which shows that the Haar system represents / a.e . (as well as in L 2(I));
moreover, each approximation of / a t the resolution 2~
appears as a par-
tial sum of the Haar expansion. Th e decompositions w e will develo p have
similar features and, in addition, the spanning sets consist of smooth func-
tions. Thes e set s can then be used to study space s in which smoothnes s is
intrinsic (fo r example , Sobole v or Lipschitz spaces) , whic h cannot be done
with the Haar system.
For thos e familia r wit h probabilit y theory , w e observe tha t th e above
sequence {f
i s one of the simples t example s o f a martingale. Le t
^ b e the a-algebr a o f subsets o f / generate d b y the intervals {/ j j,
k = 0, 1, ... , 2" - 1. The n {^ } form s an increasing family of cr-algebras.
Note that f
= E[f\^v] (th e expectation of / conditione d on ^ ); in partic-
ular fv i s i^-integrable, and the martingale property £[^
+ 1
|^ ] = fv holds .
We will not discuss further the interplay between martingale theory and har-
monic analysis , bu t we would like to point ou t th e influenc e o f martingale
theory in motivating the discrete representation formulae we will study.
There is a formula due to Calderon which relates Littlewood-Paley theory
to the construction of certain spanning set s as alluded to above. Ther e are
many variant s of the Caldero n formula . Her e w e wil l outlin e th e versio n
that uses the Poisson kernel since this version is most directly linked to the
classical Littlewood-Pale y theor y described above. Later , in §1, §2, and §3,
however, w e shall us e a version o f the Caldero n formul a tha t employ s a
compactly supported kernel in place of the Poisson kernel.
To describe the Calderon formul a w e begin by selecting a ° function
p whic h i s real, radial , wit h Supp p c B {(0) = {x R
\x\ 1}, no t
identically zero and satisfying / (p = 0. By normalizing (p we can assume
(0.9) / 0(tl)e~
= -l,
where 1 = (1, 0, ... , 0) and the Fourier transform is defined by
/(«) = / f{x)e-
This gives us the following Caldero n formula identity (cf. (0.2)) :
(0.10) f( X) = J" ^{t0-(p i*f)(y)}9t{x-y)dy4l,
where, as usual, tp t{x) = t cp{t x). Thi s follows formally from (0.9): since
Pt(£) = e~'^ , the Fourier transform of the right-hand side of (0.10) is
(0.11) / M)HZ\)e~
Then th e chang e o f variables { = t\£\ allow s us to use (0.9), sinc e 0 i s
radial, and yields (0.10).
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