Abstract: | Let C0, T] denote the space of real-valued continuous functions on the interval 0, T] with an analogue w
ϕ of Wiener measure and for a partition 0 = t
0 < t
1 < ... < t
n
< t
n+1 = T of 0, T], let X
n
: C0, T] → ℝ
n+1 and X
n+1: C0, T] → ℝ
n+2 be given by X
n
(x) = (x(t
0), x(t
1), ..., x(t
n
)) and X
n+1(x) = (x(t
0), x(t
1), ..., x(t
n+1)), respectively.
In this paper, using a simple formula for the conditional w
ϕ-integral of functions on C0, T] with the conditioning function X
n+1, we derive a simple formula for the conditional w
ϕ-integral of the functions with the conditioning function X
n
. As applications of the formula with the function X
n
, we evaluate the conditional w
ϕ-integral of the functions of the form F
m
(x) = ∫0
T
(x(t))
m
for x ∈ C0, T] and for any positive integer m. Moreover, with the conditioning X
n
, we evaluate the conditional w
ϕ-integral of the functions in a Banach algebra
which is an analogue of the Cameron and Storvick’s Banach algebra
. Finally, we derive the conditional analytic Feynman w
ϕ-integrals of the functions in
.
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