CHAPTER 2
Wavelets an d squar e function s
The purpose of this lecture is to understand function s o n the real line by break -
ing them up into pieces which have the attributes of position and scale. Thes e pieces
will b e parameterize d b y intervals , whic h ar e natura l object s havin g positio n an d
scale: th e center of an interval determines the position an d the length o f an interva l
determines th e scale . Th e basi c grou p actin g o n positio n an d scal e i s th e grou p
spanned b y translation s an d dilation s o f th e rea l line , an d thi s grou p wil l pla y a n
ubiquitous - while a t time s implici t - role i n ou r discussions .
Naturally, w e will parameterize th e piece s o f a function b y th e interval s wher e
the piece s ar e localized . W e shall introduc e a meanin g fo r th e notio n " a functio n
localized t o a n interval " tha t allow s fo r mor e genera l function s tha n th e obviou s
examples o f function s supporte d o n th e interval . W e wil l allo w th e functio n t o
have "tails " outsid e th e specifie d interval . On e o f th e clea r advantage s o f thi s
more general notio n o f localization i s that i t allow s both a function an d it s Fourie r
transform t o b e localized .
We cal l a functio n fi rapidly decayin g o f order s N an d K i f ther e exist s a
constant C suc h tha t th e functio n j satisfies th e estimat e
\p{x)\c{\ + \x\r
N
and al l the derivative s o f / u p t o orde r K satisf y th e sam e estimat e
\4ik)(x)\C(l + \x\)- N
We shal l no t b e specifi c abou t th e value s o f the parameter s T V and K\ admit -
tedly a sourc e o f discomfor t fo r beginner s i n th e subject . Bu t i t wil l fre e u s fro m
many less essential details and hopefully giv e a clearer view on the main points. An y
argument w e will engage i n will assume N an d K t o be large enough. Occasionall y
we wil l modif y bum p functions , a procedur e whic h ofte n lower s th e constant s N
and if , bu t thi s wil l b e n o har m i f N an d K ar e larg e enoug h t o begi n with .
Any Schwart z functio n p G S(R) provide s a n exampl e o f a rapidl y decayin g
function.
As th e valu e o f th e constan t C i s lef t undetermine d i n th e abov e definition ,
any translat e o r rescale d translat e o f a rapidl y decayin g functio n i s agai n rapidl y
decaying. T o have a meaningful concep t o f position an d scal e of a rapidly decayin g
function, on e need s t o fix th e constan t C an d translat e an d rescal e th e deca y es -
timate fo r th e function . Thi s lead s t o th e notio n o f bump functio n adapte d t o a n
interval, a s in [38] .
A bump functio n adapte d t o a n interva l / = [a , b) i s a function satisfyin g
|0(x)|C|J|-
1
/
2
(l
+
^ = ^ i ) "
J V
5
Previous Page Next Page