6 INTRODUCTION where {y/ Q } (a s well as {p Q }) is fixed. I n fact, that y/ Q 's are all translates and dilates of a fixed i// e S? (similarl y for (p) . Note that (0.13) has many of the properties of an orthonormal decomposition for example, the coefficien t of y/ Q i n the expansion fo r / depend s linearly o n / . Th e function s y/ Q in (0.13 ) ar e "almost orthogonal " i n som e sense , but , i n general , the y are not orthogonal. W e call th e map / {(/, (p Q )} th e " ^-transform." Th e identity (0.13 ) i s derived fro m a variant o f th e Caldero n formula , usin g a different discretizatio n procedure from the one leading to equality (0.12). Finally, in the one dimensional case, we will show how one can construct an orthonormal set of functions {(p vk ) where (as in (0.13)) fo r any two in- tegers v an d k, 9 vk {x) = 2 v '2(p{2ux-k), wit h g a ° functio n decaying rapidly at ±oo . Thi s will give us an example of a wavelet basis it has the same "organization" as the Haar system and, in addition, consists of smooth functions. A s is the case for the Haar system, this set is an orthonormal basis for L however, unlike the Haar-system it serves also as a basis for the other spaces we will have introduced. A wavelet basis inR * ca n be constructe d from thi s one by forming appropriat e tenso r products. A s we develop this material, w e will present som e o f it s mathematical application s (fo r exam- ple, to the stud y of operators) in addition, i n the last fe w chapter s of thi s monograph we will describe some applications to science and engineering.
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