П

as t - 0.*

_J _{\ dx} s in l(æ—u)

S <»-—/4®)) l _{Г d9(t)}

_d9(t) , '

1 f PD ⁱ

- , 2 \Pn / J* t2

^{+ 2 —}