A R K I V F O R M A T E M A T I K B a n d 5 n r 34
1 . 6 4 0 1 1 C o m m u n i c a t e d 8 J a n u a r y 1964 b y L . CARLESON
A d d e n d u m to " T h e W i e n e r - H o p f e q u a t i o n i n a n a l g e b r a o f B e u r l l n g "
By EDGAR ASPLUND
I n a p a p e r [1] b y the a u t h o r t h e m a i n s t a t e m e n t (Theorem 3, page 107) s t a t e s , for a kernel / in the Beurling algebra A 2 (defined on page 90) a complete set of solutions ~ in t h e dual space B 2 of A 2 to the W i e n e r - H o p f e q u a t i o n
f 7 /(x - y) of(y) dy
= ~(x) + ~0(x), ~(x) = 0 for x < 0, ~o(x) = 0 for x >~ 0,c o
u n d e r certain conditions.
1. T h e function / is h e r m i t e a n s y m m e t r i c ( / ( - x ) =/(x)) so t h a t the fourier trans- f o r m [(~) = S~r
e *r
is real valued.2. T h e function log I l - 1 1 is locally integrable, so t h a t t h e Szeg5 f a c t o r i z a t i o n
[(~e)_ 1 =eh~(~)e -~(~
can be defined (page 99); one can continue h 1 a n d h 2 a n a l y t i c a l l y to the u p p e r respectively lower half-plane; also, on e v e r y i n t e r v a l on the real axis contained in the c o m p l e m e n t of [-1(1), h t a n dh z
are in a sense defined b y L e m m a 3, p a g e 100, " l o c a l l y " in A 2.3. T h e set [-1(1) is countable, a n d if {b~} is the set of p o i n t s for which ( ~ - b ~ ) [ (~) ~< 0 in some n e i g h b o r h o o d of b~ t h e n ~ l b ~ I < ~ .
U n d e r such conditions t h e " f o r m a l solution" E(~) = c$-~1-[(1 - fl~/~)(1 -
b~/~) -1,
c real, n pos. integer, ~lfl~ [ < ~ , gives the complete set of solutions, b a c k - t r a n s - f o r m i n g t h e f o r m u l a e~(~
- i ~ ) = eh~(~-~'7)E(~- i~), ~(~ + i~) = eh'(~+~)E(~ + i~)
p r o v i d e d one can show b y m e a n s of t h e criteria of T h e o r e m 2, p a g e 95, a n d Proposi- t i o n 6, p a g e 96, t h a t ~ is t h e analytic c o n t i n u a t i o n into t h e n e g a t i v e half-plane of a t r a n s f o r m a t i o n of a function in B 2 vanishing on t h e n e g a t i v e half-axis a n d t h a t is t h e a n a l y t i c continuation into the positive half-plane of a function in B~ (the
" s u b - d u a l " of A 2) vanishing on the positive half-axis. T h e p u r p o s e of this n o t e is t o show t h a t t h e condition on v~ is superfluous. T h e application of the a b o v e cited criteria to q~ a n d v~ yietd the explicit conditions w r i t t e n out t h e first t w o rows of p a g e 107, so we will d e m o n s t r a t e here t h a t t h e second of these conditions is dispen- sable.
T h e proof of this fact follows to a g r e a t e x t e n t t h e original proof of T h e o r e m 3 in [1], a n d we here skip some of t h e details t h a t were carried out there. W e suppose t h e n t h a t E is given subject to t h e s a m e conditions as in [1] e x c e p t for the one insuring t h a t ~ is the analytic c o n t i n u a t i o n of a B~ function (line 2, p a g e 107).
523
E. ASPLUND, Addendum to " IViener-Hopf equation"
T h e n ~v E B 2 can be f o u n d satisfying ~v(x) = 0 for x < 0 a n d r - i~) = ~ ~v (x) e- ~ e- i~ dx
= eh~(r E(~ - i~), a n d the goal is t o show t h a t ~(x) = ~ / ( x - y)~v(y) dy -qJ(x) vanishes for x > O .
W e define t h e f u n c t i o n V in t h e c o m p l e m e n t of t h e real line b y
f ~ : c { f : / ( x - y ) q ~ ( y ) d y } e - ~ d x - e ~ ( : ) E ( ~ ) , I m ~ > 0 V(~) =
-f:~(x)e-~Xdx, Im~<O.
We observe t h a t t h e proof on page 108 of t h e analytic c o n t i n u a t i o n t o points outside /-l(1) on the real line carries over almost v e r b a t i m for this new definition of V. I t is t r u e t h a t ~o appears in the formulas, b u t t h a t is only superficial, since it comes f r o m e h'(~) E(~) t h a t is t r a n s f o r m e d back a n d forth, a n d this function could equally well have been k e p t till line 7 f r o m below (there are a couple of misprints on this page 108, line 9 f r o m below should begin ~{~/... a n d on lines 7 a n d 8 f r o m below one should have
... e h~(~ i~ - iO) - eh'(~+~~ + iO)...).
T h e isolated singularities off the origin are r e m o v a b l e as before, since at these t h e f u n c t i o n ehl(~)E(~) grows at m o s t as (Imp) 89 (and for t h e other t e r m s we have t h e previous estimates). Hence V(~) is a n entire function of 1/~, a n d it is clear f r o m t h e R i e m a n n - L e b e s g u e l e m m a applied t o t h e definition t h a t V h a s a zero at infinity.
T h e other zeros of V can be e s t i m a t e d b y means of J e n s e n ' s f o r m u l a and, as was t h e case with B on pages 104 a n d 105, this analysis shows t h a t V ( ~ ) = c ~ - n l - [ ( 1 - ~ / ~ ) , with c real, n n a t u r a l n u m b e r a n d ~ - I ~ I < ~ -
Noticing t h a t this means t h a t V(1/~) is an entire function of order at m o s t one, a n d t h a t if its order equals one t h e n its t y p e will be zero, we e x p a n d it in a power series, V ( $ ) = ~ vn~ -n a n d b a c k t r a n s f o r m to get ~ ( x ) = - ~ r (Vn+i(i)n+l/n!)x ~.
Using t h e g r o w t h t h e o r y for entire functions we see t h a t 5 m a y be c o n t i n u e d t o a n entire function ~(z) of order at m o s t 1, a n d t h a t if the order of ~ is 89 th e n its t y p e will be zero (the relevant g r o w t h t h e o r e m s are listed in Boas [2] as T h e o r e m 2.2.2, page 9, a n d T h e o r e m 2.2.10, page 11). N o w we k n o w t h a t the restriction of 5 to t h e positive real axis shall be a function in B ~. Hence t h e function
if
(z) ~ (~) dzu ( z ) = ; 0
is, b y Proposition 2 of [1] (page 93) b o u n d e d on t h e positive real axis, a n d since u is also of g r o w t h order at m o s t 89 minimal t y p e it is of course s t a n d a r d t h a t u is a c o n s t a n t (Theorems 1.4.2 a n d 1.4.3 of Boas [2]: Phragmdn-LindelSf t y p e theorems) a n d t h a t so is 8.
N o w we h a v e t h a t V($)=c/~ for some c o n s t a n t c a n d we will m a k e use of t h e definition of V for small negative pure i m a g i n a r y values of ~. F i x an a r b i t r a r y e > 0 a n d decompose t h e kernel f u n c t i o n / = / 1 + / 2 so t h a t [1 has c o m p a c t s u p p o r t a n d 11/211A~<e. T h e n [1 is an entire function a n d I [ , ( 0 ) - 1 1 < ~ . L e t ~ t e n d to zero along t h e negative i m a g i n a r y axis. T h e n
524
ARKIV F(JR MATEMATIK. B d 5 n r 3 4
- V(r /l(x - y)cf(y)(e -tell: - e-lr +
(/1(0) - 1) r162o o
+f:(/2~Cl))(x)e-I~lZdx
a n d i n v i e w of t h e e s t i m a t e s
f : f:/~(x-Y)cf(Y)(e-'~'X-e-'~'Y)dxdy[<~]l/~[]llcfll'
A 2~
( P r o p o s i t i o n 7 a n d T h e o r e m 2, p a g e s 96 a n d 95 r e s p e c t i v e l y of [1]), a n d
i t is clear t h a t , since e w a s a r b i t r a r y , t h e f u n c t i o n V(~) does i n f a c t v a n i s h i d e n t i c a l l y . T h e p r o o f is t h e n c o m p l e t e .
The author is grateful to professors L. Carleson and Y. Domar for encouragement to pursue this sharpening of the main result in [1].
Royal Institute of Technology, Stoctcholm, Sweden
R E F E R E N C E S
1. ASPLUND, E., The Wiener-ttopf equation in a n algebra of Beurling. Acta Math.,
108,
pp. 89- 111 (1962).2. BoAs, 1~. P., Entire Functions. New York, 1954.
Tryckt den 19 januari 1965 Uppsala 1965. AImqvist &Wiksells Boktryckeri AB
3 6 : 6 525
