Characterization of Periodic Eigenfunctions of the Fourier Transform Operator ()
1. Introduction
In this paper, we will study certain generalizations of the Dirac comb (or III functional, see [1])
(1)
where 
is the Dirac functional. We work within the context of the Schwartz theory of distributions [2] as developed in [1,3-7]. For purposes of manipulation we use “function” notation for
, 
and related functionals. Various useful proprieties of 
and 
are developed in [1,3-5].
The 
functional is used in the study of sampling, periodization, etc., see [1,4,5]. We will illustrate this process using a notation that can be generalized to an n-dimensional setting. Let 
with
, and let
. We define the lattice
and the corresponding 
-periodic Dirac comb
(2)
The Fourier transform of the 
-periodic Dirac comb is
(3)
Let 
be any univariate distribution with compact support. We can periodize 
by writing
(4)
where 
represents the convolution product, to obtain the weakly convergent Fourier series
(5)
We observe that 
has support at the points 
of the lattice
, while the Fourier transform 
has support at the points
of the lattice 
It follows that
if and only if
i.e., if and only if
(6)
Let 
be the Fourier transform operator on the space of tempered distributions. It is well known [1,4,5], that 
is linear and that
(7)
where 
denotes the identity operator on the space of tempered distributions. We are interested in tempered distributions 
such that
(8)
where 
is a scalar. Any distribution f that satisfies (8), and that we will call eigenfunction of
, must also satisfy the following equation
(9)
due to the linearity of the operator
. When
, then
. Thus the eigenvalues of the operator 
are
.
Eigenvectors of 
We first consider the eigenvectors of the discrete Fourier transform operator 
since, as we will see later, they can be used to construct all periodic eigenfunctions of the Fourier transform operator 
[8,9].
Definition 1. Let 
. The matrix
, is said to be the discrete Fourier transform operator.
It is easy to verify the operator identity
where
is the reflection operator. It is easy to verify
where 
is the 
identity matrix. In this way we see that if
then
so 
must take one of the values
.
Let 
be the multiplicity of the eigenvalue
of 
and let
(10)
be orthonormal eigenvectors of 
corresponding to the eigenvalue
Example 1. 
The matrix
has the eigenvalues
, 
with corresponding eigenvectors
We normalize these vectors to obtain
2. The Main Results
A generalized function
, is said to be an eigenfunction of the Fourier transform operator 
if
For
. We would like to characterize all periodic eigenfunctions f of the Fourier transform operator
, i.e.,
within the context of 1,2,3 dimensions.
2.1. Periodic Eigenfunctions of 
or 
Let 
be a p-periodic generalized function on
, 
, and assume that
where 
and
. The 2-periodic function
is such an eigenfunction, constructed from the eigenvector 
of
. We will now characterize all such periodic eigenfunctions.
Since 
is p-periodic, 
is represented by its weakly convergent Fourier series
(11)
We Fourier transform term by term to obtain the weakly convergent series
(12)
for the Fourier transform of
. Now since 
and
, 
must also be pperiodic with
We recognize this as the Fourier transform of
We define
and write
(13)
Now if the term
appears in the sum (13) then (since 
is p-periodic)
must also appear. Thus
for some integer
. It follows that
i.e,,
and
thus
for some
, and since 
is N-periodic, we can use (13) to write
(14)
where
is the inverse Fourier transform of the N-periodic sequence of Fourier coefficients
. Since 
we can use (12), (14) to see that
i.e., that 
is an eigenvector of the discrete Fourier transform operator 
associated with the eigenvalue
. In this way we prove the following Theorem 1. Let the generalized function 
on 
be a 
-periodic eigenfunction of the Fourier transform operator 
with eigenvalue
, or
. Then 
for some integer 
and 
has the representation
(15)
where 
is an eigenvector of the discrete Fourier transform operator 
with
Example 2. When 
we obtain the corresponding 1-periodic
with
Of course, this particular result is well known, see [1]. Our argument shows that a periodic eigenfunction of the Fourier transform operator that has one singular point per unit cell must be a scalar multiple of the Dirac comb
.
Example 3. When
, we obtain the 
-periodic eigenfunctions
and
from the eigenvectors 
and 
for
. It is easy to verify that
Characterization of periodic eigenfunctions of 
on 
Let 
be a bivariate generalized function and assume that 
is an eigenfunction of
, i.e.,
with 
or
, (and
). Assume further that 
is 
-periodic, i.e.,
Here 
are linearly independent vectors in
.
We simplify the analysis by rotating the coordinate system as necessary so as to place a shortest vector from the lattice 
along the positive x-axis. We can and do further assume with no loss of generality that 
have the form
where
(16)
(17)
(18)
(19)
The dual vectors then have the representation
and
has the Fourier transform
where
. Now since 
is 
-periodic, 
can be represented by the weakly convergent Fourier series
(20)
We Fourier transform the series (20) to obtain the weakly convergent series
(21)
From (21), we see that the support of 
lies on the lattice 
and since
, 
must also be 
-periodic so we can write
(22)
where
is a primitive unit cell associated with the lattice
, where 
are affine coordinates, and 
is the bivariate convolution product. Using the bivariate inverse Fourier transform, we see that
We define
(23)
and write
(24)
Now 
is 
-periodic, so if 
for some integers
, then the term
equals the term
and the term
equals the term
for some integers
. From the supports of these 
-functions we see that
i.e.,
for some 
. Likewise, we see in turn that
for some
, and analogously
Finally,
for some
. Using these expressions we can now write
where, in view of (16)-(19)
and
From (21), (23) we also have
(25)
(26)
We will now consider separately the cases
.
Case 
When 
the vectors 
are orthogonal and 
has the corresponding periods
along the x-axis and y-axis, respectively. The function 
is represented by the synthesis equation
(27)
and by using (24) and (26), in turn we write
In this way we conclude that
(28)
Thus 
must be an eigenvector of the bivariate discrete Fourier transform 
associated with the eigenvalue
, (
, or
). Since 
is an 
-periodic sequence of complex numbers, we can write
Case 
We observe that
Since 
is 
-periodic, then 
is also 
-periodic. Thus 
has the periods
along the x-axis and the y-axis, respectively, a situation covered by the analysis from the 
case. In this way we prove Theorem 2. Let the generalized function 
on 
be an 
-periodic eigenfunction of the Fourier transform operator 
with eigenvalue
, or
. Assume that the linearly independent periods 
from 
have been chosen as small as possible subject to the constraint that
. Then there are positive integers 
such that
and there is a nonnegative integer 
such that 
is orthogonal to
with
The generalized function 
is 
-periodic and there is an orthogonal transformation 
such that
is 
-periodic with the representation
Here 
is an eigenfunction of 
with
for 
Note that the 
normalized eigenfunctions 
denoted by
(29)
with 
of 
serve as an orthonormal basis for the 
dimensional space 
of 
-periodic discrete real valued functions. Here (29) has the corresponding eigenvalue
Theorem 3. Let the generalized function 
on 
be an 
-periodic eigenfunction of the Fourier transform operator 
with eigenvalue
, or
. Assume that the linearly independent periods 
from 
have been chosen as small as possible subject to the constraint that 
. Then there are positive integers 
such that
and there are nonnegative integers
such that
,
and
are pairwisely orthogonal with
where
The generalized function 
is 
-periodic, and there is an orthogonal transformation 
such that
is
-periodic with the representation
(30)
Here
where
for
and
2.2. Some Quasiperiodic Eigenfunctions of the Fourier Transform Operator on 
In this section we will construct some quasiperiodic eigenfunctions of the Fourier transform operator. A generalized function 
is said to be quasiperiodic if the Fourier transform 
is a weighted sum of Dirac 
functionals with isolated support [10].
Lemma 1 Let 
be linearly independent vectors in
. If
and 
is distinct from
, then
(31)
(32)
are eigenfunctions of the Fourier transform operator 
associated with
, respectively.
Quasiperiodic eigenfunctions of 
on 
with m-fold rotational symmetry.
Let
(33)
for some 
and let
(34)
where 
be the vertices of a regular 
with center at the origin. The parameter 
has been chosen so that
for each
. Thus
(with
) where
is a quarter turn rotation. We will use this fact to generate quasiperiodic eigenfunctions of 
on 
with rotational symmetry.
We will now construct a family of quasiperiodic eigenfunctions of 
that have rotational symmetry. Let
, and 
be given by (34), let 
be given by (33), and let
(35)
and
(36)
(with
). Figures 1 and 2 show representations of such eigenfunctions with 
and 
respectively. Filled circles correspond to negatively scaled Dirac
’s, and unfilled circles correspond to positively scaled Dirac
’s. The radius of each circle is proportional to the square root of the modulus of the scale factor for the corresponding
. By construction,
3. Representation of Some Quasiperiodic Eigenfunctions