Characterization of Periodic Eigenfunctions of the Fourier Transform Operator

Let the generalized function (tempered distribution) f on be a p-periodic eigenfunction of the Fourier transform operator , i.e.,       , f x p f x f f      , for some   . We show that 1, , 1, or , i i      that p N  for some and that 1,2, ,  N  f has the representation     1



  , , for some    .We show that 1, , 1, or , i i for some and that 1, 2, , where  is the Dirac functional and  is an eigenfunction of the discrete Fourier transform operator N  with We generalize this result to -periodic eigenfunctions of on and to -periodic eigenfunctions of on .

Introduction
In this paper, we will study certain generalizations of the Dirac comb (or III functional, see [1]) where  is the Dirac functional.We work within the context of the Schwartz theory of distributions [2] as developed in [1,[3][4][5][6][7].For purposes of manipulation we use "function" notation for  , and related functionals.Various useful proprieties of III  and are developed in [1,[3][4][5].

III
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 The Fourier transform of the -periodic Dirac comb is Let g be any univariate distribution with compact support.We can periodize g by writing where  represents the convolution product, to obtain the weakly convergent Fourier series We observe that Let be the Fourier transform operator on the space of tempered distributions.It is well known [1,4,5], that is linear and that where denotes the identity operator on the space of tempered distributions.We are interested in tempered distributions such that where  is a scalar.Any distribution f that satisfies (8), and that we will call eigenfunction of , must also satisfy the following equation due to the linearity of the operator .When  4 n  , then 4 4 . Thus the eigenvalues of the operator are 1, .

Eigenvectors of N 
We first consider the eigenvectors of the discrete Fourier transform operator N  since, as we will see later, they can be used to construct all periodic eigenfunctions of the Fourier transform operator [8,9].
, is said to be the discrete Fourier transform operator.
It is easy to verify the operator identity where 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 : where N I is the N N  identity matrix.In this way we see that if , be orthonormal eigenvectors of N  corresponding to the eigenvalue   , 0,1,2,3    with corresponding eigenvectors We normalize these vectors to obtain

The Main Results
  is such an eigenfunction, constructed from the eigenvector 4,0,2 f of 4 .We will now characterize all such periodic eigenfunctions.


: for the Fourier transform of f .Now since F f   and 0   , F must also be p-periodic with where 1, i We recognize this as the Fourier transform of i.e., that  is an eigenvector of the discrete Fourier transform operator N  associated with the eigenvalue N  .In this way we prove the following Theorem 1.Let the generalized function f on be a -periodic eigenfunction of the Fourier transform operator with eigenvalue Then p N for some integer and where  is an eigenvector of the discrete Fourier we obtain the corresponding 1-periodic 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

Characterization of periodic eigenfunctions of on
Let f be a bivariate generalized function and assume that f is an eigenfunction of , 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 1 2 , a a along the positive x-axis.We can and do further assume with no loss of generality that have the form Open Access AJCM C. DE SOUZA, D. W. KAMMLER 308 The dual vectors then have the representation f can be represented by the weakly convergent Fourier series We Fourier transform the series (20) to obtain the weakly convergent series , .
From ( 21), we see that the support of F lies on the lattice , F must also be -periodic so we can write , and write and the term   for some integers 1 2 1 2 , , , n n n n     .From the supports of these  -functions we see that for some , and analogously 0, 1, 2, for some .Using these expressions we can now write where, in view of ( 16)-( 19) From (21), (23) we also have We will now consider separately the cases 0, and by using ( 24) and ( 26), in turn we write In this way we conclude that Thus  must be an eigenvector of the bivariate discrete Fourier transform , .
, and along the x-axis and the y-axis, respectively, a situation covered by the analysis from the case.In this way we prove 0 M  Theorem 2. Let the generalized function f on be an -periodic eigenfunction of the Fourier transform operator with eigenvalue . Assume that the linearly independent periods 1 2 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 The generalized function f is -periodic and there is an orthogonal transformation such that , : serve as an orthonormal basis for the 1 2 dimensional space      , or i  .Assume that the linearly independent periods

N N N
3 .Then there are positive integers and there are nonnegative integers The generalized function is -periodic, and there is an orthogonal transformation such that ,0,0 , 0, ,0 , 0,0, , , , , e , , is a quarter turn rotation.We will use this fact to generate quasiperiodic eigenfunctions of on with rotational symmetry.
In this section we will construct some quasiperiodic eigenfunctions of the Fourier transform operator.A generalized function f is said to be quasiperiodic if the Fourier transform f  is a weighted sum of Dirac  functionals with isolated support [10].
x-axis and y-axis, respectively.The function  is represented by the synthesis equation periodic discrete real valued functions.Here (29) has the corresponding eigenvalue

Theorem 3 .
Let the generalized function on be an 1 2 3 -periodic eigenfunction of the Fourier transform been chosen as small as possible subject to the constraint that , ,

2.1. Periodic Eigenfunctions of or  
primitive unit cell associated with the lattice 1 2