A SIMPLE DEFINITION OF THE FEYNMAN INTEGRAL, WITH APPLICATIONS
NOTATION: If F(x) = G(x) s-almost everywhere on C [a,b] and also for every
x s D [a,b] , we shal l write F(x) = G(x) .
From Remark 2 of [6] we have tha t i f v e L
2
[ a , b ] and x e D [ a , b ] then
b b
(2.2) J v(t)dx(t ) = J v ( t ) x ' ( t ) d t .
a a
Thus if v e L
p
[ a , b ] and {cp } and [Y ] are two C.O.N, sequences of B.V., then
[cp}b [V)b
J v(t)dx(t ) = J v(t)dx(t )
a a
for x€D[a,b ] , and hence
Ccp)b [Y)b
j expCi J v(t)dx(t)]dx.(v) = J exp{i J v(t)ft(t)}dH(v) .
T
a
T
a
L 2 L 2
We now introduce the space S . We shall prove later that
SdS / S / S' 3 S" .
DEFINITIONS say FeS* = S*[L2] iff there exists a "eft such that
- « v b
(2.3) F(x) = J expCi E J v.(t)dx (t)}dn(v) .
L2 a
NOTE:It follows from the definitions of S and S and the properties of the
P.W.Z. integra l on D tha t S ^ S .
LEMMA 2 . 1 . If F e s and F i s given by (2-3) with H eft i t follows tha t H i s
uniquely determined by F .
PROOF:Since F e S , i t follows tha t F e S . Hence from Theorem 2.1 of [k]9 K eff? i s
uniquely determined by the equation
v b
F ( x ) « J exp{i S f v.(t)dx.(t)]d*(v ) .
v Ma
J
°
L 2
But since FeS , (2-3) holds for some ^ eft with the stronger equivalence " = "
and thus this K is uniquely defined. In the Banach algebra S we defined ||F|| = ||K||
and so for FeS , we define J|F||= ||H||.
LEMMA 2.2. The space S = S (L ) is a Banach algebra.
The proof is identical with the proof of Theorems 2.2 and 2.3 of [k] except that
" » " must be replaced by " = " .
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