Some remarks on coefficients of double Fourier series

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

T. M

(Poznań)

Some remarks on coefficients of double Fourier series

Let co{t) be a positive function defined in (0, 1>. An increasing sequence {%} of positive integers is called со-lacunary if it satisfies the two follow­

ing conditions (cf. [ 2 ])

oo

N

( 1 ) JF co(l/W'*) = 0{co(l{nN)}, ^ п ксоЩпк) = 0{nNco(l/nN)}

fc = JV f c = l

(JV = l , 2 , . . . ; w i > l ) . We say that the trigonometric series

OO

( 2 ) A amitn соs wii x соs % у

£,?■=l

is (co1; со2)-lacunarу if the sequences {m*} and {w?-} are oox- and a) 2 -lacunary, respectively.

Denote by Q(s,t) a positive function, defined and bounded for positive s , t < u /2 . Suppose that f ( x , y ) is 27i-periodic in each variable separately. Write

Г Г

The class of all functions / Lebesgue-integrable with pth power over the square Q = [ 0 , 2 тг; 0 , 2 тг], such that

{ / / l 4 ?*/(® , y)\pdxdy\',v < CQ(h, k)

for positive h , Тс <л:/ 2 , G being a constant depending only on /, will be denoted by Ap(Q). Analogously we define the class H ^ (Q ) of 27t- periodic functions f ( x , y) essentially bounded in the square Q and satis­

fying the inequality

esssupl у )I < CQ(h,lc).

Q

3 — P r a c e M a te m a ty c z n e X I (1967)

34 T. M a r k i e w i c z

Let co1, co2 be positive functions bounded in Q. If sup У)\ = ° (со 1 (Д)co 2 (fc)} as h , k - > 0

X,yeQ

we shall say that / belongs to the class Н(сог, a>2).

The symbols G,Ck (fc = 0 , 1 , 2 , . . . ) used below will denote some positive constants.

T

1. Let co1(t) and co2(t) be moduli of continuity satisfying (1).

Then the continuous function f ( x , y), with a double (wj, oo2)-lacunary Fourier series ( 2 ), belongs to the class H(co1, co2) if and only if its Fourier coeffi­

cients атьП. are of order 0 {co 1 (l/m i)co 2 (l/%)}.

Proof. S u f f i c i e n c y . It is easy to verify that, in virtue of (1),

Choose, for each h, k, 0 < h < 1 / т 1? 0 < Тс < 1 / щ , and integers M

= M(h)j N = N{k) such that l/m M+1 < h < 1 /mM, l / n N+1 < к < l / n N . We assumed that

F { x , y ) = A(f l 2kf { x —h , y - k ) has the Fourier series

OO

absolutely and uniformly convergent. Hence

T A$l,2kf(®-h,y-k) 4

г = 1

j=N+l

i= l j= 1

oo

N

00

i=MĄ~ 1 /=JV+1

Therefore

v

Applying (1) and the well-known inequalities ([4], p. I l l ) wi(^) ^

a)i(ti)

t2 tx

т 2 (У ^ 2

^2 ^1

and when h < t 2,

we have

$1 < C1hkmM(o1 1

mM n^a>2 ¹

% N

c m . со^ 11п^)

1/тм l / nN

^ „ , 7 «hW co2{k)

< C2h k — — — ---— = C2co1(h) co2(k ) ;

h к

Л1 oo

S l <C„* y j n < 0 , k ? ę ^ J - ± - )

Jj“ ' mi '

J

+ 1 \ nil l / mM \ nN+ll Ci (o1(h)co2(k).

Similarly, we obtain

$ 3 < С&о)г{Ь) ы2{к), $ 4 < Ceco1(h)co2(k) , and the proof of sufficiency is completed.

N e c e s s i t y . Observing that

4 ^ат.уП. = i- JJ A % .'njn. f ^ x - - ^ — , y ~ ~ ^ ) ^ ^ s i n y y d x d y and

^тг/т^тг/и.,-/j ж --- , у ---

2 m,- 2 m 7 11 2 т * /

2 \ 2 % we have

hence

dxdy,

It is easily seen that in case of necessity the assumption of lacunarity can be omitted. Theorem 1 is also true for general (cox, co2)-lacunary Fourier series of the form (15) of [3]. In case of one variable the result was obtained by Rubinstein ([2], Th. 2 . 2 ).

T heorem 2. Let 1 < p < 2 , l/p-j-l/q = 1. Suppose that f belongs to Л|^(12) and has the Fourier series

OO

«m.nCOS m x cosny m,n=o

(3)

36 T. M a r k i e w i c z

such that for some г, у > 0 , the numbers \am>n\m rn 11 are non-increasing in m, n ^ 1, separately. Then

am,n = 0 { ( - i - ) as m ^ l , n ^ l . l\ mn I \ 2m 2n / J

I n particular, if Q(s,t) = saf (a, /3 > 0) we have

(& rn. W . --

I ma+1/qn^+1,q 1

P ro o f. W rite F( x, y) = A ^ <tlkf ( x —h, y —k). The function F ( x , y ) has the Fo urie r series

4 ^ {0>m,n sin mh sin nk) sin mx sin n y . m,n=l

In view of Hausdorff-Young theorem ([5], p. 101)

OO X ^

j JV1 |4amnsinm^sin^|aJ < j— JJ ИйЬл/О» — h, y —k)\v dxdy\ .

B y the assumption,

{J/ y - m vdxdy)Vv < CQ(h, k),

Taking h = --- , к ^{TC TC} 2N л TC

0 <

<

2 , we get / 4 1

\q \

i \ l

and observing that sin0 > —

for

TC

w 1 1 \am>nmn\q < 11 ¹ 4 amn sin mh sin nkf

' ' m = [ M / 2 ] + l « .= [iV /2 ] + l m = l w = i

< CqQq

\ 2M 7 2N Furthe r,

M N '

{ ~ w w f 1 1

i = [ M / 2 ] + l n = [ V / 2 ] + l

Hence

. „ = о { ( - к Г 0 Н , Н }

l\JfY/ \ 2 M ' 2NJ)

A similar theorem can be proved for general Fourier series (see [3], 7.2;

for one variable see [1], p. 428).

T

3. Let f e H (2)(Q) and let (3) be convergent to f(oc,y) for all pairs (%, y). Suppose that for some r , p > 0 , am>nm~rn~11 are non-nega­

tive and non-increasing in m , n ^ l , separately. Then

Proof. By the assumption,

d$l, 2 kf(oc—2h, 2 / —27b) = 0{Q(h, Щ for all {x,y).

It is easy to check that

oo

A{$, 2 kf{®— 2 h, у —Щ = 16 E am>n (sin mh sin nJc)2 cos mx cos ny.

m ,n = l

Therefore,

C,Q(h, ft) » l 4 * U / ( 0 - 2 7 ( , 0 —2ft)j

M N

> 16 I E am> n (sin mh sin n Jc)2.

m = [ M / 2 ] + l

n=[NI2] + l If h — tu /2М, Jc = n/2N we have

^

M N

E E a**.»“ 2»2 <

*

m=[MI2] + l n=[Nl2]+l

Beasoning now as in the proof of Theorem 2 we obtain the desired con­

clusion.

The last theorem is a slight generalization of a result given in [3], 7.4.

7C TC \

~ Ш ’ 2 N /

R eferences

[1] А. А. К он ю ш к ов , О классах Липшица, Известия Акад. Наук СССР, 21 (1957), рр. 423-448.

[2] А. И. Р у б и н ш т е й н , Об со-лакунарных рядах и о функциях классов Н ю, Мат. Сб. 65 (107), 2 (1964), рр. 239-271.

[3] R. T a b e r sk i, On double integrals and Fourier series, Ann. Polon. Math.

15 (1964), pp. 97-115.

[4] А. Ф. Тиман, Теория приближения функций действительного перемен­

ного, Москва 1960.

[5] A. Z y g m u n d , Trigonometric series, I, I I , Cambridge 1959.

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK