Graphics Reference
In-Depth Information
Let f,g ΠL
2
([a,b]). Define the
inner product of f and g
, denoted by ·f,gÒ,
Definition.
by
b
<>=
() ()
Ú
fg
,
ftgtdt
.
a
Since this notation depends on the interval [a,b], we shall write ·,Ò
[a,b]
if there is any
confusion.
Fact 4.
·f,gÒ is an inner product on the vector space L
2
([a,b]) and the usual length
and distance function associated to an inner product agree with the norm || ||
2
and
distance function d
2
as defined above.
There is also an L
•
([a,b]) space of function but its definition is special and so we
treat it separately. Because it involves some often seen terminology, it is worth defin-
ing here. The general definition would apply to “measurable” functions and everything
would be defined up to a set of measure 0, but too avoid technicalities with measure
that have not been satisfactorily dealt with in this topic, we shall restrict out defini-
tions to continuous functions.
Definition.
Let [a,b] be an interval in
R
, either bounded or unbounded. Define the
L
•
space
, L
•
([a,b]), to be the set of bounded continuous functions on [a,b]. Its ele-
ments are called
L
•
functions
. The
L
•
norm
on L
•
([a,b]), denoted by || ||
•
, is defined
by
({}
|| ||
f
=
sup
f x
.
•
Œ
[
]
xab
,
If f,g ΠL
•
([a,b]), define the
L
•
distance between f and g
, denoted by d
•
(f,g), by
()
=-
dfg
,
||
fg
||
.
•
•
With these definitions everything we said earlier about the L
p
spaces holds here.
The function || ||
•
satisfies the four metric properties listed in Fact 2 above, the func-
tion d
•
(f,g) defines a metric on the equivalence classes of functions with respect to
the relation ~ , where
f
~
g
if and only if
||
f
-
g
||
=
0
•
and the resulting metric space is complete.
21.5
Fourier Series
This section presents the relevant definitions and some basic theorems. Proofs are
omitted.
We start with some additional motivation. Recall that every element
v
of a vector
space can be expressed as a linear combination of basis vectors
v
j
, that is,
