Vol. 26, No. 1, 1968

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ISSN: 0030-8730
Generalized Ilstow and Feynman integrals

David Lee Skoug

Vol. 26 (1968), No. 1, 171–192
Abstract

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)]
(1.1)

is M Ilstow and limiting M Feynman integrable on C[a,b]. We then apply these results to the functional

              ∫ b
F(t,ξ,x ) = exp( 𝜃[t− s,x(s)+ ξ]ds)σ[x(t)+ ξ]
a

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

(t,ξ,q) ∈ (0,t0)⊗ R1 ⊗ {R1 − {0}}

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)
(1.2)

Mathematical Subject Classification
Primary: 28.46
Milestones
Received: 28 April 1967
Published: 1 July 1968
Authors
David Lee Skoug