Graphics Reference
In-Depth Information
n
Â
a
jj
j
v
=
v
.
=
1
In general, to determine the coefficients a
j
one would have to solve some linear equa-
tions, but if the
v
j
are an orthonormal basis, then
n
Â
1
(
)
=
vv
•
=
a
v v
•
a
.
k
j
j
k
k
j
=
If the
v
j
are only an orthogonal basis, then
vv
vv
•
•
k
a
k
=
.
k
k
In the case of infinite dimensional vector spaces we would get infinite sums, so
that one has to be a little careful here. However, one can make sense out of such sums
by introducing the notion of convergence. For example, polynomials of degree n such
as
n
Â
0
j
()
=
px
ax
j
j
=
form an n-dimensional vector space with basis the polynomials x
j
. As n goes to infin-
ity, one can ask which functions can be represented by (convergent) power series.
The trigonometric functions sin nt , cos nt , and e
int
are periodic L
2
functions of
period 2p. They are also orthogonal sets of functions, but not of unit length (with
respect to the L
2
inner product).
21.5.1
Theorem.
Let I be any interval of length 2p.
(1) ·sin mt,sin ntÒ=0 fm π n,
=p if m = n.
(2) ·cos mt,cos ntÒ=0 fm π n,
=p if
m = n .
(3) ·sin mt,cos ntÒ=0.
(4) ·e
imt
,e
int
Ò=0 if
m π n,
= 2p
if
m = n.
Proof.
See [Spie69].
Therefore,
if
one could represent a function f(t) by a series in the form
•
Â
i
nt
ae
,
(21.10)
n
n
=-•
then, assuming one can do the integration on a term-by-term basis, one could, using
the above argument, solve for a
n
, namely,
