2. THE SPACES OF FUNCTIONALS S AND S
In this section we shall show that S (with the proper norm) is a Banach algebra
and show how it is related to the Banach algebras S and S' introduced in [k]. We
shall also define a new Banach algebra S which is intermediate between S' and S ,
and which is closely related to S .
TERMINOLOGY: We shall say that two functionals F(x) and G(x) are equal s-almost
everywhere if for each p0 the equation F(px)=G(px) holds for almost all xeC [a,b]
, in other words, if F(x) =G(x) except for a scale-invariant null set. We denote this
equivalence relation between functionals by F W G . (Our measure in C is Wiener
measure.)
The definition of S also involves the P.W.Z. (Paley-Wiener-Zygmund) integral [12]
which is defined as follows.
DEFINITION:Let cp ,cp ,... be a C.O.N, (complete orthonormal) set of real functions of
n b
bounded variation on [a,b] . Let v€L [a,b] and v (t)= II cp.(t) J v(s)cp.(s)ds .
n
j=l
3
a
3
Then the P.W.Z. integral is defined by
b {cp3 "b b
P v(s)dx(s) = J v(s)dx(s) = lim J v (s)dx(s)
a a n-*00 a n
for all xec[a,b] for which the above limit exists.
NOTE:It was shown in [12] that this integral exists for almost all xec[a,b] and is
essentially independent of the choice of cp ,cp ,... Moreover if v is of bounded
variation, it is essentially equivalent to the Riemann - Stieltjes integral. Clearly
"almost all" may be replaced by "s-almost all" in this statement.
v
DEFINITION:Let S = S(L ) be the space of functionals expressable in the form
v b
(2.1) F(x) = f exptiS J v.(x)dx.(t)}d*(v)
T
v 0=1
a
L2
for s-almost all xec [a,b] , where H eft = ft [L ] . (Note, it is assumed that the
b d
P.W.Z. integral J v.(t)dx.(t) is based on the same CO.N. sequence [cp 3 for all
choices of v and x ).
k
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