Fourier's integral theorem expresses a linear transformation, say , when applied to the space of square integrable functions. From this perspective one has

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Furthermore, this transformation is one-to-one because Fourier's theorem says that its inverse is given by

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That maps square integrable functions into square integrable functions is verified by the following computation, which gives rise to Parseval's identity: For we have

Thus we obtain Parseval's identity (= ``completeness relation'', see Eq.(1.17) on page ). The only proviso is (a) that the function be square-inegrable and (b) that its Fourier transform be given by the Fourier transform integral.

**Remark 1:** The fact that the Fourier transform
is a one-to-one linear transformation from the linear space
to the linear space
is summarized by saying that the Fourier transform is an ``isomorphism''.

**Remark 2:**
The line of reasoning leading to Parseval's identity also leads to

whenever .

**Remark 3:**
The above two remarks imply that the
Fourier transform is a *unitary* transformation in
. Unitary transformations are ``isometries''
because they preserve lengths and inner products. One says, therefore, that the space of
functions defined on the spatial domain is ``*isometric*'' to the
space of functions defined on the Fourier domain. Thus the Fourier
transform operator is a linear isometric mapping. This fact is
depicted by Figure 2.6

Note, however, that even though the Fourier transform and its inverse,

take square integrable functions into square integrable functions, the ``basis elements'' are

Thus they do not belong to . Nevertheless linear combinations such as Eq.(2.46) are square integrable, and that is what counts.

The Fourier transform, call it , is a linear one-to-one operator from the space of square-integrable functions onto itself. Indeed,

Note that here and are viewed as points on the common domain of and .

- (a)
- Consider the linear operator
and its eigenvalue equation
- (b)
- Identify the operator
? What are its eigenvalues?
- (c)
- What are the eigenvalues of

Recall that the Fourier
transform
is a linear one-to-one transformation from
onto itself.

Let
be an element of
.

Let
, the Fourier transform of
, be defined by

It is clear that

are square-integable functions, i.e. elements of .

Consider the SUBSPACE spanned by these vectors, namely

- (a)
- Show that
is finite dimensional.

What is ?

(Hint: Compute , etc. in terms of ) - (b)
- Exhibit a basis for .
- (c)
- It is evident that
is a (unitary) transformation on
.

Find the representation matrix of , relative to the basis found in part b). - (d)
- Find the secular determinant, the eigenvalues and the corresponding eigenvectors of .
- (e)
- For , exhibit an alternative basis which consists entirely of eigenvectors of , each one labelled by its respective eigenvalue.
- (f)
- What can you say about the eigenvalues of viewed as a transformation on as compared to which acts on a finite-dimensional vector space?

Suppose we define for a square-integrable function and its Fourier transform

the

and the

- (a)
- Show that
*independent*of the function , and determine the value of this - (b)
- Determine the equivalent width and the equivalent Fourier width for
the unnormalized Gaussian

Consider the *auto-correlation*
^{22}

of the function whose Fourier transform is

Compute the Fourier transform of the auto correlation function and thereby show that it equals the ``spectral intensity'' (a.k.a. power spectrum) of whenever is a real-valued function. This equality is known as the

Consider a linear time-invariant system. Assume its response to a specific driving force, say , can be written as

Here , the ``unit impulse response' (a.k.a. ``Green's function'', as developed in CHAPTER 4 and used in Section 4.2.1), is a function which characterizes the system completely. The system is said to be

(Here the bar means complex conjugate.) In that case the system response to a generic forcing function is

A system characterized by such a unit impulse response is called a

- (a)
- Compute the total energy
- (b)
- Consider the family of forcing functions

- ...\space
^{22} - Not to be confused with the convolution integral Eq.(2.56) on page