10 R.H. CAMERCN AND D.A. STORVICK
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COROLLARY 1:If
F,GeS•K-
and F(x) = G(x) for all binary polygonal functions in
C [a,b] , then
F(x) = G(x) .
COROLLARY 2: If F e § and F is defined only on pV , then there exists an extension
FeS such that F (x) = F(x) on D . Moreover F is essentially unique in the
sense that if F , F «S and F (x) = F (x) = F(x) on D , then
F (x) = F (x) .
Finally, if H is associated with F by (l.*0, it follows that H is associated with
F* by (2.3).
PROOF:Let
* - v b
F(x)=Jvexp[i S J v.(t)3x.(t)}d^(v)
L2 j=l a
J J
"* vr , *
for every xe C [a,b] for which the right member exists. By the definition of S ,
we have F eS . By (2.2) we have F (x) = F(x) for xeD [a,b] . Thus the existence
of F satisfying the required conditions has been established. The uniqueness follows
from Corollary 1.
LEMMA 2.k. If FeS then the measure * is uniquely determined by equation (l.U) on
DV .
PROOF:Let F. be the restriction of F to D . Then F,eS and by Corollary 2 of
Theorem 2.1, there exists an essentially unique F eS such that F, (x) = F (x) on D .
Since the measure defining F is unique, the measure * e 7ft satisfying (l.k) is unique.
NOTATION:If FeS , we define ||F||S ||H|| , where K is associated with F = F(x) by
(l.k) for xeD .
It follows from Lemma (2.1+) that for FeS , the measure * is uniquely determined
by F and it is clear that ||F|| is a norm for S if we identify elements of S which
v
are equal on D .
LEMMA 2.5:The space S = S(Lp) is a Banach algebra where elements of S that are
v
equal on D are considered equivalent.
The proof is identical with the proofs of Theorems 2.2 and 2.3 of [k].
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