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/2 h(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) = f Q (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 fc A,. k . i/=l A:= 0
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