On Absolutely Convergent Fourier Series
C. W. O~NEWEER
1. Introduction
Let T denote the circle group and let Z denote the group of integers. We shall consider functions f which are integrable on T and we shall denote their Fourier coefficients by f(n), where n E Z. F o r fl > 0 the set of all f E L I ( T ) such t h a t
~=_oo I](n)1 ~ < ~ will be denoted by A(fi). Among the classical results in t h e theory of absolutely convergent Fourier series are the following theorems [6, Vol. 1, Chapter VI, 3].
T~EOREM 1 (Bernstein). I f f e Lip ~ for some a > 89 then f c A(1).
THEOREM 2 (Zygmund). I f f is of bounded variation on T ( f C BV) and i f f e L i p a for some a > 0 , then f e A ( 1 ) .
Attempts to generalize these theorems have led to the following.
THEOREM 1A (Szs I f f E Lip ~ for some ~ with 0 < o~ ~_ l, then f C A(fl}
for all fl such that fl > 2/(2a q- 1).
THEOREM 1B (Hardy). I f f E L i p ~ for some ~ with 0 < o ~ < 1 , then.
~o<l,i<r162 < oo for all fl such that fl > (1 -- 2a)/2.
Definition 1. Let f be a function defined on T a n d for r ~ 1, let
n - - 1
v,Efl = sap( _5o -
where the supremum is taken over all finite partitions 0 ~ x0 < xl < . . . < x, < 2~v of T. The function f is of r-bounded variation
(re
r-BV) if Vr[f] < oo,52 C. W. O N N E W E E E
THEOREM 2A ( H i r s c h m a n [2]).
and i f f E Lip s for some ~ > 0 ,
I f f E r - B V for some then f E A(1).
r with 1 < r < 2,
I t is well-known t h a t each of t h e foregoing t h e o r e m s is t h e best possible in a certain sense [6, Vol. 1]. I n [4, Exercise 1.6.6] K a t z n e l s o n gave a new a n d v e r y simple example of a f u n c t i o n in Lip 89 t h a t does n o t belong to A(1). I n t h e re- m a i n d e r of this section we shall give a simple extension of K a t z n e l s o n ' s e x a m p l e which can be used to show t h a t all t h e previous theorems are sharp. W e first give t h e definition of t h e so-called R u d i n - S h a p i r o polynomials P , ( x ) a n d Q,(x). L e t
P o ( x ) = Q o ( x ) =
1, and for m > _ 0 ,Pm+I(X) = Pm(X) ~- e~2"%Qm(x)
Next, let f~+l(X) : P m + l ( X ) - - Pro(x)let
a n d Qm+x(x) = Pro(X) -- ei2mxQm(x).
a n d for each ~ w i t h 0 < ~ < 1, let
g~(x) = ~ 2-k(~+~)fk(x).
k = l
I t follows i m m e d i a t e l y from t h e definition of g~ t h a t g~(n) = 0 if n _< 0 a n d t h a t O~(n) = e ( n ) 2 -k(~+89 if 2 k - x _ < n < 2 k for some k > l a n d w i t h e ( n ) = - b l .
A p r o o f similar to t h e one given b y K a t z n e l s o n for t h e case ~ = 89 yields t h e following.
THEOREM 3. For each ~ with 0 < ~ < 1 we have (i) g~ E Lip ~ and g~ ~ Lip y for any y > ~, (ii) gc~ E 0r
(ill) g~ ~ A(2/(2~ ~- 1)), (iv) ~ 2 = i n ('~ = ~ .
2. Convolution functions
T h r o u g h o u t this section we shall denote the conjugate of a n u m b e r p > 1 b y q, t h a t is, l i p -~ 1/q = 1. F o r f , g E L I ( T ) t h e convolution f 9 g is defined b y
( f * g ) ( x ) =
f f(x
- t ) g ( t ) d t .T
T h e n ( f 9 g) ^ (n) = ](n)~(n) for all n E Z. The following t h e o r e m is due to M. Riesz (6, Vol. 1, page 251].
THEORE~ 4. A continuous function f has an absolutely convergent Fourier series
~if and only i f there exist functions g, h
E L2(T )
such thatf
= g 9 h.ON ABSOLUTELY CONVERGENT FOURIER SERIES 53 T h e next t h e o r e m gives a partial extension of this result.
THEOREM 4A. I f g, h E Lp(T) for some p with 1 < p ~ 2, then g 9 h E A ( p / ( 2 p -- 2)).
Proof. I t follows from Y o u n g ' s i n e q u a l i t y a n d t h e H a u s d o r f f - Y o u n g inequality t h a t
t h a t is, g 9 h E A(q/2) = A ( p / ( 2 p -- 2)).
W e n e x t show t h a t T h e o r e m 4A is sharp.
THEOREM 5. For every p with 1 < p ~< 2 there exist functions g, h E L~,(T) such that g * h ~ A(fi) for any fi < p/(2p -- 2).
Proof. W e define the functions g and h b y
{ ~ l / ' l o g n ) - i if n > 1 ,
~(n) = h(n) = if n < 1.
Clearly, ~(n) ~ 0 as n--> ~ a n d
(~(n)) pnp-2 ~ ~ np-2-P/q(log n) - p : ~ n - l ( l o g n ) - p < ( ~ ,
n=2 n=2 n--2
because p > 1. A t h e o r e m due to H a r d y a n d L i t t l e w o o d [6, Vol. 2, page 129] im- plies t h a t g, and hence also h, belongs to L e ( T ). F u r t h e r m o r e , if fl < p/(2p -- 2) t h e n
((g , h)~ (n)[~ = ~ (nl/qlog n)-2~ = oo,
r t ~ -- oo n ~ 2
because 2fl/q < 1. Therefore, g 9 h ~ A(fl).
T~EOREM 6. I f g E Lp(T) with l < p ~__ 2 and i f h E Lip ~ with 0 < o~ < 1, then g , h C A(fi) for all fl such that 2p/(2ap Jr 319-- 2 ) < 8"
Proof. F i r s t choose fi such t h a t 2p/(2~p + 3p -- 2) < fi < q. T h e n Y o u n g ' s i n e q u a l i t y implies t h a t
. . . . - - q . . . . q . . . .
54: C. W . O N ~ E W E E R
Since fi > 2p/(2~p ~- 3p - - 2), we have flq/(q -- fl) > 2/(2~ ~- 1). Hence, The- orem 1A implies t h a t B is finite. Also, t h e H a u s d o r f f - Y o u n g i n e q u a l i t y implies t h a t A is finite. Therefore, g 9 h E A(fi).
Choosing fi = 1 in Theorem 6 we o b t a i n t h e following corollary. I t shows h o w we can ameliorate functions in Lip ~ w i t h 0 < ~ < ~ _ ~, which are n o t necessarily in A(1), into functions in A(1) b y m e a n s of the convolution operator.
COROLLAI~Y l. Let g E Lp(T), 1 < p _~ 2, and h C Lip ~, 0 < a < 89 I f ( 2 a ~ - 1)p > 2 , then g . h E A ( 1 ) .
W e n o w show to w h a t e x t e n t T h e o r e m 6 a n d Corollary 1 are the best possible.
THEOREM 7. Let p and ~ satisfy the conditions 1 < p < 2 and 0 < ~ < l i p . Then, (i) for all ~ with 0 < ~ < ~ there exist functions g and h with g E L~(T) and h E L i p ~ l and such that g , h ~ A ( 2 p / ( 2 ~ p - ~ 3 p - - 2)), (ii) for all p~ with i < P l < P there exist functions g and h with g E L ~ ( T ) and h E L i p a and such that g 9 h ~ A ( 2 p / ( 2 a p -~ 3p - - 2)).
Proof. (i) I f y is d e f i n e d b y y = ~ + l/q, t h e n 2/(2y + 1) = 2p/(2ap + 3p - - 2).
L e t h ~ g~l, t h e n , according to Theorem 3(i), h C Lip ~1. L e t g be d e f i n e d b y g(n) ~ {~ -k(~-~') if 2 k - l ~ n < k n _ ~ 0 . ~ l 2k f ~ s~ ' i f
T h e n ~ ( n ) ~ 0 as n--~ ~ a n d
k ~ l k = 0
because 1 - - ( y - - ~ l ) p J r P - - 2 ~ - ( ~ l - - a ) P < 0. Thus, g E L e ( T ) . F u r t h e r - more, i f n E Z a n d if 2 k - ~ n < 2 k for some k > l , t h e n
g(n)h(n) = 2-k(Y-~)s(n)2- k(~+89 ~- s(n)2- k(:, +89 : ~(n),
t h a t is, g 9 h = g~. Since, according to T h e o r e m 3(iii), gr ~ A ( 2 p / ( 2 a p ~- 3p - - 2)), we h a v e established (i).
(ii) The p r o o f of (ii) is similar to t h e p r o o f of (i). I n this case t h e functions g a n d h are chosen as follows. L e t h = g~ a n d let ~(n) = 0 if n < 0 a n d let ~(n) = 2 -~(y-~) if 2 k-~ < n < 2 k for some k > 0 a n d w i t h y - ~ a - ~ - 1/q. T h e n it is clear t h a t the functions g a n d h satisfy t h e conditions m e n t i o n e d in (ii).
R e m a r k 1. The following case of Theorem 7 is of special interest. F o r each p such t h a t l < p < 2 a n d each ~ such t h a t 0 < ~ < ( 2 - - p ) / 2 p there exist functions g a n d h w i t h g E Lp(T), h E L i p s a n d g . h ~A(1). This improves
O N A B S O L U T E L Y C O N V E R G E N T F O U R I E R S E R I E S 55 a r e s u l t of M. a n d S. I z u m i [3, T h e o r e m 3] w h o p r o v e d for e a c h p w i t h 1 < p < 2 a n d e a c h s w i t h s > 2 t h e e x i s t e n c e o f f u n c t i o n s g in Lp(T) a n d h in Ls(T) such t h a t g . h S A ( 1 ) .
3. Multipliers of type
(ll,(Zn), Ip(Z"))
I n this section we shall d e f i n e a collection o f f u n c t i o n s on t h e n - d i m e n s i o n a l t o r u s T". W e shall use t h e s e f u n c t i o n s t o show t h a t c e r t a i n results of H a h n [1]
a b o u t p - m u l t i p l i e r s on T ~ are t h e best possible. F u r t h e r m o r e , for n ---- 1 t h e s e new f u n c t i o n s will be t h e same as t h e f u n c t i o n s g~ w h i c h were d e f i n e d in S e c t i o n 1.
T h r o u g h o u t this section we shall use t h e n o t a t i o n x = (x 0, x l , . . . , xn_l) for x in T " a n d m = (mo, m 1 . . . . , m , _ l ) for m in Z n.
Definition 2 [2]. A b o u n d e d a n d m e a s u r a b l e f u n c t i o n f d e f i n e d on T n is a p - m u l t i p l i e r , 1 _< p _< 0% if for e v e r y f u n c t i o n F in /p(Zn), t h e f u n c t i o n T ( f ) F is a g a i n in / / Z ' ) , w h e r e T ( f ) F is d e f i n e d b y
T ( f ) F ( m ) ---- ~, F ( m -- k)](k).
k E Z a
T h e set o f 19-multipliers will be d e n o t e d b y M~.
Definition 3 [1, page 327]. L e t cr be a positive real n u m b e r a n d let a , be t h e largest i n t e g e r less t h a n a. F o r 1 < p _< oe, L i p (~, p) is t h e class o f all func- tions f d e f i n e d on T " such t h a t for Ikl < ~ . we h a v e (O/ax)kf E L p ( T n) a n d for
I/el ~ ~ , we h a v e
/ O \ k f t
w h e r e for each h a n d x in T " we set
Obviously, if P l >--P2, t h e n L i p ( ~ , p l ) C Lip (~,p2);
L i p ( ~ , oo) c L i p ( ~ , p ) for all p > _ 1 a n d for all ~ > 0 . following [1, T h e o r e m s 12' a n d 20].
= O(Ihl ~-~*) if ~ - - a , < 1,
= O(Ih[) if ~ - - ~ , = 1, Ahf(x) = f ( x 4- h) -- f(x).
so, in p a r t i c u l a r , H a h n p r o v e d t h e
THEOREM 8.
(a) I f 1 <19 < 2 and a > n/p, then Lip(o~,p) C M r for 1 < r ~ oo.
(b) / f p > 2 and a > nip then L i p (a, 19) c M r for 2t)/(t 9 + 2) < r <
<_ 2p/(p 2).
(c) / f n i p < a <__ n/2, then Lip(c~,p) c M r for 2n/(n 4- 2 ~ ) < r <
< 2n/(n - - 2a).
5 6 C. W . O N N t g W E E I ~
W e shall p r o v e t h a t t h e s e results are s h a r p in t h e sense t h a t for p >_ 2 we c a n n o t replace ~ > nip b y cr >_ nip in T h e o r e m 8(a) a n d (b), w h e r e a s t h e con- clusion o f T h e o r e m 8(c) does n o t hold for r = 2n/(n ~- 2a) or r = 2n/(n - - 2~).
W e do n o t k n o w w h e t h e r t h e conclusion o f T h e o r e m 8(a) holds if 1 < p < 2 a n d ~ : nip.
THEOREM 9.
(a) I f p > 2 and i,f ~ = n/p, then L i p ( ~ , oo) dgM2p/(p+2);
L i p (hi2, oo) dg M 1.
(b) I f 0 < a < n/2, then Lip (c~, m) ~ M2~/(,+2~).
in particular,
I n o r d e r to p r o v e T h e o r e m 9 we f i r s t define f u n c t i o n s h~(x) for each c~ > 0.
F o r c o n v e n i e n c e we shall w r i t e ~ for 2 ~ a n d r for e x p (2~i/~). F o r i = 0, 1, 2 , . . . a n d l = 0, 1 . . . ~ - - 1 we d e f i n e t h e t r i g o n o m e t r i c p o l y n o m i a l s P~t(x) i n d u c t i v e l y . L e t Poo(x) . . . Po~_l(X) = 1 for all x E T ". N e x t , a s s u m e t h a t t h e p o l y n o m i a l s Pkt(x) h a v e been d e f i n e d for some k >_ 0 a n d all l w i t h 0 <
_< l < ~. E a c h j w i t h 0 < j < fi has a u n i q u e r e p r e s e n t a t i o n o f t h e f o r m J = Jo q- 2jl -}- . . . Jr- 2~-1J~-~,
w i t h j, e {0, 1}. L e t j = (Jo . . . . , j , _ ~ ) E Z" a n d let ] - x =-joxo q- . . . q- j,_~X,_x.
N e x t , for l w i t h 0 _ < I < ~ we define Pk+xl(X) b y
~ - 1
P k + l / ( X ) = ~
(o~eii"x2kpkj(x).
j = o
Since
we h a v e for a r b i t r a r y c o m p l e x n u m b e r s
E I E
t = o 1 = o
T h e r e f o r e ,
2_1 ~-1 ~-1
=-~ eo~ = {2 i f / = 0 '
j i f l = 1 , 2 , . . . , ~ - - 1,
CO , 9 9 9 , Cn--1
= ~ Y. Icjl ~.
j:O
,.#,~q.~,zk-lp i,,~]2 = ~ ~ ip~_lj(X)[2 = ~k+l.
E IPk,(x)l = = E [ E ~=~ - - k - - l j ~ . ~ , I
~=0 t = 0 j = 0 1 = 0
H e n c e , for each k > 0 we h a v e
lipko(x)ll ~ < ~(k§
Also, I/3ko(m)l = 1 i f m = ( m o . . . ran_l) w i t h 0 _ _ < m i < 2 k for i = 0 , 1 . . . . , n - - 1, a n d Pko(m) = 0 otherwise. F o r k > 1 let fk(x) = Pko(X) - - Pk_io(x), a n d for ~ > 0 let
O ~ A B S O L U T E L Y C O N V E R G E N T F O U R I E R S E R I E S 57
k ~ l
W e can show t h a t h a E L i p (~, ~ ) . T h e p r o o f requires a long a n d t e d i o u s com- p u t a t i o n w h i c h we shall omit. W e o n l y o b s e r v e t h a t we n e e d a n n - d i m e n s i o n a l v e r s i o n o f B e r n s t c i n ' s i n e q u a l i t y : if f is a t r i g o n o m e t r i c p o l y n o m i a l o n T ~ o i degree k, t h a t is,
f(x) = ~ cieq'~,
j E Z n
w i t h m a x / ( I j01 + Ijll + 9 9 9 + I j , - l l ) =
k,
t h e n for each of t h e first o r d e r p a r t i a l d e r i v a t i v e s o f f we h a v eof <kllfll .
Proof of Theorem
9. (a) Consider t h e f u n c t i o n F w h i c h is d e f i n e d on Z" b y F ( 0 ) ~ - - F ( 0 . . . . , 0 ) ~ - 1 a n dF ( m ) =
0 for m # 0 a n d m C Z " . Clearly,F E lr(Zn).
W e shall p r o v e t h a th,/p ~M2v/(p+2 ).
F o r each m E Z = we h a v eT(h,/p)F(m) = ~ F ( m -- k)f~,/e(k) = h,/~(m).
kC~ Z n
Also,
i~,/v(k) 12p/(p+2) = ~ (2k, __ 2(k-1),)2 ,k(p+2)/2v.
:p/(v+2)
k E Z n k = l
8 9 2k"2 -k" = ~ ,
k = l
t h a t is,
T(h~/p)F
( ~ 1 2 p / ( p + 2 ) . T h e r e f o r e ,hn/p ~ M2p/(p+2).
Since
M2p/(v+2)
= M2e/(v-2) we also h a v ehn/~
C M2v/(p_2).(b) F o r each ~ such t h a t 0 < ~ < n / 2 we h a v e
oo
i/~,(k) 12n/(n+2~) > 89 ~ 2kn2--,,k(2~+,)/2n. 2n/(Za+n) = ~ .
k E Z n k = l
T h e r e f o r e , an a r g u m e n t as in (a) shows t h a t
h a ~ M2n/(n+2a),
a n d hence also,ha f~ M2n/(n
2~).Remark
2. F o r each n t h e f u n c t i o nhn/2
p r o v i d e s an e x a m p l e of a f u n c t i o n in L i p (n/2, ~ ) w h i c h does n o t h a v e an a b s o l u t e l y c o n v e r g e n t F o u r i e r series.H e n c e t h e n - d i m e n s i o n a l v e r s i o n o f T h e o r e m 1 is sharp. This a n d r e l a t e d results were established b y W a i n g e r [5].
58 e, w , OI~I~EWEER
R e m a r k 3. T h e f u n c t i o n s go as d e f i n e d in Section 1 also p r o v i d e n e w e x a m p l e s t h a t s h o w t h a t several of t h e results o f H i r s c h m a n on p - m u l t i p l i e r s c a n n o t be i m p r o v e d as was a l r e a d y p o i n t e d o u t b y H i r s c h m a n in [2].
T h e a u t h o r w o u l d like to t h a n k P r o f e s s o r L.-S. H a h n for a n u m b e r of helpful c o n v e r s a t i o n s on t h e s u b j e c t o f this paper.
R e l e r e n c e s
1. HAI~N, L.-S., On multipliers of p-integrable functions, Trans. Amer. Math. Soc. 128 (1967), 321--335.
2. HIRSCHMAN, I. I. JR., On multiplier transformations, Duke Math. J. 26 (1959), 221--242.
3. IzvMI, M. and S.-I., Absolute convergence of Fourier series of convolution functions, J. of Approx. Th. 1 (1968), 103--109.
4. KATZNELSON, Y . , A n Introduction to Harmonic Analysis, J. Wiley and Sons, Inc., New York, 1968.
5. WAINGER, S., Special trigonometric series in k-dimensions, Memoirs Amer. Math. Soc. 59 (1965).
6. ZrGMUND, A., Trigonometric Series, Vols. 1 and 2, 2nd ed., Cambridge Univ. Press, New York, 1959.
Received January 10, 1973 C. W. ONNEWEER
Dep. of Mathematics and Statistics The University of New Mexico Albuquerque, New Mexico 87131 USA
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