INTRODUCTION

3

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

v

k(x) = 2

u/2h(2vx

- 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

=

\kl~v,

(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

= [0, 1] and examine some of the features

of ho w an / e

L{(I)

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

2V

- 1.

For any subinterval J c I let

fj =

\T\L

f{x)dx

be the mean of / ove r / . Als o put

2 , , - l

(0.5) / , ( * ) = £ / / Xj (*);

fc=0

fv i s an approximation of / "a t the resolution 2~

v

." 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(*) - /„(*)}.

i/=0

But a simple argument shows that

(0-7) /, +i-/.= Etf'V*)*,,*'

A:=0

where (/ , g) denote s the usual inner product f fg. Thus ,

(0-8) / = // + £ E/.V fcA,.k.

i/=l A:= 0