6
R.H.CAMERON andD.A. STORVICK
NOTATION:
A binary polygonal function in C[a,b] is one which is linear on each interval
[a+ (-i-Kb-a) , a +-^-(b-a) ] for j = l , . . . , 2 m .
The m binary polygonal approximation [x] tothefunction x€c[a,b] isdefinedby
[x]m(t)=x(t) when t-a+ MS^Sl ,
k = 0
l,...,2m
Wjt) islinear on [a+ US^lS! , a
+
^ ] , k -
1 ) 2
^».
DEFINITION:A functional F will be called continuous at x with respect tobinary
polygonal approximation if
limP([x] ) = F(x)
m
-»oo
where [x] isthe m binary polygonal approximation of x .
The Haar functions area C.O.N, seton [0,1] , andaredefined asfollows [11; p kk]:
Xv"'(t) =1 on [0,1]
0
x(1)(t)
H
0
,(«/
1 on [0,1/2)
-1 on (1/2,1]
0 at l/2 ,
J2 on [0,1/0
XW (t ) ={-72 on (lA,l/2]
0 elsewhere on [0,1]
, , J% [V2.3A)
x(2,(t)
A-Jz (3A,H
1 I 0 elsewhere on [0,1]
and in general for k = l , 2 , . . . , 2 ; n = l , 2 , . . .
f ^ / ? r2k-2 2k-lx
+
/
2 on
^ K L
-5T
x(k)(t) .
An /2k-l
" ^ on ( j
2k
n+1
2 , 2'
0 elsewhere on [0,1]
REMAR K 1: Let h(t) be a step-function on [a,b] and let f(t) be continuous on
[a,b] , and let f(t) = 0 when t = a and when t = b and whenever h(t) has a dis-
continuity at t . Then
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