4 INTRODUCTION 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~ v 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 l/ }(^=l 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 v = 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 n : \x\ 1} , no t identically zero and satisfying / (p = 0. By normalizing (p we can assume rOO (0.9) / 0(tl)e~ t dt = -l, Jo where 1 = (1, 0, ... , 0) and the Fourier transform is defined by /(«) = / f{x)e- ix *dx. 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 /*oo (0.11) / M)HZ\)e~ tm m)dt. Jo 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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