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Department of Mathematics and Statistics, Villanova University, Villanova, PA, USA
5.1 Definition and examples
For a given function f such that
, the Fourier transform of f is defined, for each real number ω, by



(5.1)
The idea behind this definition is that, for each value of ω, the value of
captures the component of f that has the frequency ω∕(2π) (and period 2π∕ω ).

Example 5.1.
The Fourier transform of a Gaussian. Let
, for some positive constant A > 0. Then we have



(5.2)
To prove this, we first need the following fact.
Lemma 5.2.
For A ≠ 0, we have
.

Proof.
Squaring the integral, we get
Taking square roots proves the lemma.

Now to compute the Fourier transform for this example. For each ω,
This establishes the result we were after. □

Observe that if we take
, then
and
, a constant multiple of f itself. In the language of linear algebra, this function f is an eigenvector, or eigenfunction, of the Fourier transform.



Additional examples are considered in the exercises. Let us look at some basic properties of the Fourier transform.
5.2 Properties and applications
Additivity. Because the integral of a sum of functions is equal to the sum of the integrals of the functions separately, it follows that

for all integrable functions f and g and every real number ω.

(5.3)
Constant multiples. Because the integral of c ⋅ f is equal to c times the integral of f, we see that

for all integrable functions f, all (complex) numbers c, and every real number ω.

(5.4)
Proposition 5.3.
The Fourier transform acts as a linear transformation on the space of all absolutely integrable functions. That is, for two such functions f and g and any constants α and β, we get that


(5.5)
Shifting/translation. For an integrable function f and fixed real number α, let
. (So the graph of g is the graph of f shifted or translated to the right by α units.) Then



(5.6)
Proof.
For each ω, we get


Since the graph of g is a simple translation of the graph of f, the magnitude of the component of g at any given frequency is the same as that of f. However, the components occur at different places in the two signals, so there is a phase shift or delay in the Fourier transform. Also, the fixed translation α encompasses more cycles of a wave at a higher frequency than at a lower one. (For instance, an interval of width α = 2π contains two cycles of the wave y = sin(2x), but only one cycle of the wave y = sin(x).) Therefore, the larger the value of α is relative to the wavelength 2π∕ω the larger will be the phase delay in the transform. That is, the phase delay in the transform is proportional to ω, which explains the factor of e −i ω α in the transform of the shifted function g.
Shifting/modulation. For a given function f and a fixed real number ω 0, let
. (So h “modulates” f by multiplying f by a periodic function of a fixed frequency ω 0∕2π.) Then



(5.7)
Proof.
For each ω, we get


These two shifting properties show that a translation of f results in a modulation of
while a modulation of f produces a translation of
.


Scaling. For a given function f and a fixed real number a ≠ 0, let ϕ(x) = f(ax). Then


(5.8)
Proof.
Assume that a > 0 for now. For each ω, we get


A similar argument applies when a < 0. (Why do we get a factor of 1∕ | a | in this case?)
Even and odd functions. A function f, defined on the real line, is even if
for every x. Similarly, a function g is odd if
for every x. For example, the cosine function is even while the sine function is odd.


Using Euler’s formula (4.4),
, we may write the Fourier transform of a suitable real-valued function f as



Now, if f is even, then, for fixed ω, the function
is odd, whence
. Thus, an even function has a real-valued Fourier transform.


Similarly, if f is odd, then
is also odd for each fixed ω. Thus,
. It follows that an odd function has a purely imaginary Fourier transform.


Transform of the complex conjugate. For a complex-number-valued function f defined on the real line
, the complex conjugate of f is the function
defined by




(5.9)
To uncover the relationship between the Fourier transform of
and that of f, let ω be an arbitrary real number. Then we have
This proves the following proposition.


Proposition 5.4.
For an integrable function f defined on the real line, and for every real number ω,


(5.10)
Example 5.5.
Let
2 sin(ω)∕ω. Now let ϕ(x) = f(ax), where a > 0. That is,
, which is the same as
. This agrees with the scaling result (5.8).



Example 5.6.
Let
. Now let
5.6), we get

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Example 5.7.
As an application of the shifting/modulation property (5.7), observe that
Thus, for any suitable f, take h(x) = f(x)cos(ω 0 x). That is,
It follows that




For a specific example, let
. With h(x) = f(x)cos(ω 0 x), we get

