THE SUMMABILITY , 1 OF FOURIER SERIES

BY Fu CHENG HSIANG

1. Let f(x) be a function integrable over (0, 2r). in the sense of Lebesgueand periodic with period 2r. Let

(1)

and write

.f(x) ao + (a,, cos nx + b, sin nx),

(2) ,I,(t) (u) du,

where (u) f(x + u) + f(x u) 2f(x). Hahn [1] proved that

(3) lim

is not sufficient for the series (1) to be summable (C, 1) at the point x thoughthe condition (3) implies the summability (C, 1 -t- 7) for every > 0. Prasad[3] improved this result as follows. The convergence of the integral

q,(u.__) du()u

is not sufficient for the summability (C, 1) of (1) at the point x.be noted that (4) is more stringent than(3), see [3].

It should

2. The aim of this note is to develop the above theorem in another direction.We prove the following

THEOREM. The convergence of the integral

(5) , (t) o(u___) duU1+

for a positive number is not sufficient for the Fourier series at the point x to besummable (C, 1 ), if > 7/(1 + 7). The condition implies however the sum-mability (C, 1) of the series at x.

We require a number of lemmas.

LEMMA 1. Let > O. If the integral

(u) du(6)U

Received September 26, 1945.

43

44 FI CHENG HSIANG

converges, then the necessary and su2cient condition for the existence of (5) is

(7) lim,I,(t)

t.-,o

Let 0 < di < t; the result is easily seen from the following equation

-" (I,(u) du.u- ’(u)du [u--’ (u)], + (1 + n) u-

The existence of ,(t) (7 > O)implies

(t)

LEMMA 2.

(8)

In fact, integration by parts gives

(t) u-’-" ,(u)u+" du .(t) (1 -1- r/) u"e,(u) du.

Then the result follows by observing (t) o(1) as -- 0.

LEMMA 3 [2]. Let r.(x) be the n-th Cesdro mean of order , of the Fourier series(1) at the point x, then

(9) a:(x) s ,(t)(t) dr,

where 2(t) 2(t) q- 12(t),

(10)

AndifO < v < l, then

(tl)

1 sin (n; t),(t)() sin

r(n + , + 1)I’(, -- 1) F(n -- 1)’

sin O; t) sin + +

(t)

_An,

(12)

(13)

d

I(t)

d

<_ An,1< An-"

<A+ a_.,n,t nt

SUMMABILITY (C, 1 {!) OF FOURIER SERIES 45

LEMMA 4.

the integral

If (I)(t) is an integral, then on writing

cos(n; t)= eos((n

(14) e(t)cos (n; t)._, dtsin

is equal to

2(0 < < 1),

(15) A (I)(t) cos nt sin nt

as n ---, where A and B are independent of n and bounded for 0 < < 1.

In fact, since the function

1 1h(t)

(2 sin)-’ ?-’

is of bounded variation in the interval (0, ), we have

(t)h(t) cos (n; t) dt 0

Then the result follows by setting

A cos 2-r, B=sin -3. We have

l fo )(sintnt)nr(t d 2__ (t)

sinnt--ntcsnt sinntdtn

2 fo ()(s’__m_n) i f )nr -t-- dtr

sin 2nt dt.

By Lemma 2, (t) o(t/’), afortiori, O(t) o(t). Hence the first term iso(1) s n -- by Fejr’s theorem. Further, (t)/t/ being bounded, (t)/tmust be integrable. By Riemnn-Lebesgue’s theorem, the second term is o(1).The Fourier series (1) is therefore (C, 1) und the second part of the theorem isproved.

4. We are in a position to prove the first part of the theorem.the integral of (9) by parts and using (13), we have

(16) r(a:(x) s) 0(n’-) O(t)’(t) dt.

Integrating

FU CHENG HSIANG

It follows from (12) that

(t) 2’(t) dt o(n dr) o(1)

by observing &(t) o(t). And by (13),

,b(t)ft,(t) dt 0 dt q- 0/n In nt In

’+(t) dt)nut

1 dr)q- O(f" 1

Combining these results, we have

a. (x) s-- o(1)-l- I,(17)

f ,b(t) _, dt

where

o(1).

It + I.

cos sin (n;t),b(t) dt

(2 sin )3-’

The integral 12 is numerically not greater than a constant multiple of

f (f," )__(t) dt o ("++)-2 dt1/n

3-+

Hence

(t8) t, o(n’-"+"’).

By Lemma 4, there are constants A and B such that

(19) I o(1) -4- nA ,(t)t’-’ cos nt dt q- nB e(t)t+- sin nt dt.

SUMMABILITY (C, [ e) OF FOURIER SERIES 47

5. To show that the series (1) is not summable (C, 1 e) for every> 7/(1 -b y), we may assume that e < 7, so that there is a rational number

p/q satisfying

11+ q

We have o prove thnt the condition of our theorem does not involve the sum-inability (C, 1 prt/q) of (t) at the point .

Choose the positive integer s such that s(p + q)e > 2. Write

K s(p -+- q), n. [,!1,-3/2

C,

and define

F(t) c. sin n,t

F(O) O.

The function F(t) is continuous in (0, -).

F(t) sin nt sin"ti ’ .k

dt c.tl

-lk

S+ S+ S.We shall show ha ’,S and S

In ghe firsg place, we have

1- cos 2n.t dt2S c.-, t_,

(rn-;/ < < --1/ 2, 3, -’)n._ ;u 1,

Letting p/q, we have then

dt -F -. c. f’n"-’sinn’tsinn"tdt_,ll-ek

n.t sin n,t dt

O(n;’).

,k

n, f’"ek (n,_ n;’) c, + ,r’ " cos 2n.t dt,

where t, is a value of the interval of integration, by the second theorem of mean.Hence

ek ek7l" eK 71" eK-3/2n.S, k ’ c, (1 q- o(1)) ’2-

48 FI CHENG HSIANG

Remembering K > 2, we obtain

(20) n: .Secondly, using the second theorem of mean,

-/

f"" 2 sin n.t sin n.t dt ’-

-1/kwhere n-

[cos (n. n.) + cos (n. -t-- n.)t] dr,

Hence the integral is equal to

Consequently, we have

n;S 0 n;-’ c. n’,,’-"

2KObserving (n./n,) < v- < 4-K, we see that the summation is less than

which is bounded for v.

(21)

Finally,

Hence

"" ’-’ dt 0 n-’)(n._ O(nT"

(22) n:S o(1).

Combining the results (20), (21), nd (22), we obtain

sin n,t(23) n: F(t) ’t_, dt= n:(S + S + $3)--+

The above analysis also establishes the following relation

-lilt

c, f"""-’ ’-’ cos n,t sin n,t dt O(n’).

SUMMABILITY (C, 1 e) OF FOURIER SERIES 49

Thus we have

(24)

by observing that

COS rtF(t) ,; dt 0(1),

sin 2n, dt o(l).

6. On the interval (0, r), if we can show that F(t)/t is integrable and thatt/F(t) is absolutely continuous, then Theorem 1 is established. In fact, underthese circumstances, if we put

t+"F(t) ,b(t),

then ’(t) o(t) exists almost everywhere, o(t) is integrable in Lebese sense,and t--"(t) tends to zero as 0. But since kt + , from the relations(16), (17), (18), (20), (21), and (22)it follows that

so that the corresponding Fourier series (1) is not summable (C, 1 ) t thepoint x, though the integrM (5) exists by Lemm 1.

-1/k7. On the interval rn/ < < ’n,_ we have

d (t) (1 + n)t"c, sin nt +(25) d

Hence--a/k

To f’,(t) dt < 3 c,n,,1

where

which is negative and therefore ](t) is integrable.$ < l, then since

Furthermore, if 0 < 6’ <

(t__) c, sin n,tI+

50 FU CHENG HSIANG

we have, on writing A (r/) *’,

f !t___)dt < c f.....

sinn’tdt

which tends to ero 5 O. Therefore the integral

--’ dt(t)

exists.Our theorem is thus completely proved.

BIBLIOGRAPHY

1. H. HAHN, Ueber Fejdr’s Summierwng der Fourierschen Reihe, Jahresbericht der DeutschenMathematiker-Vereinigung, vol. 25(1916), pp. 359-366.

2. G. H. HARDY AND J. E. LITTLEWOOD, Notes on the theory of series (VII)" On Young’s con-vergence criterion for Fourier series, Proceedings of the London Mathematical Society(2), vol. 28(1928-29), pp. 301-311.

3. B. N. PRASAD, On the summability (C, 1) of Fourier series, Mathematische Zeitschrift,vol. 40(1935-36), pp. 496-502.

NATIONAL UNIVERSITY OF CHEKIANG, MEITAN, KWEICHOW, CHINA.