Graphics Reference
In-Depth Information
•
Â
()
=
i
n
q
u
1,
q
a e
.
(21.9)
n
n
=-•
The amazing thing is that all of this sort of works. It led Fourier to what are now
called Fourier series expansions for functions. Although we may be tempted to con-
clude that we can express an arbitrary function f(q) in the form (21.9), this is
not
quite
correct. The problem is with the convergence of the infinite series. Theorems 21.5.3
and 21.5.7 will shortly carefully state what
is
correct.
This concludes our bird's eye view of what led up to Fourier series. See [Seel66]
for more details. There is one more topic to be dealt with before we are ready to deal
with the subject of Fourier series itself.
The L
p
Function Spaces
21.4
In this section we introduce some function spaces and operators that play an impor-
tant role in functional analysis. We will also encounter some common terminology
that is worth knowing. However, the real motivation for getting into this topic is that
this chapter deals with operators that map functions to other functions and it is very
useful to have some notation for describing the domains of these operators and also
to learn about some of the properties of these function spaces. For simplicity we
restrict our discussion to functions of one variable, but similar definitions and facts
hold for functions of more variables.
Definition.
Let [a,b] be an interval in
R
, either bounded or unbounded, and let 1 £
p <•. Define L
p
([a,b]), the
L
p
space
of functions on [a,b], as follows:
L
1
([a,b]) is the set of integrable functions on [a,b] .
L
p
([a,b]), 1 < p <•, is the set of functions f on [a,b] whose integral Ú
a
|f|
p
converges.
Functions in L
p
([a,b]) are called
L
p
functions
.
Note 1.
The definition of L
p
space assumed Riemann integration. If we had used the
Lebesgue integral, then we would not have had to make a special definition for p = 1.
Unfortunately, integrability does not imply absolute integrability in the case of the
Riemann integral, ergo the special case.
Note 2.
We do not have to restrict ourselves to real-valued functions. Everything we
say about L
p
([a,b]) would hold for complex-valued functions
if
p > 1. We could intro-
duce notation to distinguish between the two cases, but in what we will be doing here
it does not seem worthwhile to do so.
The function spaces L
p
([a,b]) have a great many interesting properties. Proofs of
the facts we list below can, for example, be found in [Nata61].
L
p
([a, b]) is an infinite dimensional vector space.
Fact 1.
The
L
p
norm
on L
p
([a,b]), denoted by || ||
p
, is defined by
Definition.
