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 C° ° 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.