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
![](https://www.scirp.org/html/5-1100287\166eae1b-440a-419a-b3e1-4e6c8382646d.jpg)
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
![](https://www.scirp.org/html/5-1100287\b2e38744-58c6-414a-8c7d-d7ebcb0c147d.jpg)
if and only if
![](https://www.scirp.org/html/5-1100287\dcbe5afc-ad10-487f-af46-ec89d8b355da.jpg)
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 ![](https://www.scirp.org/html/5-1100287\6bca7f40-7700-41a6-b499-a936cf677746.jpg)
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
![](https://www.scirp.org/html/5-1100287\6d77a871-2fad-4bb2-95b4-96aac7a0915c.jpg)
, is said to be the discrete Fourier transform operator.
It is easy to verify the operator identity
![](https://www.scirp.org/html/5-1100287\2e3b0956-2d13-4add-bf5e-acac8b158814.jpg)
where
![](https://www.scirp.org/html/5-1100287\15ecad07-f27d-4cec-b0c8-5543b3520734.jpg)
is the reflection operator. It is easy to verify
![](https://www.scirp.org/html/5-1100287\3ea3f8e3-d98d-40c8-af11-69735d9915b3.jpg)
where
is the
identity matrix. In this way we see that if
![](https://www.scirp.org/html/5-1100287\54703d37-559c-45fb-813a-e58f4f419f7d.jpg)
then
![](https://www.scirp.org/html/5-1100287\7ba058ca-7197-4af3-b8bc-82a36ec6d334.jpg)
so
must take one of the values
.
Let
be the multiplicity of the eigenvalue
![](https://www.scirp.org/html/5-1100287\f7862155-6765-4274-92d7-1b83d7e12e4d.jpg)
of
and let
(10)
be orthonormal eigenvectors of
corresponding to the eigenvalue
![](https://www.scirp.org/html/5-1100287\c99e80bf-722a-40de-b34f-04f5064c4c73.jpg)
Example 1. ![](https://www.scirp.org/html/5-1100287\bff94875-bfcd-4808-a47f-752d9f1b6895.jpg)
The matrix
![](https://www.scirp.org/html/5-1100287\9cc0689b-0918-48a4-956a-73942afe88c6.jpg)
has the eigenvalues
,
with corresponding eigenvectors
![](https://www.scirp.org/html/5-1100287\135dae6e-3e13-4d4f-beb4-63bfff951a12.jpg)
We normalize these vectors to obtain
![](https://www.scirp.org/html/5-1100287\fc9ccb94-3f01-4221-96d6-926c93c71466.jpg)
![](https://www.scirp.org/html/5-1100287\fd642b96-b9c2-414a-8971-a4d37da35d68.jpg)
2. The Main Results
A generalized function
, is said to be an eigenfunction of the Fourier transform operator
if
![](https://www.scirp.org/html/5-1100287\7575c762-5962-4b27-ae95-ca51c1d9758f.jpg)
For
. We would like to characterize all periodic eigenfunctions f of the Fourier transform operator
, i.e.,
![](https://www.scirp.org/html/5-1100287\d9cbfd33-a7a5-402b-86a3-da0e3fbd6fa9.jpg)
within the context of 1,2,3 dimensions.
2.1. Periodic Eigenfunctions of
or ![](https://www.scirp.org/html/5-1100287\91d2493b-64e6-49c5-9a4d-ebfa3876d244.jpg)
Let
be a p-periodic generalized function on
,
, and assume that
![](https://www.scirp.org/html/5-1100287\ac5db952-6c5c-494c-be47-d148b171a4c1.jpg)
where
and
. The 2-periodic function
![](https://www.scirp.org/html/5-1100287\77e24508-9a45-4daa-a7a0-f358d5bb582d.jpg)
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
![](https://www.scirp.org/html/5-1100287\4677fb5e-bf53-42af-9a38-e6e98a12f1a8.jpg)
We recognize this as the Fourier transform of
![](https://www.scirp.org/html/5-1100287\165e8c83-8318-428a-a508-cd6ba0549d5c.jpg)
We define
![](https://www.scirp.org/html/5-1100287\ba9ba17b-4e9c-4dee-ac70-364265102f18.jpg)
and write
(13)
Now if the term
![](https://www.scirp.org/html/5-1100287\f736f579-2b16-4bc3-b06c-b98af93d0c46.jpg)
appears in the sum (13) then (since
is p-periodic)
![](https://www.scirp.org/html/5-1100287\4920ae57-f393-4757-81a5-fa5dab97a0ec.jpg)
must also appear. Thus
![](https://www.scirp.org/html/5-1100287\cbc2fcbf-7cc2-401c-86d4-e286b546c063.jpg)
for some integer
. It follows that
![](https://www.scirp.org/html/5-1100287\ac9705c3-8662-4dc4-a5af-041e973e1ee3.jpg)
i.e,,
![](https://www.scirp.org/html/5-1100287\1f63d58b-e9c8-4916-a697-65b63f238475.jpg)
and
![](https://www.scirp.org/html/5-1100287\c6860cba-d4b4-4ad7-b4a0-74ce33430404.jpg)
thus
![](https://www.scirp.org/html/5-1100287\17772c7d-3558-4650-be9a-ef3d2b77ca3f.jpg)
for some
, and since
is N-periodic, we can use (13) to write
(14)
where
![](https://www.scirp.org/html/5-1100287\5cb55344-d045-40c7-ae2a-4542a2357a06.jpg)
is the inverse Fourier transform of the N-periodic sequence of Fourier coefficients
. Since
we can use (12), (14) to see that
![](https://www.scirp.org/html/5-1100287\899f10b6-de43-462a-a94e-6fb17efe58f3.jpg)
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
![](https://www.scirp.org/html/5-1100287\c6803898-1b89-427a-ae7b-5b39a4e18666.jpg)
Example 2. When
we obtain the corresponding 1-periodic
![](https://www.scirp.org/html/5-1100287\80bf3355-90a9-41a2-b7f1-69596857c9e3.jpg)
with
![](https://www.scirp.org/html/5-1100287\661d7eff-9c72-4ef6-82f3-cc1a48f0f495.jpg)
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
![](https://www.scirp.org/html/5-1100287\dd2c9dee-e93c-4719-a342-b2a08673e7ac.jpg)
and
![](https://www.scirp.org/html/5-1100287\4e69b279-4256-442f-aa64-472a3dc4660f.jpg)
from the eigenvectors
and
for
. It is easy to verify that
![](https://www.scirp.org/html/5-1100287\4a0691e2-235b-46a2-923c-449212c40e04.jpg)
Characterization of periodic eigenfunctions of
on ![](https://www.scirp.org/html/5-1100287\03963234-4916-4ec0-bdc5-26e974bb8ba2.jpg)
Let
be a bivariate generalized function and assume that
is an eigenfunction of
, i.e.,
![](https://www.scirp.org/html/5-1100287\ca696fec-75d2-4d2a-aed0-d6f4034e0e05.jpg)
with
or
, (and
). Assume further that
is
-periodic, i.e.,
![](https://www.scirp.org/html/5-1100287\294b5ea0-cd57-445b-970b-ff741a0095f6.jpg)
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
![](https://www.scirp.org/html/5-1100287\2cf0f3b9-c647-4e37-aa8e-018a1f29f02a.jpg)
where
(16)
(17)
(18)
(19)
The dual vectors then have the representation
![](https://www.scirp.org/html/5-1100287\10d37333-2b2e-42b3-b2d6-d683de3a7e3d.jpg)
and
![](https://www.scirp.org/html/5-1100287\9b84fc6a-996f-4545-add6-66cf599da911.jpg)
has the Fourier transform
![](https://www.scirp.org/html/5-1100287\1b10b430-17e6-46fc-8bd1-78598b631681.jpg)
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
![](https://www.scirp.org/html/5-1100287\e37820e4-abf0-4452-8281-4b5c9662103e.jpg)
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
![](https://www.scirp.org/html/5-1100287\4c507a9c-b209-43e9-bf28-6d64b9a5fae9.jpg)
We define
(23)
and write
(24)
Now
is
-periodic, so if
for some integers
, then the term
![](https://www.scirp.org/html/5-1100287\e4d93f57-a082-42c6-b92c-57ccb3ecb9aa.jpg)
equals the term
![](https://www.scirp.org/html/5-1100287\bd5aa148-74da-4e7f-89a8-934c0900c46d.jpg)
and the term
![](https://www.scirp.org/html/5-1100287\e6c9a606-8700-4499-8bbb-19a2eb2490ae.jpg)
equals the term
![](https://www.scirp.org/html/5-1100287\e4756e92-e3e3-45ba-bd67-340c4d0e3dcc.jpg)
for some integers
. From the supports of these
-functions we see that
![](https://www.scirp.org/html/5-1100287\8df9ada5-e9db-4e83-8263-d09f4e939199.jpg)
i.e.,
![](https://www.scirp.org/html/5-1100287\389791ca-21af-4d29-bf8a-131c570754d5.jpg)
for some
. Likewise, we see in turn that
![](https://www.scirp.org/html/5-1100287\0b3f2d14-0511-4fb1-bc1e-5e56b9221986.jpg)
for some
, and analogously
![](https://www.scirp.org/html/5-1100287\38852548-6b6d-4a43-a8c9-8bc2c024bc63.jpg)
Finally,
![](https://www.scirp.org/html/5-1100287\1ccde264-515d-4300-aa2f-5b46b98ce92b.jpg)
for some
. Using these expressions we can now write
![](https://www.scirp.org/html/5-1100287\62b9d13b-7818-483d-86cd-7de00855ddc4.jpg)
where, in view of (16)-(19)
![](https://www.scirp.org/html/5-1100287\96e934da-133d-4f82-854f-ac45d119b46f.jpg)
and
![](https://www.scirp.org/html/5-1100287\57d86122-c707-4804-923d-0f7272b594e8.jpg)
From (21), (23) we also have
(25)
(26)
We will now consider separately the cases
.
Case ![](https://www.scirp.org/html/5-1100287\7b05d418-28f3-413d-81f8-f2e45a4e023e.jpg)
When
the vectors
are orthogonal and
has the corresponding periods
![](https://www.scirp.org/html/5-1100287\5be8e628-db39-42d9-9556-c57d2dc22cfb.jpg)
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
![](https://www.scirp.org/html/5-1100287\5a1d018e-09c1-468e-a243-b2263ae15384.jpg)
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
![](https://www.scirp.org/html/5-1100287\bd919045-76aa-44d4-b0dc-4d7fc2027847.jpg)
Case ![](https://www.scirp.org/html/5-1100287\15ff616d-7144-405c-83a3-c2101aa1079d.jpg)
We observe that
![](https://www.scirp.org/html/5-1100287\d329d67b-de8c-4016-bc9e-7adf10248d6b.jpg)
Since
is
-periodic, then
is also
-periodic. Thus
has the periods
![](https://www.scirp.org/html/5-1100287\1aef0dcf-125e-47d8-a086-f21f097ce9c4.jpg)
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
![](https://www.scirp.org/html/5-1100287\c9aade36-e8f9-4960-9c25-77030f4f33e4.jpg)
and there is a nonnegative integer
such that
is orthogonal to
![](https://www.scirp.org/html/5-1100287\410b16fe-81a0-4cf5-80f3-47d6586d95e5.jpg)
with
![](https://www.scirp.org/html/5-1100287\90bf5c00-95c3-4f47-88e1-3027f31a510c.jpg)
The generalized function
is
-periodic and there is an orthogonal transformation
such that
![](https://www.scirp.org/html/5-1100287\fb1d6a4c-36a2-4130-801b-009a4f73f942.jpg)
is
-periodic with the representation
![](https://www.scirp.org/html/5-1100287\e7bf3f34-b2eb-4836-9c21-a5b578ad8829.jpg)
Here
is an eigenfunction of
with
![](https://www.scirp.org/html/5-1100287\0cbf3b8b-b94d-4a53-9f05-0c9190e6b231.jpg)
for ![](https://www.scirp.org/html/5-1100287\dd0919c5-c757-4ea3-b820-d1d2a4f70d97.jpg)
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
![](https://www.scirp.org/html/5-1100287\dd724326-53e7-4e2a-9f7e-7c9d0606d766.jpg)
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
![](https://www.scirp.org/html/5-1100287\548a922b-248f-4454-bd31-483e163825f1.jpg)
and there are nonnegative integers
![](https://www.scirp.org/html/5-1100287\3135be6d-4d82-41d0-ab61-c7281bb6efaa.jpg)
such that
,
![](https://www.scirp.org/html/5-1100287\11a0d6cf-16ac-4baf-b0ae-4398f5db181c.jpg)
and
![](https://www.scirp.org/html/5-1100287\1edd0678-e9b0-4089-8947-f9c7ea4adbfb.jpg)
are pairwisely orthogonal with
![](https://www.scirp.org/html/5-1100287\a9e1858a-9354-4356-8552-270b8273e343.jpg)
where
![](https://www.scirp.org/html/5-1100287\a52a7fa2-7665-432e-96ca-5f4afa4421ce.jpg)
![](https://www.scirp.org/html/5-1100287\dc553a7f-fe29-4ff7-8ef9-f641720df044.jpg)
The generalized function
is
-periodic, and there is an orthogonal transformation
such that
![](https://www.scirp.org/html/5-1100287\7e0047b2-1aa6-4c93-8962-9e1380585530.jpg)
is
![](https://www.scirp.org/html/5-1100287\eacf4ad3-c0f8-4641-9d94-b06dbd86181e.jpg)
-periodic with the representation
(30)
Here
![](https://www.scirp.org/html/5-1100287\1dd653d0-8592-4866-92dc-c17b1b59877b.jpg)
where
![](https://www.scirp.org/html/5-1100287\e0929562-2d68-41be-bae0-a23c91bcc16d.jpg)
for
![](https://www.scirp.org/html/5-1100287\4eb840d1-7805-403b-9fcb-fa2037328114.jpg)
and
![](https://www.scirp.org/html/5-1100287\d20be42d-7a55-4a09-8463-3c0aa7b672f5.jpg)
2.2. Some Quasiperiodic Eigenfunctions of the Fourier Transform Operator on ![](https://www.scirp.org/html/5-1100287\d24c7762-4a42-4a59-8a2f-ab30b570b4a0.jpg)
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
![](https://www.scirp.org/html/5-1100287\b3de420b-b86d-4b1c-949d-9c6a1f2c2223.jpg)
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
![](https://www.scirp.org/html/5-1100287\0c5a1418-a3db-41ea-8b53-af85089ee90c.jpg)
for each
. Thus
![](https://www.scirp.org/html/5-1100287\465a955a-ad86-419e-bd7a-7eb6a3f81932.jpg)
(with
) where
![](https://www.scirp.org/html/5-1100287\8da8055d-9371-46f1-9273-fbe3e60587c1.jpg)
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,
![](https://www.scirp.org/html/5-1100287\9dd594bf-3c09-463d-b2ae-e09416541778.jpg)
3. Representation of Some Quasiperiodic Eigenfunctions