Let C[a,b] denote the space of
continuous functions x(t) defined on [a,b] ∋ x(a) = 0. This space is called Wiener
space. Using the Wiener integral we define, for each nonnegative integer M, what we
call the M Ilstow, M complex Wiener, M Feynman, limiting M complex Wiener, and
limiting M Feynman integrals of a functional F(x) on C[a,b] and show various
relationships which exist between these integrals. In particular we give necessary and
sufficient conditions for a finite dimensional functional F(x) to be M Ilstow
integrable on C[a,b].
We consider the set of linear functionals x(t1),⋯,x(tn) where a = t0 < t1 < ⋯ < tn = b
and obtain conditions on gj(u) ∋ the functional
![F(x) = g1[x(t1)]⋅⋅⋅gn[x(tn)]](a200x.png) | (1.1) |
is M Ilstow and limiting M Feynman integrable on C[a,b]. We then apply these
results to the functional
where 0 ≦ t ≦ t0, −∞ < ξ < ∞ and x ∈ C[0,t0] and show that for appropriate
functions 𝜃(t,ξ) and σ(ξ), the limiting M Feynman integral Ĝ(t,ξ,q) of F(t,ξ,x)
exists for
and satisfies there the integral equation
![∫
ˆ −iq 1∕2 ∞ qi(ξ−u)2
G (t,ξ,q) = (2πt) − ∞σ(u)exp( 2t )du
∫ t
+ (−2iπq)1∕2 (t − s)−1∕2ds
∫ ∞ 0
× 𝜃(s,u)Gˆ(s,u,q)exp(qi(ξ−u)2) du.
−∞ 2(t−s)](a203x.png) | (1.2) |
|