8 1. A CLASS OF FUNCTION SPACES
LEMMA
1.1.4. Let E G S(s+,e,r), suppose that F — {/;}^
0
G E, and set
g = X ^ o 2lX\fi\. Then for any X e+ there is C so that
l{Vis}~olU
C\\F\\B.
In particular, g G Lr,\oc, and there is C so that for all R 0
(l +
R)Wr+d\\ghAB(o,R))C\\F\\E.
PROOF.
It is sufficient to assume that /* 0. We have
oo oo
{S0Jg}?=0 =
J22tX^jfi}T=o
=
J22i\S+Y{SiJfi}f=0.
i=0 i=0
But by Definition 1.1.1(a)
US+YiSijfi^oWs C2*+\\{6iJfi}?=0\\B C2^\\F\\E,
whence
i=0
oo
C   F    ^ 2 ^
( A

£
+
)
= C\\F\\%
i=Q
where K is the constant in (1.1.2).
This implies that g G Z/r,ioc since otherwise Mr^g would be identically +oo,
which is impossible.
Moreover, for all x G 5(0, R) we have
HSIIMB(O,K))
C(l
+
R)N/r+dMr,dg(x),
so that by the lattice property of
E1,
\\g\\LAB(o,R))\\{SoMB(0,R))}T=o\\E ^ ^ ( l + ^ ^ ^ I K ^ M ^ f l ^ o l ^ .
Dividing by the positive number
\\{SOJX(^(^A))}^O\\EI
a n d applying Definition
1.1.1(b), we obtain for R 1
(1 + R)Wr+d)\\g\\LAB(o,R)) C\\{50dMr,dg}™=0\\E C\\{60d9}?=0\\E .
This proves the lemma. •
We now define the spaces which will be our main object of study. Recall that if
a distribution / G S' is such that supp^ 7 / is compact, and if the order of Tf is m,
then by the PaleyWienerSchwartz theorem, / is an entire function of exponential
type, whose restriction to R ^ satisfies \f{x)\ (7(1 + rr)m. See e.g. Hormander
[26], Theorem 7.3.1. It follows that Mr4f(x) is finite for all x G
RN
for d m.
DEFINITION
1.1.5. Suppose that £+, £_, r 0, and let E G S(e+,e,r). The
space YL(E) consists of all functions / G Lr,ioc which have a representation
oo
(ii7) / = £ / * ,
i=0
converging in Lrioc, such that
(118) {/i}£o
E
°° .
(1.1.9) fiGS',
supPTfiCB(0,2i+1),
i e N
0
.