ABSTRACT

Let the generalized function (tempered distribution) on be a p-periodic eigenfunction of the Fourier transform operator, i.e., , for some. We show that that for some and that has the representation where is the Dirac functional and is an eigenfunction of the discrete Fourier transform operator with We generalize this result to -periodic eigenfunctions of on and to -periodic eigenfunctions of on.

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

REFERENCES