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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