A R K I V F O R M A T E M A T I K B a n d 7 nr 30 1.67 11 3 C o m m u n i c a t e d 13 S e p t e m b e r 1967 b y LENNA~T CARLESON
An extremal problem related to Kolmogoroff's inequality for bounded functions
By YNGVE DOMAR
A B S T R A C T
L e t A a n d B be positive n u m b e r s a n d m a n d n positive integers, m < n. T h e n t h e r e is for c o m p l e x v a l u e d f u n c t i o n s ~0 o n R w i t h sufficient differentiability a n d b o u n d e d n c s s p r o p e r t i e s a r e p r e s e n t a - t i o n
~o (rn) = ~ ( n ) - ) t - v l + ~-X-vs,
w h e r e r l a n d v2 are b o u n d e d Borel m e a s u r e s w i t h vl a b s o l u t e l y c o n t i n u o u s , s u c h t h a t t h e r e exists a f u n c t i o n V w i t h IV `n)] < A a n d ]9] ~<B o n R a n d s a t i s f y i n g
cf(m) (O) = A L [ dvl l + B ~ l dv2 I.
T h i s r e s u l t is f o r m u l a t e d a n d p r o v e d in a general s e t t i n g also applicable to derivatives of frac- t i o n a l order. N e c e s s a r y a n d sufficient conditions are given in order t h a t t h e m e a s u r e s a n d t h e o p t i m a l f u n c t i o n s h a v e t h e s a m e essential p r o p e r t i e s as t h o s e w h i c h occur in t h e p a r t i c u l a r ease s t a t e d above.
1. We denote by
M(R)
the Banach space of bounded Borel measures on R and b yAC(R)
the subspace ofM(R)
consisting of all measures which are absolutely con- tinuous with respect to the Lebesgue measure. The Fourier-Stieltjes transform /2 of a measure/tEM(R)
is defined b y the relationp ( t ) =
f d/t(x),
for every t on the dual R. Convolution of elements in
M(R)
is defined in the usual w a y such t h a t it corresponds to pointwise multiplication of the Fourier-Sticltjes transforms.L e t / t l a n d / t 2 be given elements in
M(R),
and tt o a third given element such t h a t there exist elements 1 and v2 inM(R)
such t h a t/to =/tl"eVl+#2~ev2. (1)
We assume that there exist a real number a and measures 00 and a2 in
AC(R)
such t h a t the three relationsY. DOMAR, Extremal problem related to Kolmogoroff's inequality
pl(t) + 0, (2)
p2(t) = fq(t) ~z(t), (3)
p0(t) = pl(t) d0(t) (4}
all hold, if [ t [ ~ a.
H denotes the set of all pairs of b o u n d e d Borcl m e a s u r e s {vl, v2}, which satisfy (1).
L denotes t h e set of all pairs of b o u n d e d Borel m e a s u r e s {Vl, ~2} such t h a t
/A~I-)(- y I +/~2-X-~) 2 =
0
W e finally f o r m t h e class K of all pairs of functions {91,w in L~(R), such t h a t w i t h t h e usual definition of convolution b e t w e e n elements in L~C(R) a n d AC(R),
91 ~ vl +92 * v2 = 0, for e v e r y vl C AC(R) a n d v2 EAC(R) such t h a t {vl, v2} EL.
T h e o r e m 1. 1 ~ I /
{'~1, ~,~}~H
then VlCAC(R ).2 ~ I/{91, 92} E K, then 9z is continuous, after a change in a set o/Lebesgue measure O.
3 ~ With this assumption on 92, we/orm/or any {91, 9z} e K and {~, r2} e H the/unc- tional
F(91, 9 2 ) = F(91, 92, vl, v2) = f 91( x)v~(x)dx+ ~ 9 2 ( - x ) d v 2 ( x ) .
JR JR
lts value does not depend on the choice o/{vl, vz}.
4 ~ Let (A, B) be a/ixed pair o/positive numbers and let K(A, B) denote the subset of all {91, 92} e K such that 119111~ <~ A, ]19~[] ~ <~ B. Then there exists a {~F1, 1F2 } e K ( A, B) such that
] F(91, 92) 1 < F(uIV'l, ~F2), /or every {91,
92}eK(A,
B).5 ~ There exists a {vl, v~}EH such that
Id~(x) l.
(6)JR JR
Before we give t h e proof of T h e o r e m 1, we shall discuss a p a r t i c u l a r l y i m p o r t a n t e x a m p l e a n d m a k e some general c o m m e n t s .
L e t m a n d n be integers such t h a t 0 < m < n. I t is e a s y to see t h a t all our a s s u m p - tions are fulfilled if we choose #1,/~z and tt0 such t h a t
fq(t) = e -t~ (it) '~, ft2(t) = e -t*, po(t) = e-t' (it)rn.
434
ARKIV FOR MATEMATIK. Bd 7 nr 30 Using the definition of K we find t h a t K consists of all pairs of bounded functions of the form {~v("),g}, where g is absolutely continuous together with its n - 1 first
derivatives. Let us then form the continuous function gO = g(n) -)(- ~)1 -~-~) "~- Y2'
where {Yl' v2} EH. Convoluting this with an a r b i t r a r y measure # with compact sup- port and high differentiability properties, it is easy to see, b y using (1) and partial integration, t h a t we obtain
go-X-# =iv ++#(r"),
where #(m) denotes the measure which, in the distribution sense, is the
m-th
derivative of the measure #. Hence g0 =g(~). I f we substitute x = 0 in g(m) we obtaing(m>(0) = F ( g (~), g, ~1, v~).
Hence our theorem shows t h a t there is for every pair (A, B) of positive numbers a representation
g<m)(o)= fRg(n)(-x)~;(x)dx + f g(-x) d~2(x)
such t h a t equality can be attained in the resulting inequality
IIv( >II <A, Ilgll+--< B.
I n this particular case, the value of (6) is known, since the optimal functions g have been found b y Kolmogoroff [6], see also Bang [2]. As was shown in Bang's paper, the inequality given b y Kolmogoroff is v e r y closely related to the well-known 9 inequalities b y Bernstein and Bohr. As can be seen for instance in Achiezer [1, w167 74, 86] these two inequalities can be proved and generalized using representations which are similar to our formula (5). The starting point for our investigations was an a t t e m p t to prove and generalize Kolmogoroff's inequality using similar ideas. We shall in section 3 show t h a t the representation obtained in Theorem 1 can give a direct infor- mation on the possibility of such generalizations.
I t should finally be mentioned t h a t the ideas in this paper have connections with questions on minimal extrapolations of Fourier-Stieltjes transforms (see for instance [3] and Herz [4, Theorem 4.1]). I t is possible to p u t certain parts of these two theories into a common framework. Generalizations in the same direction as those in HSrman- der [5] are also possible.
2. Proo/ o/ Theorem 1.
10. I f {vl, v2 } E H, then b y (1)
Po = Pl~l +P2~2.
"z. DOMX~,
Extremal problem related to Kolmogoroff's inequality
B y (2), (3) and (4) this relation implies t h a t if it] ~>a
~o(t) = ~(t) -t-d2(t)r (7)
T h e difference
~3(t)
between t h e left a n d right m e m b e r s of (7) is a Fourier-Stieltjes t r a n s f o r m which vanishes for large t, hence the corresponding measure v3 belongs t oAC(R).
W e thus h a v e t h e relation'Pl = 0-0 --~)3 - - 0-2-)(- ~2,
where 0-0, v3 a n d 0-2 belong to
AC(R).
B u t t h e convolution of a measure inAC(R)
with a measure in M ( R ) belongs to A C(R). Hence 0-2 -)r ~2 E A C(R) a n d as a consequence
vl EAC(R).
2% W e choose a measure v E A C ( R ) such t h a t ~(t)=1 if It[ ~ a . L e t (~1, ~2} EK- F o r a n y / ~
EAC(R)
t h e pair((0-2-a2~r)~, v ~ - ~ }
lies in
L,
a n d b o t h measures belong toAC(R).
Hence b y the definition ofK,
~01-)(- (0" 2 - - 0-2-)(- ~2) -)(-/~ ~- ~92-)(- (V-X-~ - - ~ ) = 0.
A r e a r r a n g e m e n t gives
This implies t h a t
9?2 = 9)2 -)~'v -{- (i01-)~" 0-2 --(~1 "~ ( 0-2-)(- ~))
a l m o s t everywhere. Since v, 0-2 a n d 0-2~v all belong to
AC(R),
the r i g h t - h a n d m e m - ber is continuous which proves t h e assertion 2 ~ .3 ~ W e f o r m t h e B a n a c h space X of all pairs {vl, v2} where vl
EAC(R)
a n d v2 EM(R), with t h e n o r ma n d with t h e v e c t o r operations defined in an obvious way. L is a linear subspace in X a n d H a hyperplane, parallel to L. F o r e v e r y given (~1, ~ 2 ) E K ,
represents a linear functional on
X,
which vanishes for e v e r y {vl, v2} EL such t h a tv2 EAC(R).
I f n o w (vl, v2} EL is a r b i t r a r y a n d does n o t annihilate the functional, it is in view of 1 ~ and 2 ~ possible to find a measure rEAC(R)
with its s u p p o r t c o n c e n t r a t e d t o a n e i g h b o r h o o d of x = 0 and such t h a t t h e functional, applied to {Vl-~V, v2~v}, does n o t vanish. B u t this element belongs to L a n d v2-)(-vEAC(R)
which gives us a contra- diction. Hence t h e functional vanishes on L a n d t h u s t h e value is c o n s t a n t on H.4". W i t h t h e notions i n t r o d u c e d above we let d denote the distance between H a n d
L,
i.e.436
ARKIY F6R MATEMATIK. B d 7 nr 3 0
O b v i o u s l y ]-~(~01, ~V 2, vx, v2)] ~<d, (9)
if {~vl,~2}eK(A, B), {v~, v 2 } e H .
W e k n o w f r o m t h e H a h n - B a n a c h t h e o r e m t h a t there exists a b o u n d e d linear func- tional
G(Vl, v~)
on X, with n o r m 1, vanishing on L, a n d t a k i n g t h e value d on H. G is also a linear functional with n o r m ~< 1, on t h e closed subspace of X, consisting of t h e pairs {Vl, v2} where b o t h v 1 a n d v2 belong toAC(R).
T h e dual of this space is well known, a n d we o b t a i n f r o m this t h a t there exist b o u n d e d measurable functions~ 1 a n d 1F~ such t h a t
G ( v , =
f ( - x) v; (x) dx § f ) r 2 ( - x) v; (x) dx, (10)
if vl a n d
v2 E AC(R).
ObviouslyII~r~ll| II~II|
Since G vanishes on L, t h e definition of K shows t h a t {~F1, 1F2}EK, in particular t h a t we can assume ~F2 to be continuous. Since b y (9)
I X~(~01' ~92) I ~ d = G(~)I, T2) , if
{cf~,cf2)eK(A , B), {v~, va)eH,
4 ~ is p r o v e d if we can show t h a t
for e v e r y {~1, v2} EX.
W e can, of course, because of (10) restrict ourselves to t h e case when v 1 = 0 . I n the case when ~ has a c o m p a c t support, v2 belongs to
AC(R),
hence b y (10) t h e relation (11) is true, a n d t h u s we can also restrict ourselves to t h e case w h e n ~2 vanishes on t h e set {tilt I < a } . T h e n ( - a 2 ~ - v ~ ,v2}eL,
hencewhich gives us G( - a 2 ~ 2 , ~2) =0,
G(O, v2) = G(a~ ~ v2, O) = ~ ~ 1 ( - x) (as ~ ~ ) ' (z)
dx.
JR
(12)N o w let {Tn}~ r be a sequence of n o n - n e g a t i v e measures in
AC(R),
all with t o t a l mass 1 a n d with supports contained in t h e set{x]]x] <~l/n}.
T h e n v~ in (12) can be e x c h a n g e d t o v2~-Tn. B y a well-known p r o p e r t y of convolutions with measures inAC(R),
applied to t h e r i g h t - h a n d m e m b e r of (12), we see t h a t lim G(0, v2 ~ Tn) = G ( 0 , v2), n-)r162Y. DOMAR, Extremal problem related to Kolmogoroff's inequality
Since v 2 ~ - ~ belong to A C(R), t h e left-hand m e m b e r of this relation can be r e p r e s e n t e d using the f o r m u l a (10), a n d this finally gives, b y s t a n d a r d a r g u m e n t s ,
?
G(O, v~) =.I W2( - x) dye(x),
hence t h e desired result.5 ~ W e h a v e to p r o v e t h a t t h e r e exists an element {vx, v2} E H which gives t h e mini- m u m in (8). W e can find a w e a k l y c o n v e r g e n t sequence of pairs of measures, converg- ing to a m e a s u r e {v ~ v~ where b o t h v ~ a n d v ~ belong to M(R). I t is e a s y to see, using t h e F o u r i e r - S t i e l t j e s t r a n s f o r m s , t h a t t h e relation ( 1 ) w h i c h d e t e r m i n e s H , can be w r i t t e n
f~
~0 (Y) = ~t 1 (y - x) dVl(X) + ~2 (Y - x) du2 (x), (13) for e v e r y y E R, where/to,/~1 a n d ~2 are the functions
e -~#i ( i = 0 , 1 , 2 ) .
B u t these functions belong to Co(R ) a n d hence (13) has to be fulfilled for t h e limiting m e a s u r e s {v ~ v~ H e n c e {v ~ v2 ~ } e l l , a n d it is obvious t h a t it m u s t realize t h e infimum.
3. W e n o w r e t u r n t o t h e case w h e n f i l - e - t ~ ( i t ) ~, ft2 =e-t~, fto=e-t~(it) m, where m a n d n are integers such t h a t 0 < m < n . This case was briefly discussed in the last p a r t of w 1. U n d e r this a s s u m p t i o n K o l m o g o r o f f [6] has f o u n d t h e o p t i m a l pairs {~F1, ~F2}. Disregarding a c o n s t a n t f a c t o r with m o d u l u s 1, t h e y are f o u n d a m o n g t h e functions given b y t h e relation
U121=
A sign (sin (bx+c)),where b a n d c are real, b # 0 , a n d with t h e corresponding ~F~ d e t e r m i n e d as t h e primi- tive function of ~F 1 of order n a n d w i t h m e a n value 0.
A certain lack of s y m m e t r y in t h e functions {~1, ~ 2 } a p p e a r s w h e n m a n d n varies.
This can, however, be overcome, for e v e r y fixed set {m, n, A, B}, b y changing t h e functions
/~1
a n d / 2 2 toe t~ (it)n ei~, t
a n d e-t, (it)m etZ~ t.
respectively with t h e real n u m b e r s fil a n d f12 s u i t a b l y chosen. After such a change, which of course only corresponds to translations in the functions {~1, ~2} in t h e class considered, we see t h a t the o p t i m a l pair {Y~I, 92} has t h e following properties, for some h > 0 :
1 ~ ~ F I = A sign (sin hx-),
2 ~ ~F2(2nz/h ) = B ( - 1 ) n (nEZ),
while - B < ~ 2 ( x ) < B , if x ~ - 2 n z / h (nEZ).
438
ARKIV FOR MATEMATIK. B d 7 n r 30 A pair of functions {~1, ~ } with these properties is said to belong to the class E(A, B, h).
A natural problem is now to investigate if pairs in the class E(A, B, h) occurs as optimal pairs in other cases.
Let us then first make some assumptions on/20,/21 and/22. We assume that/21 ~:0 ahnost everywhere, and that/20//21 and/22//21 EL1 except for some bounded interval.
Finally we assume t h a t
/2~ (t + hn)//2~(t + hn) * 0, almost everywhere. Then the following theorem holds.
Theorem 2. Suppose that {W~, ~T'2}EK N E(A, B, h). Necessary and su//icient in order that {~T'I, ~F2} is an optimal pair, in the sense o/ Theorem 1, is that there exists a pair {~1, v2} EH, such that
~ (x)sign (sin ~ ) = I ~ (x)I (xER),
while v~ is a discrete measure composed o/ non-negative pointmasses at x = 4 n z/h, n EZ, and non-positive point masses at x = (4n § 2)z/h, n EZ.
{vl, v2} is then uniquely determined by the relations
/~0 (t -~- nh)//21 (t ~- nh) = ~2 (t) ~ /22 (t + n h ) / / ~ 1 (t ~- nh), (14)
oQ oo
~1 =/~0//~1 - ~2/22//21. (15)
Proo/ o/ Theorem 2. The sufficiency is a direct consequence of the conditions on ~1 and ~ which show that
[ F((pl, ~ ) ] = ] F ( ~ i , 92, "Pl, ?J2) l ~
F(W1, ~F2,
"k' 1, ~)2) = I F(Vl, V2) I,if {~91, q~2} EK(A, B).
To prove the necessity we assume t h a t {~1, ~2} EK N E(A, B, h) is an optimal pair. Then b y Theorem 1, 4 ~ and 5 ~ there exists an optimal {Vl, v2}EH, which obviously has to fulfil the conditions on the signs of v~ and v2.
I t remains to prove the formulas (14) and (15). (15) is a direct consequence of (1).
I t shows, together with the conditions on/20,/21 and/22, t h a t ~1 ELl(R) . Hence we can assume t h a t v~ is continuous. The condition on the sign variation of v~ then implies t h a t
v/(2nze/h) = 0 (nEZ).
p 9 A
But vl is the inverse Fourier transform of the function vl in the L 1 sense. Hence the periodic function
#l(t +nh),
- o o
which locally belongs to L 1, has its Fourier coefficients determined b y the values v~(2n z/h) in such a way t h a t they, too, must vanish. Hence
Y. DOM.A.R, Extremal problem related to Kolmogoroff's inequality
~x (t + nh) =O, a l m o s t everywhere.
A direct s u m m a t i o n of (15) t h e n gives (14).
T h e o r e m 2 is applicable in t h e case w h e n / t l , / t 2 and re0 are given b y t h e relations /21 (t) = e -t~ (it) ~' e ~ t,
/ 2 ~ ( t ) = e t 2 ,
/20 (t) = e -t* (it) ~ e t~*t, w h e r e (~1, ill, (~2 a n d flz are real a n d
0 < ~ 2 < ~ 1 - - 1 . (it)~J, 7" = 1, 2, is t h e n to be i n t e r p r e t e d as
exl~ ~ i sign t + log I t I) ~J"
I f al, a2, A a n d B are given, it is possible to show t h a t we can choose h, fil a n d f12 such t h a t there is a pair of functions {q~l, qJ2} E K N E ( A , B, h). T h e p r o b l e m is to decide w h e t h e r this pair is a n o p t i m a l one.
I f b o t h al a n d a2 are integers this is t h e case, which follows f r o m K o l m o g o r o f f ' s result. T h e only new in this case is t h e existence a n d t h e explicit f o r m of t h e o p t i m a l pair {YI' V2}"
T h e case w h e n some aj is n o t a n integer corresponds t o t h e p r o b l e m for fractional d e r i v a t i v e s (see B a n g [2]). I n this case it is e a s y to see f r o m (14) t h a t ~e(t) is a n a l y t i c within t h e period ( - h / 2 , h/2) e x c e p t a t t = 0 , where it has a singularity of a t y p e which excludes t h e possibility of t h e d e m a n d e d sign-variation in d v~. H e n c e {1F1, ~F~}
is n o t o p t i m a l in this case.
This completes a discussion of B a n g [2] where he c a m e t o t h e s a m e conclusion in t h e case 0 < a S < ~1 = 1. I n t h a t case, assuming fll = fi2 = 0, t h e o p t i m a l pairs {~F1, ~F 2 } a r e explicitly known. T h e y consist essentially of all {LF', ~F } w i t h II ~F' II ~o < A, I1 ~F II oo ~ B , where ~ is absolutely continuous a n d where
2 B t F ( x ) = - B , if x ~ - ~ ,
2 B W'(x) = A, if - - - < x < 0 .
A
B y m e a n s of T h e o r e m 1 we can easily give t h e necessary a n d sufficient condition o n /20, fulfilling (4) with given
/21 = (it) e -~', 22 ~ e-t2 440
ARKIV F/)R MATEMATIK. Bd 7 n r 30 i n order t h a t all these f u n c t i o n s {y/, ~o} give o p t i m a l pairs. E a s y a r g u m e n t s show t h a t t h e c o n d i t i o n is t h a t
~o = / x l * v ,
where v is a n a b s o l u t e l y c o n t i n u o u s n o t necessarily b o u n d e d m e a s u r e , such t h a t for some c
~ ' = c o n s t a n t ~ c , if x < 0 , 2 B v'>~c, if 0 < x < - -
A '
2 B v' is ~ c, b o u n d e d a n d decreasing, if x > ~ - .
I n p a r t i c u l a r we o b t a i n B a n g ' s case w h e n for some d > 0, a n d for 0 < ~ < 1,
{
v t = 0 , x < 0 ,r x>0.
R E F E R E N C E S
1. ACHIEZER, 1~. I., Vorlesungen fiber Approximationstheorie. Berlin 1953 (1947).
2. BANO, T., Une in6galit6 de Kolmogoroff et les functions presqueperiodiques. Danske Vid. Selsk Mat.-Fys. Medd. X I X , 4 (1941).
3. DOMAR, Y., On the uniqueness of minimal extrapolations. Ark. Mat. 4, 19-29 (1959).
4. HERZ, C. S., The spectral theory of bounded functions. Trans. Am. Math. Soc. 94, 181-232 (1960).
5. H ~ R M A N D E R , L., A n e w p r o o f a n d a generalization of a n inequality of B o h r . M a t h , S e a n d . 2, 3 3 - 4 5 (1954).
6. K O L N I O O O R O F F , A . N., O n inequalities b e t w e e n u p p e r b o u n d s of consecutive derivatives of a n arbitrary function defined o n a n infinite interval. U 6 e n y e Zapiski M o s k o v . Gos. U n i v . M a t e m a t i k a 30, 3 - 1 6 (1939); A m e r . M a t h . Soe. Translation 4.
Tryckt den 9 april 1968
Uppsala 1968. Almqvist & Wiksells Boktryckeri AB
