1.1. DEFINITIONS AND BASIC PROPERTIE S 13
Now let 0 r 1 and e+  N/r + N = t 0. By (1.1.17) for R 1
2
itf(lr)/r f 1/^)1 dx
^B(0,H )
2iJV(ir)/r
s u p
 / . (
X
)  1  / l/^a;)rdx
a e £ ( 0 , f l ) JB(0,R)
C ( l + iJ)W
r + d
(
1

r
) inf M . ^ M ^ / , ) ^ )
1
 / /i(x)
r
dx
C{1 + iJ)"/'+ d inf M ^ f M ^ J W .
xeB(0,R)
Thus for each J? 1 and each x G B(0,R)
,_ n JB(OM)
It follows as before that
OO « . /
R^iN/r+d)j2
\My)\dyc(y2Z02~iKL)
*ll{/i}£ob,
i = 0 JB(0,R)
which is (1.1.20).
Now let / G ^ ( ^ ) , and suppose that / = YliLo /* ^s a representation satisfying
(1.1.8), (1.1.10) and (1.1.11), and converging in Sf. By (1.1.20) the series converges
in Li,ioc (£r,ioc if r 1 by Lemma 1.1.4), and consequently the distribution / can be
identified with a function in Li,ioc It follows that / G YL(E), and Y(E) C YL(E).
To prove the converse inclusion, let / G YL(E), and suppose that / = Yl^Lo fi
is a representation satisfying (1.1.8) and (1.1.9), and converging in Lr,ioc • It follows
from (1.1.20) that the series converges to / in S'.
We have to prove that / has a representation / = YliLodi satisfying (1.1.10)
and (1.1.11), and such that {#j]LolU C\\{fj}^0\\E' Let I/J0 £ S be such that
s u p p ^ o C B(0, 2) and J^o(£) = 1 for f G £(0,1). Set
tlj(x) = 2(
j + 1 ) i
Vo(2
j + 1
x) 
2jNil0@jz)
for j G N, so that ^ i ( 0 = ^ o ( 2 ^ "
1
0  ^o(2"^), E ; = o ^ i ( 0 =
l o n
B(0, 2*+1), and supp J ^ 
c
B(0, 2^+2) \ 5(0,2''). Then
/* = /** (S*=0^i) = ]£/i*^
j = 0
This gives, if the change in order of summation is justified,
oo oo i oo oo oo
(1.1.21) / = J2fi = EE/ ** = EE/ ** = E *
i=0 i=Q j=0 j=0 i=j j=0
with gj = Ys'Zj fi * ilj • But Y^Lj fi converges in S', so g5 = (X X j /*) * ^j
a n d
hence (1.1.10) and (1.1.11) are satisfied.
We can write {g3}f=0 = {J2Zo fi+j*^j}j^o = Et~o{/*+i *Vj}^0 L e t V b e a
function in S such that Fip{Ji) = 1 on B(0,
22+7+1).
Then fi+j*ipj — fc+j * (ilj*p),