Graphics Reference
In-Depth Information
If f Œ L 2 ([-p,p]) and if ·f,e int Ò=0 for all n π 0, then f = 0.
21.5.8
Theorem.
Proof.
See [Nata61].
Rather than using the functions e int one often chooses normalized functions for
a basis for L 2 ([-p,p]), such as, for example, the functions e 2pint . Orthonormal bases
are, after all, the nicest types of bases. Additionally, one does not have to restrict
oneself to these exponential type functions. Other variations of these functions are
often used as an orthonormal basis. The specific choice of basis is dictated by what
is most convenient for a particular application.
This is as far as we shall take the subject of Fourier series here.
21.6
The Fourier Transform
Fourier series have to do with representing functions (periodic ones to be precise) as
series. The Fourier transform, on the other hand, attempts to represent functions as
integrals. One advantage Fourier integrals have is that they can represent fairly arbi-
trary functions, not just periodic ones. Since the domain of functions may now be all
of R , the standard trick for turning a function into a periodic one would not work.
Our discussion of the Fourier transform will concentrate on functions of one
variable and only briefly mention the two-variable case.
Definition.
If f, G: R Æ R , then
Ú
() =
() -
2p i
ux
Fu
fxe
dx
(21.16)
-•
is called the Fourier transform of f(x) and
Ú
() =
()
2p i
ux
gx
Gue
du
-•
is called the inverse Fourier transform of G . The functions F and g will be denoted by
FT(f) and FT -1 (G), respectively.
Note 1. There is no uniform agreement in the literature as to what is called the
Fourier transform, although all the various definitions have the form
Ú
() =
()
i
bux
Fu
a
fxe
dx
-•
for suitable constants a and b.
Note 2. The formula for the Fourier transform should remind the reader of the
definition of the Fourier coefficients of a function. The similarity is not accidental.
Actually we could have also used the Laplace equation to motivate the definition. See
[Seel66]. In this case, rather than the region being a disk, we would be dealing with
a halfplane that has a function specified along its edge. If we were to work through
Search WWH ::




Custom Search