2 Introduction
are unique . Th e vector s i n B ar e no t linearl y independent ,
so a linea r expansio n woul d no t b e unique .
Th e vector s i n A al l hav e lengt h 1. Th e vector s i n B al l
have lengt h y | .
Th e vector s i n A ar e orthogona l (i.e. , thei r do t produc t i s
zero). Th e vector s i n B ar e no t orthogonal .
Th e coefficient s {ci}?
=1
fo r a linea r expansio n i n th e se -
quence A ca n b e compute d easil y usin g th e do t product .
Specifically, le t x = c\e\ + c^^ wher e th e vector s i n A ar e
denoted and e2 - Th e coefficient s ar e compute d b y th e
dot product s ci x e\ fo r i = 1, 2. Thi s i s a property o f or-
thonormal bases . I f we check, however , w e find tha t on e way
to writ e a : as a linea r combinatio n o f th e vector s i n B ca n
be foun d usin g th e coefficient s forme d b y takin g th e sam e
dot products . I n othe r words , i f we denote th e vector s o f B
by / i , /2, h
a n
d give n som e x £ R
2
w e tak e di = x fi fo r
i = 1, 2,3, then x = di/ i + ^2/2 + ^3/3 - Eve n withou t bein g
orthogonal o r linearl y independent , th e se t B retain s on e of
the extremel y usefu l feature s o f a n orthonorma l basis .
Bot h A an d B satisf y a propert y whic h i s know n a s Parse -
val's identity fo r orthonorma l bases . I n th e cas e of sequenc e
A, thi s i s a versio n o f th e Pythagorea n Theorem . Le t \\x\\
denote th e length , o r norm , o f th e vecto r x. Then ,
E $
= £#
Again, w e se e tha t eve n thoug h B i s no t eve n a basis , an d
certainly no t a n orthonorma l basi s fo r R 2, i t maintain s a n
important characteristi c o f a n ONB .
Both sequence s A an d B ar e example s o f a particula r typ e o f
frame, calle d a Parseva l frame , fo r th e vecto r spac e R
2.
Sequenc e B
demonstrates tha t man y o f the properties of an orthonormal basi s ca n
be achieve d b y nonbases . Thi s i s exactl y th e motivatio n behin d th e
study o f frames . Frame s ar e mor e genera l tha n orthonorma l bases ,
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