2 R.H. CAMERON and D.A. STORVICK
DEFINITION:Let q/O be a given real number and let F(x) be a functional defined on
a subset of C [a,b] containing all the polygonal elements of C [a,b] . Let a ,o ~ ,...
be a sequence of subdivisions such that norm c r -*0 and let {\ ] be a sequence of
complex numbers -with &e X 0 such that X -*-iq . Then if the integral in the right
hand side of (1.2) exists for all n and if the following limit exists and is indepent-
ent of the choice of the sequences {CJ} and {X } , we say that the sequential Feynman
integral -with parameter q exists and is given by
Sfq X b - r - 2 - -
(1.2) J FOOdxslim Ya
x
Jvm exV{--f J |^(t,a ?)|| dt}F(X( (•)V ? ) )dl ,
n -*00 n5 n __ n a
H
where
(1-3)
V,X ^ 2ji}
k
l
x
k k"l
We note that m depends on a and m is the number of subintervals in a . We
emphasize that the Lebesgue integral on the right side of (1.2) exists for all n .
In section 3 we shall show that the sequential Feynman integral exists for a broad
class of functionals S which we now define.
DEFINITION:Let D[a,b] be the class of elements x€c[a,b] such that x is
absolutely continuous on [a,b] and
xf
€L_[a,b] . Let D = X D .
d 1
v
DEFINITION:Let % = #?(Lp[a,b]) be the class of complex measures of f i n i t e variatio n
defined on B(Lp) , the Borel measurable subsets of L
p
[a,b] . We set ||M-||=var H .
(in thi s paper, L
?
always means rea l L
?
) .
v v
DEFINITION:The functional F defined on a subset of C that contains D is said
to be an element of S = S(Lp) if there exists a measure K e #? such that for x6D ,
v b dx.(t)
(1.10 F(5) s J * expCiZ J v (t) (_L_)dt}dH(?) .
L2
We sha3JL show in Corollaries I and II of our fundamental existence theorem, Theorem 3*lj
that the sequential Feynman integral of each functional F S exists and satisfies
sf
v b
J qF(x)to= J expt-ir- E J [v (t)]2dt}d*(v) .
v ^ j=i
a
J
L2
In section 2 we shall show that with the proper norm, S is a Banach algebra, and
shall show its relationship to other Banach algebras. In particular we shall show that
S contains
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