A SIMPLE DEFINITION OF THE FEYNMAN INTEGRAL* WITH APPLICATIONS 7
b
J h(t)df(t)=0.
a
b b
For by integration by parts J h(t)df(t) = - J f(t)dh(t)=0 , since f is zero whenever
a a
h has a jump.
REMARK 2: Let h(t) be a step-function on [a,b] and let f(t) and g(t) be
continuous on [a,b] . Let f(t)=g(t) when t = a and when t=b and whenever h(t)
has a discontinuity at t . Then
b b
J h(t)df(t) = J h(t)dg(t) .
a a
REMARK 3: Let h(t) be a step-function on [0,1] whose discontinuities occur only
at the points o, -A- , -1-,-i- ,...,£- . Then
1 m-1 2n , 1
(2.U) h(t) = I h(s)ds + £ £ X W (t) J XW(s)h(s)ds
0 n=0 k=l n O n
for all t on [0,1] except at the points t = 0 ,
0 n= 0 k= l n O n
2m 2m '•••' 2m *
This can be easily seen because the Haar functions are a C.O.N, set. Consequently
if all the terms of the orthogonal development of h are included, it converges in the
L? mean to h . But all the non-vanishing terms in the development are included in the
(k)
right member of (2A),since h is orthogonal to X when nm . Then (2.U) holds
n
for almost all t in [a,b] , and since both members of (2.U) are continuous except at
, k=0,l,...,2 , it follows that (2.U) holds on [0,1] except at these points.
LEMMA. 2.3. Let veL^a,!)] . If
b
(2.5) F(x) = / v(t)d x(t) ,
a
then for s-almost every xec[a,b] and every xeD[a,b] , F(x) is continuous with
respect to binary polygonal approximation (continuous B.P.A.).
PROOF Case I. Let a = 0 and b = l and assume that the P.W.Z. integral is given in
terms of the Haar Functions
Let
1 m-1 2n ,,
N
1
v(t) ^ J
v
(s)ds + E Z X(k)(t) J X(k)(s)v(s)ds ,
0 n=0 k=l n O n
m-1 2 /, . 1 , .
xm(t) = x(l) + E S X
W
(t) I
XW(s)dx(s)
,
then
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