It is useful t o consider on e o f the simples t bases , th e Haa r system , in
order to motivate man y aspect s of our representation . First , th e wa y th e
Haar system is organized serves as a model fo r some of the features of our
spanning sets. Fo r simplicity let us consider the one dimensional case. Thi s
system is generated by the functio n
(I i f O x ± ,
h{x) = { .
I - 1 i f \ x 1.
For an y tw o integer s v, k w e the n let h
k(x) = 2
- k) . The
functions h
v k
ar e calle d th e Haa r function s an d th e collectio n {h
u k
v, k e Z , is called th e Haa r system . Clearly , eac h h
u k
i s supported in
the dyadic interval I
v k
(k + 1)2"*"]. I t is not hard to see that the
Haar system forms an orthonormal basis for L (E). Le t us focus our atten-
tion on the unit interval / = I0
= [0, 1] and examine some of the features
of ho w an / e
i s represented by the Haar system. Fo r this purpose
we only need those h
v k
tha t are supported in / (w e also need to include
the characteristic functio n of this interval); thus the range of the indices is
v 0 and 0 k
- 1.
For any subinterval J c I let
fj =
be the mean of / ove r / . Als o put
2 , , - l
(0.5) / , ( * ) = £ / / Xj (*);
fv i s an approximation of / "a t the resolution 2~
." It follows immediately
from the Lebesgue differentiation theorem that lim^ ^ fv(x) = f(x) almos t
everywhere. Thus , for almost every x ,
(0.6) f(x) = fQ(x) + f){/„+l(*) - /„(*)}.
But a simple argument shows that
(0-7) /, +i-/.= Etf'V*)*,,*'
where (/ , g) denote s the usual inner product f fg. Thus ,
(0-8) / = // + £ E/.V fcA,.k.
i/=l A:= 0
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