Periodic Eigenfunctions of the Fourier Transform Operator

PERIODIC EIGENFUNCTIONS OF THE FOURIER TRANSFORM OPERATOR

COMLAN DE SOUZA AND DAVID W. KAMMLER

Abstract. Let the generalized function (tempered distribution) f on R be a p-periodic eigenfunction of the Fourier transform operator F, i.e., f(x + p) = f(x), Ff = λf for some λ ∈ C. We show that λ = 1, −i, −1, or + i, that √ p = N for some N = 1, 2,..., and that f has the representation ∞ N−1 X X n f(x) = γ[n]δ x − − mp p m=−∞ n=0 where δ is the Dirac functional and γ is an eigenfunction of the discrete Fourier transform operator FN with N−1 1 X λ (F γ)[k] = γ[n]e−2πikn/N = √ γ[k], k = 0, 1,...,N−1. N N n=0 N

We generalize this result to p1, p2-periodic eigenfunctions of F 2 3 on R and to p1, p2, p3-periodic eigenfunctions of F on R . We present numerous illustrations of such periodic eigenfunctions of F that are deﬁned on the plane R2, and show how these can be used to create interesting quasiperiodic eigenfunctions of F having m-fold rotational symmetry within this context.

1. Introduction In this paper we will study certain generalizations of the Dirac comb (or X functional, see [2, p. 117]) ∞ X (1) X(x) := δ(x − n) n=−∞

Date: April 28, 2011. 1991 Mathematics Subject Classiﬁcation. Primary 42C15, 42B99; Secondary 42A16, 42A75. Key words and phrases. periodic, eigenfunction, Fourier Transform operator. 1 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 2 where δ is the Dirac functional. We work within the context of the Schwartz theory of distributions [10] as developed in [13], [8, Chapter 4, Chapter 8], [3, Chapter 7], [2, Chapter 3]. For purposes of manipulation we use “function" notation for δ, X and related functionals. More precise functional notation, e.g., δ{φ} := φ(0), φ ∈ S(R) ∞ X X{φ} := φ(n), φ ∈ S(R) n=−∞ is used where appropriate. Here

∞ m (n) (2) S(R) := {φ ∈ C (R) : sup |x φ (x)| < ∞, m, n = 0, 1, 2,...} is the linear space of univariate Schwartz functions. Various useful proprieties of δ and X are developed in [13], [8, Chapter 4, Chapter 8], [3, Chapter 7], [2, Chapter 3]. For example, x (3) δ = |a|δ(x), a > 0 or a < 0. a The Fourier transform of the Dirac comb can be obtained by using the Poisson sum formula ∞ ∞ X X ∧ φ(n) = φ (k), φ ∈ S(R) n=−∞ k=−∞ where we use the superscript ∧ for the Fourier transform Z ∞ φ∧(s) := e−2πisxφ(x)dx. −∞ Thus Z ∞ X∧{φ} := X∧(x)φ(x)dx Integral notation for functional −∞ Z ∞ := X(s)φ∧(s)ds, Deﬁnition of Fourier transform −∞ ∞ X =: φ∧(k), Action of X k=−∞ ∞ X = φ(n), Poisson sum formula n=−∞ =: X{φ}, φ ∈ S(R), Action of X. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 3

We assume the knowledge of the remaining properties of the δ functional such as, the sifting property Z ∞ (4) δ(x − b)φ(x)dx = φ(b), b ∈ R, φ ∈ S(R) −∞ where the “integral” refers to the action of the functional δ(x − b), and the identity (5) g(x)δ(x − a) = g(a)δ(x − a) (which holds when, g, g0 , g00 ,... are all continuous functions of slow growth at ±∞). The X functional is used in the study of sampling, periodization, etc., see [8, Chapter 5], [3, Chapters 7–9], [2, Chapter 7]. We will illustrate this process using a notation that can be generalized to an 1 n-dimensional setting. Let a1 ∈ R with a1 6= 0, and let A1 := . We a1 deﬁne the lattice

La1 := {na1 : n ∈ Z} and the corresponding a1-periodic Dirac comb X (6) grida1 (x) := δ(x − a).

a∈La1 We use (3) and (1) to write ∞ X grida1 (x) := δ(x − na1) n=−∞ ∞ 1 X x = δ − n |a | a 1 n=−∞ 1 1 x = X , |a1| a1 and thereby ﬁnd ∧ grida1 (s) = X(a1s) ∞ X = δ(a1s − k) k=−∞ 1 k = X s − |a1| a1

(7) = |A1| gridA1 (s). PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 4

Let g be any univariate distribution with compact support. We can periodize g by writing ∞ X f(x) : = g(x − na1) n=−∞ ∞ X = δ(x − na1) ∗ g(x) n=−∞

(8) = grida1 (x) ∗ g(x) , where ∗ represents the convolution product. We then use (8),(7) and (5) as we write ∧ ∧ f (s) = |A1| gridA1 (s) g (s) ∞ X ∧ = |A1| δ(s − kA1) g (s) k=−∞ ∞ X ∧ = |A1| g (kA1)δ(s − kA1) , k=−∞ and thereby obtain the weakly convergent Fourier series ∞ X ∧ 2πikA1x (9) f(x) = |A1| g (kA1)e . k=−∞

We observe that grida1 has support at the points na1, n = 0, ±1, ±2,... of the lattice La1 , while the Fourier transform |A1| gridA has support n 1 at the points , n = 0, ±1, ±2,... of the lattice LA1 . It follows that a1 ∧ grida1 = grida1 if and only if

a1 = ±1, i.e., if and only if

(10) grida1 = X. Let F be the Fourier transform operator on the space of tempered distributions. It is well known [2], [3], [8], that F is linear and that (11) F 4 = I, where I denotes the identity operator on the space of tempered distributions. We are interested in tempered distributions f such that (12) Ff = λf, PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 5 where λ is a scalar. Any distribution f that satisﬁes (12), and that we will call eigenfunction of F, must also satisfy the following equation (13) F nf = λnf for any positive integer n, due to the linearity of the operator F. When n = 4, then F 4f = λ4f If = λ4f f = λ4f. Thus the eigenvalues of the operator F are 1, −1, i, −i.

2. Preliminaries We would like to characterize all periodic eigenfunctions f of the Fourier transform operator F, i.e., Ff = λf, f 6= 0, within the context of 1,2,3 dimensions. Such eigenfunctions are really tempered distributions, but we will continue to use “function” nomen- clature.

2.1. Periodic eigenfunctions of F on R.

2.1.1. Eigenfunctions of FN . We ﬁrst consider the periodic eigenfunctions of the discrete Fourier transform operator FN since, as we will see later, they can be used to construct all periodic eigenfunctions of the Fourier transform operator F. Deﬁnition 2.1. Let N = 1, 2,... . The matrix 1 1 1 ... 1  1 ω ω2 . . . ωN−1 1  2 4 2N−2  F := 1 ω ω . . . ω  , ω := e−2πi/N N N . . .  . . .  1 ωN−1 ω2N−2 . . . ω(N−1)(N−1) is said to be the discrete Fourier transform operator. In component form we have N−1 1 X (14) (F f)[k] := e−2πikn/N f[n], k = 0, 1, 2,...,N − 1. N N n=0

We often use (14) in a context where both f and FN f are N-periodic functions on Z, see [3, Chapters 1,4,6], and we say that f, Ff are functions on PN . In such cases, the summation can be taken over any N consecutive integers, and we allow k to take all values from Z. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 6

It is easy to verify the operator identity 1 F 2 = R N N N where 1 0 0 ... 0 0 0 0 0 ... 0 1 0 0 0 ... 1 0 R := . . .  N . . .  0 0 1 ... 0 0 0 1 0 ... 0 0 is the reﬂection operator, i.e.,

(RN f)[n] := f[−n], n = 0, 1,...,N − 1, (with arguments taken modulo N). We then ﬁnd 1 2 1 1 F 4 = R = R2 = I N N N N 2 N N 2 N where IN is the N × N identity matrix. In this way we see that if

FN f = λf, f 6= 0, then 4 1 λ − 2 = 0, N √ √ so λ must take one of the values ±1/ N, ±i/ N. The columns (and rows) of FN are pairwise orhogonal with

T 1 T F F = I = F F . N N N N N N N Thus FN is normal so there is an orthonormal basis for C consisting of eigenvectors of FN , [12, p. 226]. Now if 1 FN f = ±√ f, f 6= 0, N then 2 2 1 RN f = (NF )f = N ±√ f = f, N N i.e., f is even, and if i FN f = ±√ f, f 6= 0, N then 2 2 i RN f = (NF )f = N ±√ f = −f, N N PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 7 i.e., f is odd. By using Euler’s identity we can write

FN = CN − iSN where 1 nk N−1 (15) CN := cos 2π N N n,k=0 1 nk N−1 (16) SN := sin 2π N N n,k=0 are symmetric real matrices. We easily verify that

CN RN = CN = RN CN

SN RN = −SN = RN SN , and thereby conclude that if 1 FN f = ±√ f N then 1 CN f = ±√ f, N and if i FN f = ±√ f N then 1 SN f = ∓√ f. N

Since CN , SN are real and symmetric, this guarantes that we can find N a real orthonormal basis for C consisting of eigenvectors of FN . We produce such a basis by finding the eigenvectors of the real matrices 1 CN , SN that correspond to the real eigenvalues ±√ . N In March 1972, J.H. McClellan and T.W. Parks [7, III, p. 67–68], while students in the electrical engineering department of Rice Uni- versity, studied the eigenvector decomposition of the discrete Fourier transform operator FN . They constructed an orthogonal basis for the N space C consisting of eigenvectors of FN . They managed√ to determine√ the multiplicities of the four possible eigenvalues, ±1/ N, ±i/ N of FN . A few years later Auslander and Tolimieri [1, Theorem I.1.2’, p. 856] derived an alternative algebraic argument, to find the multiplicities of the eigenvalues of FN . Kammler [3, Ex. 5.17] simplified PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 8 their trace argument that uses the known value of the Gauss sum  1 N ≡ 1 mod 4 N−1 √  X 2 0 N ≡ 2 mod 4 N trF := e2πin /N = N N i N ≡ 3 mod 4 n=0  1 + i N ≡ 0 mod 4, to find a simple formula for the sum of the eigenvalues of FN , see [3, Chapter 4, p. 215]. Let Mr(N) be the multiplicity of the eigenvalue (−i)r λ = √ N of FN , r = 0, 1, 2, 3. From the above references, we obtain the formu- las

(17) M0(N) := bN/4c + 1 ( bN/4c if N ≡ 0, 1, 2 mod 4 (18) M (N) := 1 bN/4c + 1 if N ≡ 3 mod 4,

( bN/4c if N ≡ 0, 1 mod 4 (19) M (N) := 2 bN/4c + 1 if N ≡ 2, 3 mod 4,

( bN/4c − 1 if N ≡ 0 mod 4 (20) M (N) := 3 bN/4c if N ≡ 1, 2, 3 mod 4 √ √ √ for√ the multiplicities of the eigenvalues 1/ N, −i/ N, −1/ N, and i/ N, respectively. Here b c is the ﬂoor function, i.e., bxc is the greatest integer that does not exceed x. Table 1 shows the values of these functions for selected values of N. Since the matrix FN is normal, these multiplicities are also the dimensions of the eigenspaces of the operator FN . We will now construct a real, orthonormal basis N for C of eigenvectors of FN when N = 1, 2, 3, 4, 5, 6. We will let

(21) fN,r,µ[n], µ = 1, 2,...,Mr(N) be orthonormal eigenvectors of FN corresponding to the eigenvalue (−i)r λ = √ , r = 0, 1, 2, 3. N Example 2.1. N = 1 The matrix F1 = [1] PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 9 √ √ √ √ N\λ 1/ N −i/ N −1/ N i/ N 1 1 0 0 0 2 1 0 1 0 3 1 1 1 0 4 2 1 1 0 5 2 1 1 1 6 2 1 2 1 7 2 2 2 1 8 3 2 2 1 4m m+1 m m m-1 4m+1 m+1 m m m 4m+2 m+1 m m+1 m 4m+3 m+1 m+1 m+1 m

Table 1. Dimensions of the eigenspaces of the operator FN .

has the eigenvalue λ1 = 1 corresponding to the eigenvector [1]. We set

f1,0,1[n] = 1, n ∈ Z. Example 2.2. N = 2 The matrix 1 h i F = 1 1 2 2 1 −1 √ √ has the eigenvalues λ1 = 1/ 2, λ2 = −1/ 2 with corresponding eigenvectors h 1 i h 1 i √ , √ . −1 + 2 −1 − 2 We normalize these vectors to obtain √ 1 −1 + 2 f2,0,1[0] = p √ , f2,0,1[1] = p √ , 4 − 2 2 4 − 2 2 √ 1 1 + 2 f2,2,1[0] = p √ , f2,2,1[1] = −p √ , 4 + 2 2 4 + 2 2 and then 2-periodicity extend these functions to Z. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 10 √ N fN,r,µ Nλ

1 f1,0,1 1 (1)

2 f2,0,1 1 (0.9239,0.3827)

f2,2,1 -1 (0.3827,-0.9239)

3 f3,0,1 1 (0.8881,0.3251,0.3251)

f3,1,1 -i (0,-0.7071,0.7071)

f3,2,1 -1 (-0.4597,0.6280,0.6280)

4 f4,0,1 1 (0.7071,0,0.7071,0)

f4,0,2 1 (0.5,0.5,-0.5,0.5)

f4,1,1 -i (0,-0.7071,0,0.7071)

f4,2,1 -1 (-0.5,0.5,0.5,0.5)

5 f5,0,1 1 (0.6793,-0.0910,0.5109,0.5109,-0.0910)

f5,0,2 1 (0.5120,0.5575,-0.2411,-0.2411,0.5575)

f5,1,1 -i (0,-0.6802,-0.1932,0.1932,0.6802)

f5,2,1 -1 (-0.5257,0.4253,0.4253,0.4253,0.4253)

f5,3,1 i (0,-0.1932,0.6802,-0.6802,0.1932)

6 f6,0,1 1 (0.7720,0.0116,0.3406,0.4145,0.3406,0.0116)

f6,0,2 1 (-0.3289,-0.5932,0.1787,0.3522,0.1787,-0.5932)

f6,1,1 -i (0,0.6533,0.2706,0,-0.2706,-0.6533)

f6,2,1 -1 (-0.3830,0.3241,-0.0621,0.7972,-0.0621,0.3241)

f6,2,2 -1 (-0.3862,0.2071,0.5901,-0.2620,0.5901,0.2071)

f6,3,1 i (0,0.2706,-0.6533,0,0.6533,-0.2706) Table 2. Approximate numerical values for orthonormal eigenvectors of FN when N ≤ 6.

We can numerically ﬁnd such eigenvectors when N = 3, 4, 5, 6,.... Indeed, we create the matrices CN , SN from (15),(16) and use MAT- LAB to ﬁnd orthonormal eigenvectors corresponding to the eigenvalues 1 ±√ . In this way we obtain orthonormal eigenvectors f3,r,µ for F3, N f4,r,µ for F4, f5,r,µ for F5, and f6,r,µ for F6 as shown in Table 2. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 11

3. The main mathematical results 3.1. Periodic eigenfunctions of F on R. A generalized function f, f 6= 0, on R is said to be an eigenfunction of the Fourier transform operator F if Ff = λf for λ = ±1, ±i. We will now determine all periodic eigenfunctions of F.

3.1.1. Periodic eigenfunctions of F on R. Let f be a p-periodic generalized function on R, p > 0, and assume that F := Ff = λf where λ = ±1, ±i and f 6= 0. Fig. 1 illustrates such a 2-periodic eigenfunction of F, 1 x x − 1/2 x − 2/2 x − 3/2 f(x) = X +X −X +X , 2 2 2 2 2 constructed from the eigenvector f4,0,2 of F4. We will now characterize all such periodic eigenfunctions.

f(x)

-4 -2 0 2 4 6 x

Figure 1. The 2-periodic generalized eigenfunction of the Fourier transform operator F constructed from f4,0,2.

Since f is p-periodic, f is represented by its weakly convergent Fourier series ∞ X (22) f(x) = Γ[k]e2πikx/p k=−∞ (see (9) and [3, Chapter 7, p. 440]) with Fourier coeﬃcients Γ[k] that are slowly growing as k → ±∞, i.e., the sequence |Γ[k]| , k = 0, ±1, ±2,... (1 + k2)m PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 12 is bounded for some choice of m = 1, 2,.... We Fourier transform term by term to obtain the weakly convergent series ∞ X k (23) F (s) = Γ[k]δ s − p k=−∞ for the Fourier transform of f. Now since F = λf and λ 6= 0, F must also be p- periodic with ∞ X k X F (s) = Γ[k]δ s − ∗ δ(s − mp) p 0≤k

∞ N−1 1 X X n √ (25) = √ γ[n]δ x − √ − m N N m=−∞ n=0 N where N−1 X γ[n] = Γ[k]e2πikn/N k=0 is the inverse Fourier transform of the N-periodic sequence of Fourier coeﬃcients Γ. Since F = λf we can use (23), (25) to see that λ Γ[k] = (FN γ)[k] = √ γ[k], k = 0, 1,...,N − 1, N i.e., that γ is an eigenvector of the discrete Fourier transform operator λ FN associated with the eigenvalue √ . In this way we prove the N following Theorem 3.1. Let the generalized function f on R be a p-periodic eigenfunction of the Fourier transform√ operator F with eigenvalue λ = 1, −i, −1, or +i. Then p = N for some integer N = 1, 2,... and f has the representation ∞ N−1 X X n (26) f(x) = γ[n]δ x − − mp p m=−∞ n=0 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 14 where γ is an eigenvector of the discrete Fourier transform operator FN with N−1 1 X λ (F γ)[k] = γ[n]e−2πikn/N = √ γ[k], k = 0, 1,...,N − 1. N N n=0 N Example 3.1. When N = 1 we obtain the corresponding 1-periodic ∞ X f(x) = δ(x − n) = X(x), n=−∞ with FX = X. Of course, this particular result is well known, see [2, p. 152–153]. 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 X. A graphical representation of this f is given in Fig. 2. √ Example 3.2. When N = 2, we obtain the 2-periodic eigenfunctions ∞ √ ∞ 1 X √ −1 + 2 X 1 √ f1(x) = p √ δ(x − n 2) p √ δ x − √ − n 2 4 − 2 2 n=−∞ 4 − 2 2 n=−∞ 2 √ 1 x −1 + 2 x 1 = X √ + X √ − , q √ q √ 2 2(4 − 2 2) 2 2(4 − 2 2) 2 and ∞ √ ∞ 1 X √ 1 + 2 X 1 √ f2(x) = p √ δ(x − n 2) − p √ δ x − √ − n 2 4 + 2 2 n=−∞ 4 + 2 2 n=−∞ 2 √ 1 x 1 + 2 x 1 = X √ − X √ − q √ q √ 2 2(4 + 2 2) 2 2(4 + 2 2) 2 from the eigenvectors f2,0,1 and f2,2,1 for F2 as given in Example 3.1.2. It is easy to verify that

(Ff1)(s) = f1(s), (Ff2)(s) = −f2(s).

Graphical representations of f1, f2 are shown in Fig. 2. Filled circles correspond to negatively scaled Dirac δ’s. The radius of each circle is proportional to the square root of the modulus of the scale factor for the corresponding δ. (When we illumine a circular aperture with light having a uniform intensity, the energy passing through the aperture is proportional to the area of the aperture, i.e., to the square of the radius. The scaling we have chosen for our graphs makes the “intensity” of the δ correspond to the “intensity” associated with the corresponding circular aperture.) PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 15

p2=1 λ=1

p2=2 λ=1

λ=−1

p2=3 λ=1

λ=−i

λ=−1

p2=4 λ=1

λ=1

λ=−i

λ=−1

Figure 2. Graphical representation of the p-periodic eigenfunctions, of the univariate Fourier transform operator with p2 = 1, 2, 3, 4. √ Example 3.3. When N = 3 we obtain the 3-periodic eigenfunctions √ ∞ ∞ 1 + 3 X √ 1 X 1 √ f1(x) = p √ δ(x − n 3) + p √ δ x − √ − n 3 6 + 2 3 n=−∞ 6 + 2 3 n=−∞ 3 ∞ 1 X 2 √ + p √ δ x − √ − n 3 6 + 2 3 n=−∞ 3 √ 1 + 3 x 1 x 1 = X √ + X √ − q √ q √ 3 3(6 + 2 3) 3 3(6 + 2 3) 3 1 x 2 + X √ − , q √ 3 3(6 + 2 3) 3

∞ ∞ 1 X 1 √ 1 X 2 √ f2(x) = − √ δ x − √ − n 3 + √ δ x − √ − n 3 2 n=−∞ 3 2 n=−∞ 3 1 x 1 1 x 2 = − √ X √ − + √ X √ − , 6 3 3 6 3 3 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 16

p2=5 λ=1

λ=1

λ=−i

λ=−1

λ=i

p2=6 λ=1

λ=1

λ=−i

λ=−1

λ=i

Figure 3. Graphical representation of the p-periodic eigenfunctions, of the univariate Fourier transform operator with p2 = 5, 6. and √ ∞ ∞ 1 − 3 X √ 1 X 1 √ f3(x) = p √ δ(x − n 3) + p √ δ x − √ − n 3 6 − 2 3 n=−∞ 6 − 2 3 n=−∞ 3 ∞ 1 X 2 √ + p √ δ x − √ − n 3 6 − 2 3 n=−∞ 3 √ 1 − 3 x 1 x 1 = X √ + X √ − q √ q √ 3 3(6 − 2 3) 3 3(6 − 2 3) 3 1 x 2 + X √ − q √ 3 3(6 − 2 3) 3 from the eigenvectors f3,0,1,f3,1,1 and f3,2,1 for F3. Graphical representations of f1, f2, f3 are shown in Fig. 2. With a bit of calculation, we can verify that

(Ff1)(s) = f1(s), (Ff2)(s) = −i f2(s), (Ff3)(s) = −f3(s). PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 17 √ (The eigenvalue +i does not appear when N = 3 and p = 3.) Example 3.4. When N = 4 we obtain the 2-periodic eigenfunctions

∞ ∞ 1 X 1 X f1(x) = √ δ(x − 2n) + √ δ(x − 1 − 2n) 2 n=−∞ 2 n=−∞ ∞ 1 X = √ δ(x − n) 2 n=−∞ 1 = √ X(x), 2

∞ ∞ 1 X 1 X 1 f (x) = δ(x − 2n) + δ x − − 2n 2 2 2 2 n=−∞ n=−∞ ∞ ∞ 1 X 1 X 3 − δ(x − 1 − 2n) + δ x − − 2n 2 2 2 n=−∞ n=−∞ 1 x 1 x 1 1 x 2 1 x 3 = X + X − − X − + X − , 4 2 4 2 4 4 2 4 4 2 4

∞ ∞ 1 X 1 1 X 3 f (x) = − √ δ x − − 2n + √ δ x − − 2n 3 2 2 2 n=−∞ 2 n=−∞ 1 x 1 1 x 3 = − √ X − + √ X − , 2 2 2 4 2 2 2 4 and

∞ ∞ 1 X 1 X 1 f (x) = − δ(x − 2n) + δ x − − 2n 4 2 2 2 n=−∞ n=−∞ ∞ ∞ 1 X 1 X 3 + δ(x − 1 − 2n) + δ x − − 2n 2 2 2 n=−∞ n=−∞ 1 x 1 x 1 1 x 2 1 x 3 = − X + X − + X − + X − 4 2 4 2 4 4 2 4 4 2 4

from the eigenvectors f4,0,1,f4,0,2,f4,1,1 and f4,2,1 for F4. Graphical representations of f1, f2, f3, f4 are shown in Fig. 2. We can easily verify that (Ff1)(s) = f1(s), (Ff2)(s) = f2(s),

(Ff3)(s) = −i f3(s), (Ff4)(s) = − f4(s). (The eigenvalue +i does not appear when N = 4 and p = 2, see Table 1. The eigenfunction f√1 is 1-periodic as√ well as 2-periodic.) Fig. 3 shows representations for 5-periodic and 6-periodic eigenfunctions of F constructed from the 5 eigenvectors of F5 and from the 6 eigenvectors of F6 (as given in Table 2), respectively.

3.2. Periodic eigenfunctions of F on R2. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 18

3.2.1. Eigenfunctions of FN1,N2 . We will generalize the results of Sec- tion 3.1 to 2 dimension. For this purpose, will need all doubly periodic eigenfunctions of the bivariate discrete Fourier transform operator. Let N1,N2 be positive integers and let f be a complex-valued function on Z×Z that is N1-periodic in the ﬁrst component and N2-periodic in the second, i.e.,

f[n1 + N1, n2] = f[n1, n2] for all n1, n2 ∈ Z, f[n1, n2 + N2] = f[n1, n2] for all n1, n2 ∈ Z.

We will say that f is a function on PN1,N2 . The collection of all such functions is a complex linear space of dimension N1N2.

Deﬁnition 3.2. The discrete Fourier transform operator FN1,N2 on

PN1,N2 is deﬁned by 1 N1−1 N2−1 X X −2πi(k1n1/N1+k2n2/N2) (FN1,N2 f)[k1, k2] := f[n1, n2]e , N1N2 n1=0 n2=0 for k1, k2 ∈ Z. From the deﬁnition, we immediatly obtain the tensor product factor- izations

FN1,N2 = FN1,∗ F∗,N2 = F∗,N2 FN1,∗ and F p = F p F p , p = 1, 2,... N1,N2 N1,∗ ∗,N2 where 1 N1−1 X −2πin1k1/N1 (FN1,∗f)[k1, n2] = f[n1, n2]e N1 n1=0 1 N2−1 X −2πin2k2/N2 (F∗,N2 f)[n1, k2] = f[n1, n2]e N2 n2=0 are the corresponding univariate operators that act on the ﬁrst, second coordinate, respectively. From our discussion of the univariate operator FN , we know that 1 1 F 4 = I , F 4 = I N1,∗ 2 N1 ∗,N2 2 N2 N1 N2 so we must have 1 F 4 = I N1,N2 2 2 N1,N2 N1 N2 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 19 where IN1 ,IN2 ,IN1,N2 are the identity operators for functions on PN1 ,

PN2 , PN1,N2 , respectively. In this way we see that the eigenvalues of

FN1,N2 are restricted to the values p p λ = ±1/ N1N2, ±i/ N1N2.

Suppose now that f1, f2 are eigenfunctions of FN1 , FN2 , respectively, with corresponding eigenvalues λ1, λ2. If we set

f[n1, n2] := f1[n1] f2[n2] then we can write

(FN1,N2 f)[k1, k2] = (FN1,∗F∗,N2 )f1[n1]f2[n2]

= (FN1,∗f1)[k1](F∗,N2 f2)[k2]

= λ1f1[k1] λ2f2[k2]

= λ1λ2f[k1, k2].

In this way we see that f is an eigenfunction of FN1,N2 associated with the eigenvalue λ1λ2. More generally, let

f1,1[n1], f1,2[n1], . . . , f1,N1 [n1] be an orthonormal set of eigenfunctions of FN1 with corresponding eigenvalues

λ1,1, λ1,2, . . . , λ1,N1 , respectively, and let

f2,1[n2], f2,2[n2], . . . , f2,N2 [n2] be an orthonormal set of eigenfunctions of FN2 with corresponding eigenvalues

λ2,1, λ2,2, . . . , λ2,N2 , respectively. The N1N2 functions

fr1,r2 [n1, n2] := f1,r1 [n1]f2,r2 [n2], rk = 1, 2,...,Nk, k = 1, 2 will be an orthonormal basis for PN1,N2 with

FN1,N2 fr1,r2 [k1, k2] = λ1,r1 λ2,r2 fr1,r2 [k1, k2].

Let Mr(N1,N2) be the multiplicity of the eigenvalue (−i)r λ = √ N1N2 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 20

for FN1,N2 , r = 0, 1, 2, 3. Since we know the dimensions of the eigenspaces for FN1 , FN2 as given in (17)-(20), we can determine the dimensions of the four eigenspaces of FN1,N2 by writing X Mr(N1,N2) := Mr1 (N1)Mr2 (N2), r = 0, 1, 2, 3.

r1+r2≡r (mod 4) Table 3, and Table 4 show simpliﬁed expressions for these multiplicities. Explicit values for the cases where N1 ≤ 4, N2 ≤ 4, are given in Table 5.

The N1N2 eigenfunctions

(27) fN1,r1,µ1;N2,r2,µ2 [n1, n2] := fN1,r1,µ1 [n1] fN2,r2,µ2 [n2], with µk = 1,...,Mrk (Nk), k = 1, 2 of FN1,N2 serve as an orthonormal basis for the N1N2 dimensional space PN1,N2 . Here (27) has the corresponding eigenvalue (−i)r1 (−i)r2 λ = √ √ , r1, r2 = 0, 1, 2, 3. N1 N2 3.2.2. Periodic eigenfunctions of F on R2. Deﬁnition 3.3. Let f be a function on R2. The Fourier transform operator F is deﬁned so that Ff is the functional Z ∞ Z ∞ ∧ 2 (Ff){φ} := f(s1, s2)φ (s1, s2)ds1ds2, φ ∈ S(R ) −∞ −∞ where Z ∞ Z ∞ ∧ −2πi(s1x1+s2x2) φ (s1, s2) := e φ(x1, x2)dx1dx2 −∞ −∞ is the ordinary Fourier transform of φ, and

2 ∞ 2 m n1 n2 S(R ) := {φ ∈ C (R ) : sup kxk kD1 D2 φ(x)k < ∞, m, n1, n2 = 0, 1, 2,...}, where D1,D2 represent partial derivatives with respect to x1, x2, respectively. In this bivariate case we again have φ∧∧∧∧ = φ i.e., we have F 4 = I so the eigenvalues of F are again restricted to the values λ = 1, −i, −1, or +i. We will characterize the eigenfunctions of the operator F that are doubly periodic on R2. We will adopt vector notation for the remaining of the section. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 21

1 λ = √ N1N2

N1\N2 4m2 4m2 + 1 4m1 4m1m2 + 1 4m1m2 + m1 + 1 4m1 + 1 4m1m2 + m2 + 1 4m1m2 + m1 + m2 + 1 4m1 + 2 4m1m2 + 2m2 + 1 4m1m2 + 2m2 + m1 + 1 4m1 + 3 4m1m2 + 3m2 4m1m2 + 3m2 + m1 + 1

N1\N2 4m2 + 2 4m2 + 3 4m1 4m1m2 + 2m1 + 1 4m1m2 + 3m1 4m1 + 1 4m1m2 + 2m1 + m2 + 1 4m1m2 + 3m1 + m2 + 1 4m1 + 2 4m1m2 + 2m1 + 2m2 + 2 4m1m2 + 3m1 + 2m2 + 2 4m1 + 3 4m1m2 + 3m2 + 2m1 + 2 4m1m2 + 3m1 + 3m2 + 2 i λ = − √ N1N2

N1\N2 4m2 4m2 + 1 4m1 4m1m2 4m1m2 + m1 4m1 + 1 4m1m2 + m2 4m1m2 + m1 + m2 4m1 + 2 4m1m2 + 2m2 − 1 4m1m2 + 2m2 + m1 4m1 + 3 4m1m2 + 3m2 4m1m2 + 3m2 + m1 + 1

N1\N2 4m2 + 2 4m2 + 3 4m1 4m1m2 + 2m1 − 1 4m1m2 + 3m1 4m1 + 1 4m1m2 + 2m1 + m2 4m1m2 + 3m1 + m2 + 1 4m1 + 2 4m1m2 + 2m1 + 2m2 4m1m2 + 3m1 + 2m2 + 1 4m1 + 3 4m1m2 + 3m2 + 2m1 + 1 4m1m2 + 3m1 + 3m2 + 2 1 λ = − √ N1N2

N1\N2 4m2 4m2 + 1 4m1 4m1m2 + 1 4m1m2 + m1 4m1 + 1 4m1m2 + m2 4m1m2 + m1 + m2 4m1 + 2 4m1m2 + 2m2 + 1 4m1m2 + 2m2 + m1 + 1 4m1 + 3 4m1m2 + 3m2 + 1 4m1m2 + 3m2 + m1 + 1

N1\N2 4m2 + 2 4m2 + 3 4m1 4m1m2 + 2m1 + 1 4m1m2 + 3m1 + 1 4m1 + 1 4m1m2 + 2m1 + m2 + 1 4m1m2 + 3m1 + m2 + 1 4m1 + 2 4m1m2 + 2m1 + 2m2 + 2 4m1m2 + 3m1 + 2m2 + 2 4m1 + 3 4m1m2 + 3m2 + 2m1 + 2 4m1m2 + 3m1 + 3m2 + 3

Table 3. Dimension of the eigenspace of the operator FN1,N2 .

3.2.3. Characterization of periodic eigenfunctions of F on R2. Let f be a bivariate generalized function and assume that f is an eigenfunction of F, i.e.,

F := Ff = λf PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 22

1 λ = − √ N1N2

N1\N2 4m2 4m2 + 1 4m1 4m1m2 − 2 4m1m2 + m1 − 1 4m1 + 1 4m1m2 + m2 − 1 4m1m2 + m1 + m2 4m1 + 2 4m1m2 + 2m2 − 1 4m1m2 + 2m2 + m1 4m1 + 3 4m1m2 + 3m2 − 1 4m1m2 + 3m2 + m1

N1\N2 4m2 + 2 4m2 + 3 4m1 4m1m2 + 2m1 − 1 4m1m2 + 3m1 − 1 4m1 + 1 4m1m2 + 2m1 + m2 4m1m2 + 3m1 + m2 4m1 + 2 4m1m2 + 2m1 + 2m2 4m1m2 + 3m1 + 2m2 + 1 4m1 + 3 4m1m2 + 3m2 + 2m1 + 1 4m1m2 + 3m1 + 3m2 + 2 Table 4. Dimension of the eigenspace of the operator

FN1,N2 continued.

1 1 λ = √ λ = −√ N1N2 N1N2 N1\N2 1 2 3 4 N1\N2 1 2 3 4 1 1 1 1 2 1 0 1 1 1 2 1 2 2 3 2 1 2 2 2 3 1 2 2 3 3 1 2 3 3 4 2 3 3 5 4 1 2 3 5 i i λ = −√ λ = √ N1N2 N1N2 N1\N2 1 2 3 4 N1\N2 1 2 3 4 1 0 0 1 1 1 0 0 0 0 2 0 0 1 1 2 0 0 1 1 3 1 1 2 3 3 0 1 2 2 4 1 1 3 4 4 0 1 2 2 Table 5. Dimension of the eigenspace of the operator

FN1,N2 when N1,N2 ≤ 4. with λ = 1, −i, −1, or +i, (and f 6= 0). Assume further that f is a1, a2-periodic, i.e.,

f(x + a1) = f(x), f(x + a2) = f(x).

2 Here a1, a2 are linearly independent vectors in R . We simplify the analysis by rotating the coordinate system as nec- essary so as to place a shortest vector from the lattice La1,a2 along the positive x-axis. We can and do further assume with no loss of generality that a1, a2 have the form T T a1 = (α1, 0) , a2 = (β1, β2) PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 23 where

(28) α1 > 0 2 2 2 (29) α1 ≤ β1 + β2

(30) β2 > 0

(31) 0 ≤ β1 < α1. The dual vectors then have the representation

1 T 1 T A1 = (β2, −β1) ,A2 = (0, α1) , α1β2 α1β2 and ∞ ∞ X X grida1,a2 (x) = δ(x − n1a1 − n2a2) n1=−∞ n2=−∞ ∞ ∞ X X = δ(x1 − n1α1 − n2β1, x2 − n2β2)

n1=−∞ n2=−∞ has the Fourier transform ∞ ∞ ∧ X X grida1,a2 (s) = | det (A1,A2)| δ(s − k1A1 − k2A2) k1=−∞ k2=−∞ ∞ ∞ 1 X X k1β2 k1β1 − k2α1 = δ s1 − , s2 + . α1β2 α1β2 α1β2 k1=−∞ k2=−∞

Now since f is a1, a2-periodic, f can be represented by the weakly convergent Fourier series ∞ ∞ X X 2πix.(k1A1+k2A2) (32) f(x) = Γ[k1, k2]e .

k1=−∞ k2=−∞

The Fourier coeﬃcients Γ[k1, k2], as in the one dimensional case, are slowly growing as k1, k2 → ±∞, i.e.,

|Γ[k1, k2]| 2 2 m , k1, k2 = 0, ±1, ±2,... (1 + k1 + k2) is bounded for some choice of m = 1, 2,.... We Fourier transform the series (32) to obtain the weakly convergent series

∞ ∞ X X (33) F (s) = Γ[k1, k2]δ(s − k1A1 − k2A2).

k1=−∞ k2=−∞ PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 24

From (33), we see that the support of F lies on the lattice LA1,A2 and since F = λf, F must also be a1, a2-periodic so we can write X (34) F (s) = Γ[k1, k2]δ(s − k1A1 − k2A2) ∗grida1,a2 (s) k1A1+k2A2∈U where 0 0 0 0 U := {x1a1 + x2a2 : 0 ≤ x1 < 1, 0 ≤ x2 < 1} 0 0 is a primitive unit cell associated with the lattice La1,a2 , where x1, x2 are aﬃne coordinates, and ∗ is the bivariate convolution product. Using the bivariate inverse Fourier transform, we see that X 2πi(k A +k A ).x f(x) = Γ[k , k ] e 1 1 2 2 | det (A ,A )| grid (x) 1 2 1 2 A1,A2 k1A1+k2A2∈U ∞ ∞ X X X 2πi(k1A1+k2A2).(n1A1+n2A2) = | det (A1,A2)| Γ[k1, k2]e

n1=−∞ n2=−∞ k1A1+k2A2∈U · δ(x − n1A1 − n2A2)

∞ ∞ ( 1 X X X = Γ[k1, k2] α1β2 n1=−∞ n2=−∞ k1A1+k2A2∈U

2 2 2 2 2 · e2πi{(β1 +β2 )n1k1−α1β1(n1k2+n2k1)+α1n2k2}/(α1β2 )

) n1β2 n1β1 − n2α1 · δ x1 − , x2 + . α1β2 α1β2 We deﬁne

2 2 2 2 2 X 2πi{(β1 +β2 )n1k1−α1β1(n1k2+n2k1)+α1n2k2}/(α1β2 ) (35) γ[n1, n2] := Γ[k1, k2]e

k1A1+k2A2∈U and write ∞ ∞ 1 X X n1β2 n1β1 − n2α1 (36) f(x1, x2) = γ[n1, n2]δ x1 − , x2 + . α1β2 α1β2 α1β2 n1=−∞ n2=−∞

Now f is a1, a2-periodic, so if γ[n1, n2] 6= 0 for some integers n1, n2, then n1β2 n1β1 − n2α1 γ[n1, n2]δ x1 − α1 − , x2 + α1β2 α1β2 0 0 0 0 0 n1β2 n1β1 − n2α1 = γ[n1, n2]δ x1 − , x2 + α1β2 α1β2 n1β2 n1β1 − n2α1 γ[n1, n2]δ x1 − β1 − , x2 − β2 + α1β2 α1β2 00 00 00 00 00 n1 β2 n1 β1 − n2 α1 = γ[n1 , n2 ]δ x1 − , x2 + α1β2 α1β2 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 25

0 0 00 00 for some integers n1, n2, n1 , n2 . From the supports of these δ-functions we see that 0 n1β2 n1β2 α1 + = , α1β2 α1β2 i.e., 2 0 α1 = n1 − n1 2 α1 = N1 for some N1 = 1, 2,... . Likewise, we see in turn that 0 0 n1β1 − n2α1 = n1β1 − n2α1, 0 0 (n1 − n1)β1 = (n2 − n2)α1, 2 0 α1β1 = (n2 − n2)α1, 0 α1β1 = n2 − n2 = M for some M = 0, ±1, ±2,..., and analogously 00 n1β2 n1 β2 β1 + = , α1β2 α1β2 00 α1β1 = n1 − n1 = M. Finally, 00 00 n1β1 − n2α1 n1 β1 − n2 α1 β2 − = − , α1β2 α1β2

2 00 β1 00 β2 + (n1 − n1) = n2 − n2, α1 2 β1 00 β2 + α1β1 = n2 − n2, α1 2 2 β2 + β1 = N2 for some N2 = 1, 2,.... Using these expressions we can now write √ 2 √ M N1N2 − M α1 = N1, β1 = √ , β2 = √ , and N1 N1

1 T 1 p 2 T a1 = √ (N1, 0) , a2 = √ (M, N1N2 − M ) , N1 N1

1 p 2 T 1 T A1 = ( N1N2 − M , −M) ,A2 = (0,N1) , p 2 p 2 N1(N1N2 − M ) N1(N1N2 − M ) where, in view of (28)−(31)

N1 ≤ N2, 0 ≤ M < N1 and p p ka1k = N1, ka2k = N2. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 26

From (35), (33) we also have

2 X 2πi{N2n1k1−M(n1k2+n2k1)+N1n2k2/(N2N1−M )} (37) γ[n1, n2] = Γ[k1, k2]e

k1A1+k2A2∈U

(38) ∞ ∞ X X F (s) = Γ[k1, k2]δ(s − k1A1 − k2A2)

k1=−∞ k2=−∞ ∞ ∞ X X k1 k1M − k2N1 = Γ[k1, k2]δ s1 − √ , s2 + . p 2 N1 N1(N1N2 − M ) k1=−∞ k2=−∞ We will now consider separately the cases M = 0, M > 0.

Case M = 0

When M = 0 the vectors a1, a2 are orthogonal and f has the corresponding periods p p α1 = N1, β2 = N2, along the x-axis and y-axis, respectively. The function γ is represented by the synthesis equation

N1−1 N2−1 X X 2πi(n1k1/N1+n2k2/N2) (39) γ[n1, n2] = Γ[k1, k2]e ,

k1=0 k2=0 and by using (38) and (36), in turn we write ∞ ∞ X X k1 k2 F (s1, s2) = Γ[k1, k2]δ s1 − √ , s2 − √ , N1 N2 k1=−∞ k2=−∞

= λf(s1, s2) ∞ ∞ λ X X k1 k2 = √ γ[n1, n2]δ s1 − √ , s2 − √ . N1N2 N1 N2 n1=−∞ n2=−∞ In this way we conclude that λ (40) Γ[k1, k2] = √ γ[k1, k2]. N1N2 Thus γ must be an eigenvector of the bivariate discrete Fourier trans- λ form FN1,N2 associated with the eigenvalue √ ,(λ = 1, −i, −1, or N1N2 +i). Since γ is an N1,N2-periodic sequence of complex numbers, we PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 27 can write

∞ ∞ N −1 N −1 X X X1 X2 f(x) = γ[n1, n2]

m1=−∞ m2=−∞ n1=0 n2=0 n1 √ n2 √ · δ x1 − √ − m1 N1, x2 − √ − m2 N2 . N1 N2 Case M 6= 0

We observe that

1 T a1 = √ (N1, 0) , N1 p p 2 T p T N1a2 − Ma1 = N1(M, N1N2 − M ) − M( N1, 0)

1 p 2 T = √ (0,N1 N1N2 − M ) . N1

Since f is a1, a2-periodic, then f is also a1,N1a2 − Ma1-periodic. Thus f has the periods

p 0 p 2 α1 = N1, and β2 = N1(N1N2 − M ) along the x-axis and the y-axis, respectively, a situation covered by the analysis from the M = 0 case. In this way we prove

2 Theorem 3.2. Let the generalized function f on R be an a1, a2- periodic eigenfunction of the Fourier transform operator F with eigenvalue λ = 1, −i, −1, or +i. Assume that the linearly independent pe- 2 riods a1, a2 from R have been chosen as small as possible subject to the constraint that 0 < ka1k ≤ ka2k. Then there are positive integers N1 ≤ N2 such that p p ka1k = N1, ka2k = N2 and there is a nonnegative integer M < N1 such that a1 is orthogonal to 0 a2 := N1a2 − Ma1 with q 0 0 0 2 ka2k = N2,N2 := N1(N1N2 − M ).

0 The generalized function f is a1, a2-periodic and there is an orthogonal transformation Q such that

fQ(x) := f(Qx) PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 28 √ T p 0 T is ( N1, 0) , (0, N2) -periodic with the representation 0 ∞ ∞ N −1 N −1 X X X1 X2 fQ(x) = γ[n1, n2]

m1=−∞ m2=−∞ n1=0 n2=0 q n1 p n2 0 (41) · δ x − √ − m N , x − − m N . 1 1 1 2 p 0 2 2 N1 N2

Here γ is an eigenfunction of F 0 with N1,N2 0 N1−1 N2−1 1 0 X X −2πi(k1n1/N1+k2n2/N2) (F 0 γ)[k1, k2] = 0 γ[n1, n2]e N1,N2 N1N2 n1=0 n2=0 λ = γ[k , k ] p 0 1 2 N1N2 0 for k1 = 0, 1,...,N1 − 1, k2 = 0, 1,...,N2 − 1. Example 3.5. Fig. 4−Fig. 10 display some of the doubly periodic eigenfunctions of the bivariate Fourier transform operator F that have the form (41) for several values of N1,N2 and for M = 0. The eigenfunctions γ all have the form

γ[n1, n2] := fN1,r1,µ1;N2,r2,µ2 [n1, n2] = fN1,r1,µ1 [n1] fN2,r2,µ2 [n2] using fN1,r1,µ1 [n1] and fN2,r2,µ2 [n2] from Table 2. We represent the term Aδ(x − a) from f with a circle of radius proportional to p|A| and center at a. When A < 0, we ﬁll the circle.

3.3. Periodic eigenfunctions of F on R3. We extend the analysis in the previous section, and analagously we generalize Theorem 3.1 in a three dimensional setting.

3 Theorem 3.3. Let the generalized function f on R be an a1, a2, a3- periodic eigenfunction of the Fourier transform operator F with eigenvalue λ = 1, −i, −1, or +i. Assume that the linearly independent peri- 3 ods a1, a2, a3 from R have been chosen as small as possible subject to the constraint that 0 < ka1k ≤ ka2k ≤ ka3k. Then there are positive integers N1 ≤ N2 ≤ N3 such that p p p ka1k = N1, ka2k = N2, ka3k = N3 and there are nonnegative integers 0 ≤ M1 < N1, 0 ≤ M2 < N1, 0 ≤ M3 < N1 + N2 such that a1, 0 a2 := N1a2 − M1a1, PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 29

λ=1

Figure 4. The (1, 0)T , (0, 1)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = 1, constructed from f1,0,1;1,0,1.

λ=1

√ Figure 5. The (1, 0)T , (0, 2)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = 1, constructed from f1,0,1;2,0,1. and

0 2 a3 := N1[(M1M3 − N2M2)a1 − (N1M3 − M1M2)a2 + (N1N2 − M1 )a3] PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 30

λ=−1

√ Figure 6. The (1, 0)T , (0, 2)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = −1, constructed from f1,0,1;2,2,1.

λ=1

√ √ Figure 7. The ( 2, 0)T , (0, 2)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = 1, constructed from f2,0,1;2,0,1. are pairwisely orthogonal with

0 q 0 0 q 0 ka2k = N2, ka3k = N3 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 31

λ=1

√ √ Figure 8. The ( 2, 0)T , (0, 2)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = 1, constructed from f2,2,1;2,2,1.

λ=−1

√ √ Figure 9. The ( 2, 0)T , (0, 2)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = −1, constructed from f2,0,1;2,2,1. where 0 2 N2 := N1(N1N2 − M1 ),

0 2 2 2 2 2 N3 := N1 (N1N2−M1 )[N1N2N3+2M1M2M3−(N1M3 +N2M2 +N3M1 )]. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 32

λ=−1

√ √ Figure 10. The ( 2, 0)T , (0, 2)T -periodic eigenfunction of the bivariate Fourier transform operator F with λ = −1, constructed from f2,2,1;2,0,1.

0 0 The generalized function f is a1, a2, a3-periodic, and there is an orthogonal transformation Q such that

fQ(x) := f(Qx) √ T p 0 T p 0 T is ( N1, 0, 0) , (0, N2, 0) , (0, 0, N3) -periodic with the representation

0 0 ∞ N −1 N −1 N −1 X X1 X2 X3 fQ(x) = γ[n1, n2, n3]

(42) m1=−∞,m2=−∞,m3=−∞ n1=0 n2=0 n3=0 n √ n q n q ·δ x − √ 1 − m N , x − 2 − m N 0 , x − 3 − m N 0 . 1 1 1 2 p 0 2 2 3 p 0 3 3 N1 N2 N3 Here

0 0 N −1 N −1 N −1 1 X1 X2 X3 (F 0 0 γ)[k1, k2, k3] = 0 0 γ[n1, n2, n3] N1,N2,N3 N1N2,N3 n1=0 n2=0 n3=0

0 0 −2πi(k n /N +k n /N +k n /N ) · e 1 1 1 2 2 2 3 3 3

λ = γ[k , k , k ] p 0 0 1 2 3 N1N2N3

0 0 for k1 = 0, 1,...,N1 − 1, k2 = 0, 1,...,N2 − 1, k3 = 0, 1,...,N3 − 1. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 33

3.4. Some quasiperiodic eigenfunctions of the Fourier transform operator on R2. 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 δ functionals with isolated support [11, Defini- tion 1.1, Definition 1.2, Definition 1.11, p. 27–31].

3.4.1. The rotation theorem for grids on R2. 2 Theorem 3.4. Let a1, a2 be linearly independent vectors in R with | det [a1 a2]| = 1. Then ∧ grida1,a2 = gridQa1,Qa2 where h0 −1i (43) Q := 1 0 is a 90◦ rotation. Proof. Let α, β, γ, δ ∈ R be chosen so that h i a a α β A := [ 1 2] = δ γ has det A = αγ − βδ = 1. We compute

−T h γ −δi h γ −δi A = det A −β α = −β α , h0 −1i hα βi h−δ −γi QA = 1 0 δ γ = α β , and observe that the columns of these matrices generate the same lattice. 2 Corollary 3.5. Let a1, a2 be linearly independent vectors in R and assume that ∧ grida1,a2 = grida1,a2 . Then a1, a2 can be replaced by orthonormal vectors that generate the same grid.

Proof. From the support of grida1,a2 , we choose a1, a2 to be as small as possible subject to 0 < ka1k ≤ ka2k. Since ∧ grida1,a2 = grida1,a2 , we must have | det [a1 a2]| = 1. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 34

This being the case, we must have ∧ gridQa1,Qa2 = grida1,a2 = grida1,a2 .

It follows that Qa1 is in the lattice generated by a1, a2, and since

kQa1k = ka1k, we must have

ka2k = kQa1k.

If a2 is distinct from ±Qa1, one of the vectors a1±a2, Qa1±a2 will be in the lattice and have a shorter length than ka2k. Thus a2 = ±Qa1. Corollary 3.6. If | det [a1 a2]| = 1, ∧ and grida1,a2 is distinct from grida1,a2 , then ∧ (44) f+(x) := grida1,a2 (x) + grida1,a2 (x) ∧ (45) f−(x) := grida1,a2 (x) − grida1,a2 (x) are eigenfunctions of the Fourier transform operator F associated with λ = 1, λ = −1, respectively. Example 3.6. Using the biorthogonal system T T T T a1 = (2, 0) , a2 = (0, 1/2) ,A1 = (1/2, 0) ,A2 = (0, 2) with det [a1 a2] = 1, and we use (44) and (45) to obtain the bivariate eigenfunctions

f+(x) = grida1,a2 (x)+gridA1,A2 (x), f−(x) = grida1,a2 (x)−gridA1,A2 (x) of F as shown in Fig. 11 and Fig. 12. These eigenfunctions are both (2, 0)T , (0, 2)T -periodic on R2. Example 3.7. Using the biorthogonal system √ T T T T 4 a1 = (α, 0) , a2 = (0, 1/α) ,A1 = (1/α, 0) ,A2 = (0, α) , α = 2, with det [a1 a2] = 1, and we use (44) and (45) to produce the eigenfunctions shown in Fig. 13 and Fig. 14, respectively. In this case neither of the eigenfunctions f+ and f− is periodic. They do have a discrete spectrum, however, so they are quasiperiodic. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 35

Figure 11. The periodic eigenfunction (44), of the bivariate Fourier transform F with λ = 1 constructed with T T a1 = (2, 0) and a2 = (0, 1/2) .

Figure 12. The periodic eigenfunction (45), of the bivariate Fourier transform F with λ = −1 constructed T T with a1 = (2, 0) and a2 = (0, 1/2) .

3.4.2. Quasiperiodic eigenfunctions of F on R2 with m-fold rotational symmetry. Let s 2π (46) α = 1/ sin n PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 36

Figure 13. The quasiperiodic eigenfunction (44), of the bivariate Fourier transform F with λ = 1 constructed√ T T 4 with a1 = (α, 0) and a2 = (0, 1/α) when α = 2.

Figure 14. The quasiperiodic eigenfunction (45), of the bivariate Fourier transform F with λ = −1 constructed√ T T 4 with a1 = (α, 0) and a2 = (0, 1/α) when α = 2. for some n = 3, 4,..., and let

T (47) ak = α cos (2πk/n), sin (2πk/n) , k = 0, 1, . . . , n − 1, PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 37 be the vertices of a regular n − gon with center at the origin. The parameter α has been chosen so that   2 cos 2πk/n cos 2π(k + 1)/n 2 det [ak ak+1] = α   = α sin 2π/n = 1, sin 2πk/n sin 2π(k + 1)/n for each k = 1, 2, . . . , n − 1. In view of Theorem 3.4, this allows us to see that grid∧ = grid , k = 0, 1, . . . , n − 1 ak,ak+1 Qak,Qak+1

(with an := a0) where again h0 −1i Q = 1 0 is a quarter turn rotation. We will use this fact to generate quasiperiodic eigenfunctions of F on R2 with rotational symmetry. q √ Example 3.8. When n = 3, α = 2/ 3, so that √ 1 3 a = α(1, 0)T , a = α(− , )T . 0 1 2 2

The corresponding hexagonal grida0,a1 and ∧ grida0,a1 = gridQa0,Qa1 are shown in Fig. 15. We note that grida0,a1 has 6-fold rotational symmetry and that

grida0,a1 = grida1,a2 = grida2,a3 . We form ∧ (48) f+ = grida0,a1 + grida0,a1 as shown in Fig. 16, noting that ∧ ∧ ∧∧ ∧ f+ = grida0,a1 + grida0,a1 = grida0,a1 + grida0,a1 = f+. We also form

(49) f− := grida0,a1 − grida0,a1 as shown in Fig. 17, noting that ∧ ∧ ∧∧ ∧ f− = grida0,a1 − grida0,a1 = grida0,a1 − grida0,a1 = −f−.

Neither f+ nor f− is periodic. Indeed f+ has the weighted spike 2δ(x) at the origin. There will be another δ spike with weight 2 if and only if there are integers n0, n1, k0, k1 such that

n0a0 + n1a1 = k0Qa0 + k1Qa1, PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 38

grid grid a ,a A ,A 1 2 1 2

Figure 15. The grida1,a2 and its dual gridA1,A2 from Example 3.8. i.e., h i h −1 i h i h i h i h −1 i n 2 + n √ = k 0 −1 2 + k 0 −1 √ , 0 0 1 3 0 1 0 0 1 1 0 3 h i h √ i = k 0 + k − 3 , 0 2 1 −1 i.e., √ 2n0 − n1 = −k1 3 √ n1 3 = 2k0 − k1. √ Since 3 is irrational we immediately conclude that

k1 = n1 = 0, 2n0 − n1 = 2k0 − k1 = 0, i.e.,

k0 = k1 = n0 = n1 = 0.

A similar argument applies to f−. We will now construct a family of quasiperiodic eigenfunctions of F that have rotational symmetry. Let n = 3, 4,..., and ak, k = 0, 1, 2, . . . , n − 1 be given by (47), let α be given by (46), and let

n−1 X (50) f (x) := grid (x) + grid∧ (x), n+ ak,ak+1 ak,ak+1 k=0 PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 39

Figure 16. The eigenfunction f3+ as given by (48) (having 12-fold rotational symmetry) for the Fourier transform operator F with λ = 1.

Figure 17. The eigenfunction f3− as given by (49) (having 6-fold rotational symmetry) for the Fourier transform operator F with λ = −1. and n−1 X (51) f (x) := grid (x) − grid∧ (x), n− ak,ak+1 ak,ak+1 k=0

(with an := a0.) By construction, ∧ ∧ fn+ = fn+ and fn− = −fn−. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 40

When n = 3, f3+, f3− are scalar multiples of the functions f+, f− from (48),(49), respectively. When n is divisible by 4, Qak is included in the list a0, a1, . . . , an−1 so n−1 n−1 n−1 X X X grid∧ = grid = grid . ak,ak+1 Qak,Qak+1 ak,ak+1 k=0 k=0 k=0

As a result, fn+ has n-fold rotational symmetry and

fn− = 0.

Of course, f4+ is a scalar multiple of the grid on Z × Z, see Fig. 18. Fig. 19 shows a represntation of f8+.

Figure 18. The periodic eigenfunction f4+ (having 4- fold rotational symmetry) as given by (50), for the Fourier transform operator F with λ = 1.

Example 3.9. Fig. 20−Fig. 23 show fn+, fn− for n = 5, 7. These generalized eigenfunctions of F are all quasiperiodic (but not periodic). PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 41

Figure 19. The quasiperiodic eigenfunction f8+ as given by (50) (having 8-fold rotational symmetry) for the Fourier transform operator F with λ = 1.

Figure 20. The quasiperiodic eigenfunction f5+ with 20-fold rotational symmetry for the Fourier transform operator F with λ = 1.

4. Conclusions A generalization of Theorem 3.1 to dimensions higher than 3 is not known at this point, and will be the subject of a future work. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 42

Figure 21. The quasiperiodic eigenfunction f5− with 10-fold rotational symmetry for the Fourier transform operator F with λ = −1.

Figure 22. The quasiperiodic eigenfunction f7+ with 28-fold rotational symmetry for the Fourier transform operator F with λ = 1.

References [1] Auslander, L. and R. Tolimieri, Is computing with the ﬁnite Fourier transform pure or applied mathematics?, Bull. Amer. Math. Soc., 1 (1979), 847–897. [2] Bracewell, R.N., Fourier Analysis and Imaging, Kluwer Academic, New York, 2003. [3] Kammler, D.W., A First Course In Fourier Analysis, Prentice Hall, New Jer- sey, 2000. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 43

Figure 23. The quasiperiodic eigenfunction f7− with 14-fold rotational symmetry for the Fourier transform operator F with λ = −1.

[4] Kannan, R., Minkowski’s Convex Body Theorem and Integer Programming, Mathematics of Operations Research 12 (1987), 415–440. [5] Lighthill, M.J., An Introduction to Fourier Analysis and Generalized Func- tions, Cambridge University Press, New York, 1958. [6] Lutzen, J., The Prehistory of the Theory of Distributions, Springer, New York, 1982. [7] McClellan, J.H. and T.W. Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform, IEEE Trans. Audio Electroacoust., AU-20 (1972), 66–74. [8] Osgood, B., The Fourier Transform and its Applications, Lecture notes, Stan- ford University, 2005. [9] Richards, J.I. and H.K. Youn, Theory of Distributions: A Non-technical Intro- duction, Cambridge University Press, Cambridge, 1990. [10] Schwartz, L., The´orie des Distributions, Hermann, Paris, 1950. [11] Senechal, M., Quasicrystals and Geometry, Cambridge University Press, New York, 1995. [12] Strang, G., Linear Algebra and Its Applications, Academic Press, inc., New York, 1976. [13] Strichartz, R., A Guide to Distribution Theory and Fourier Transforms, CRC press, inc., Boca Raton, 1994. [14] Zemanian, A.H, Distribution Theory and Transform Analysis, Dover publica- tions, inc., New York, 1987. PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 44

California State University at Fresno E-mail address: [email protected] Current address: Department of Mathematics, 5245 North Backer Avenue M/S PB108, Fresno, California 93740-8001

Southern Illinois University at Carbondale E-mail address: [email protected]