The (L) summability of double Fourier series.

By TARKESttWAR SIbIGI~I (2~llahabad~ India)

Summary. - In this note the author applies the method of {L) summability to the double Fourier series to obtain a summability criterion for it.

1. -Recently , BORWE:H,T [1] has cons t ruc ted a new method of summabi l i t y of an inf in i te series E a , , whose n th par t ia l sum is s , . He def ined the ser ies

a , , or the sequence t s . }, to be summable by Loga r i t hmic method of summa- bil i ty, or summable (L}, to the sum s if, for r in the in te rva l (0, 1}

lira 1 ~ s , r n ~ 8 ~

which is s imply wri t ten as s , ~ s (L) .

W e want to apply this method of summabi l i t y to a double series Eam,~,

whose (m, n)th par t i a l sum is s~ ,n . The series Ea,~,~, or the sequence ism,~t, is said to be summ~ble by L o g a r i t h m i c method of summabi l i ty , or summab le (L), to t h e sum s, if for q, r in the in te rva l (0, 1)

l im 1 ~ ~E s,~,,q"r n - - s , l l o g ( 1 - - q ) l . l l og (1 - - r ) l ,~=~,=~ m n

as q and r tend to 1 - - 0 .

2. - Le t f ( x , y) be a periodic func t ion with per iod 27: in each var iab le and in tegrab le in the sense of LEBESOUE over the square S(--7: , u : --rc,~). Le t the series

~ ) ~ 0 ~ 0

kin,. (a,~,, cos mx cos n y + b,~,, sin mx cos n y +

c,,,~ cos m x sin n y ~ d,, ,~ sin mx sin ny) ,

Annal~ d~ Matematica 17

130 T. SI~GE: The (L) summability o] double Fourier series

be called t h e double FouRI]~R series of the function given by

(2.2) k. , . . - -

-1 for m

1 for m

_1 f o r m > 0 , n > 0 ,

- - 0 , n = 0

- - 0 , n > 0 ; m > 0 , n : - 0

f(x, y), where ~m, n a r e

and the coefficients in (2.1) are given by

(2.3t a~,,~ = ~ f(u, v) cos mu cos nv du dv

and three similar expressions defining b,~,n, c,~,, and d~,,,~. Let us write

(2.4) ~ (u, v) = [f(x + u, y + v) + f(x + u, y - - v) + f(x - u, y + v)

-t- f ( ~ c - u, y - v) - - 4 f ( x , yi]

L e t f(x,) be a periodic function with period 2r: and integrable in the sense of LEBESGUE over (--7:, g) then :its FOURIER series is

{2.5) ~ ao + E (an cos nx -~ b. sin nx),

and we also write

(.9.6) ¢(u) = [f(x + u) + f ( x - u ) - 2 f ( x ) ] .

3 - In a recent note, HSAI=~G [2] applied the method of (L) summabili ty to FOCRIER series of f(x) in order to obtain the corresponding summabili ty criterion for it, his theorems are given below:

T~EORE~[ A. [2] - The necessary and sufficient condition for the FouRIEn series of f(t) to be summable (L}, at t - - x , to f(x), is

(3.1) f ~} t ) tan_ 1 0

l r s i n t ~ i l o g ( I - - r ) [ ] 1 - - r c o s / I d t ' - ° [ J

as r ~ l - - 0 .

T. SINGH: The (L) summabiti ty of ,double Fourier scric,~ 131

T~]~onE~ B [2] - I f

(3..~) t

f ~ I(u)i 0

du = o It i ~og (it [],

and

(3.3) f ] f~ (u) l du = o [[ log (t) l ] U

t

as t ~ -{- 0, for any a rb i t r a ry 8(0 < ~ < n), then the FOURIER series of f(t), at t - x, is s u m m a b l e (L) to f(x).

In a note ) / [OHA~¥ and NA~DA [3] po in ted out that the condi t ion 13.3) impl ies (3.2) and fu r the r improv ing upon the T h e o r e m B, NA~DA proved the fol lowing Theorem.

THEORE~ C. [4] - I f

7~

f ot r [lo 11 t

as t ~ 0, then the FOURIER ser ies of f(t), at t = w, is s u m m a b l e (L) to /{x).

4:. - In this note we app ly the m e t h o d of {L) summab i l i t y to the doub le FOURIER ser ies to obta in a co r r e spond ing summab i l i t y cr i ter ion. W e prove the fo l lowing T h e o r e m which ex t ends the T h e o r e m C for the case of double FOURIER series.

T I t E O R E M . - If

(4.1) O{s, t) - - ¢t (u, vt du dv U V

s t

i 1 1] = o l o g s l o g - ,

as s ~ ~ 0, and t ~ ~ 0, then the double Fourier series (2.1) is summable (L) to f (x , y}.

132 T. SIN(;~r: Th(" (L) sttJJ~mability of d',mblc k'o,wier .wries

5. PROOF OF THE THEOREM - L e t s.~,. d e n o t e the ~m, n ) t h of the s e r i e s (2.1}, t hen we k n o w tha t

s~, , , - flx, y) = ~ ., ¢ (u, v} 1 1 o o sin 2 u s i n ~ v

} 1/ = u--z- ~(u, v) s i n m u s i n n v d u d v - ~ o{ t ) . U V

o o

T h u s

~ qmr~ . . ~ { s.,..~ - - f(x, y) }

m ~ l n ~ l m T~

f j ~ ° ° ~ m s i n m u ~ r t l s i n n v 1 ~ (u, v) v, ~ du dv 7~ 2 1~ V m ~ l ~ n = i Tt

o 0

-[- o(I log (1 - - q) l" l l ° g ( 1 - - r) l )

---- ~1 i f ~(u ' v) T(q,u) T(r,v) ~ u v O 0

+ ° ( i l ° g ( 1 - - q ) l ' l l ° g ( i - r ) l ) ,

= I + o( l log (1 - q ) [ . I log (1 - - r) l ) ,

w h e r e

t q s i n u t T(q, u) - - t an -1 1 - - q COS U I"

T h u s it s u f f i c e s for o u r pu rpose , to show tha t

I - - - - - - 1 / / ¢ ~ m , 7~ 2 U V

o o

v} T(q, u) T(r, vj du dv

: o( I log (1 - - q) t" I log (1 - - r) I ) ,

p a r t i a l s u m

du dv

as q ~ 1 - - 0 a n d r ~ 1 - - O .

T. SI.x~It: The (L) .~ttmm(Ibility of double Fot~xier series 133

Now, let ~ - - 1 - - q and ~ q - - l - - r , and let us set

< - , = ( / / + / s + / / + / / ) ....... o o o ~ C o ~

¢(u, v) U V

Ttq, u) T(r, v) du dv

=h+l~+l~+h.

We have the following est imates

(52) I T(q, u) -- O(u/~), for 0 ~ u ~ ~,

T(q, ~) -- 0 and T(q, ~) -~ 0(1) ,

un i formly for 0 ~ u <: ~: a n d 0 < ' q < 1 .

Also

T(q, u) du -- T' (q, u)

q(cos u - - q) -- 1__ 2 qcos u.4_ q2

(5.3) _ [ 0 1 1 / ( 1 - - q ) ] - - O(1/~), for u ~

[ 0[(1 - - q ) / u '~] -- O(~/~t"), for u > ~.

Since

(5.4) ~ (u, v) d' ~ (u, v) u v - - d u d v

almost everywhere therefore, by (4.[), (5.2) and (5.3), we have

0 0

d2~(u, v) T(q, u) T(r, v) du dv du dv

[ ¢ dT(q, u ) /(I)(u, v) T(q, u) T(r, v ) - | ap(u, v)

- - L J d U

- - f O(u, vl T(q, u) dT(r,dv v) dv

T(r, v) du

t34 T. SINGI~: The (L) s ummabiI i ty o] double Fou~'ier series

[l x ~ ~ l ~ l = o l O g u l O g - - . u v V 0,0 J

+ o log u log v ~-~ d u o,o

+ o log ~ log T ~-~ ev 1 0~0 J

[//11] log u log - -

+ o v du dv

0 0

1 =o[,o. ~,oo~], since

f log 1 d u = ~ log e

o

Again by (5.4), we have

y f ~oI~v, I~ - - du dv T(q, u) T(r~ v) d u dv

o

-- [~(u, v) Tlq, u) £ dTiq , T(r, v) - - | O(u, v) T(r , v) du

du J

£ dT(r, V) - - / ¢(u, v) T(q, u) dv

dv 2

Jf ~ l -{- ¢(u, v) r ( q , u) T(r, v) du dv ~" ~ du dv o, .~

÷ I

tl I

0

H

r

11

÷ Cf~

Jr

~r~

I

I m~

II

~r~,

I ~

~ I,,,

~o

~

0 CJ~

7l.

r ...

. ~

Jr

~1 ~

cr~

Ii 0 E~Q

L +

+

.o

~o

136 T. SINGII: Th, e (L) summabil i ty of double Fourier series

o log u - l ° g v ] T ( q ,u ) mod ITtr , v)[ t.~,~

75

d- o log - - log --'u ~ d u

-t- o tog-~- l o g - - . v - ~ dv

7~ 7~

1 1 ( l o g ~ - l o g - ) .

This comple tes the peoof of the Theorem.

The au thor t akes this oppor tun i ty of express ing his warmes t thanks to P ro fe s so r B. N. PaASAD for his va luab le advice and gu idance in the prepa . ra t ion of this note.

R E F E R E N C E S

[1] D. BORWEIS, A Logarithmic method of summal)ility, (~gonr. London Math. Soc.~, Vol. ~3 (195S), pp. 2 ~ ~0.

[2] F, C. ]=[SA1NO, Summability (L) of l~ou~'ier series, cButl. American Math. Soc.,, Vol. 67 (1961). pp. 150-153.

[3] R. MOHANTY and .,~. ~ANDA, The summability (L) of the differentiated Fourier series, CQuart. Jour. Math. (Oxford)>>, Vol. 13 (1962), pp. 40.44.

[4] M. ~ANDA, The summability (L) of Fourier series and first differentiated Fourier series, • Quart. J-our. Math. (Oxford)~, Vo]. 13 (196-°I, pp. 229-234~.