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 op- erator F, i.e., f(x + p) = f(x); Ff = λf for some λ 2 C. We show that λ = 1; −i; −1; or + i; that p p = N for some N = 1; 2;:::; and that f has the representation 1 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 = p 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]) 1 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., δfφg := φ(0); φ 2 S(R) 1 X Xfφg := φ(n); φ 2 S(R) n=−∞ is used where appropriate. Here 1 m (n) (2) S(R) := fφ 2 C (R) : sup jx φ (x)j < 1; m; n = 0; 1; 2;:::g 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 = jajδ(x); a > 0 or a < 0: a The Fourier transform of the Dirac comb can be obtained by using the Poisson sum formula 1 1 X X ^ φ(n) = φ (k); φ 2 S(R) n=−∞ k=−∞ where we use the superscript ^ for the Fourier transform Z 1 φ^(s) := e−2πisxφ(x)dx: −∞ Thus Z 1 X^fφg := X^(x)φ(x)dx Integral notation for functional −∞ Z 1 := X(s)φ^(s)ds; Definition of Fourier transform −∞ 1 X =: φ^(k); Action of X k=−∞ 1 X = φ(n); Poisson sum formula n=−∞ =: Xfφg; φ 2 S(R); Action of X: PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 3 We assume the knowledge of the remaining properties of the δ func- tional such as, the sifting property Z 1 (4) δ(x − b)φ(x)dx = φ(b); b 2 R; φ 2 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 2 R with a1 6= 0, and let A1 := . We a1 define the lattice La1 := fna1 : n 2 Zg and the corresponding a1-periodic Dirac comb X (6) grida1 (x) := δ(x − a): a2La1 We use (3) and (1) to write 1 X grida1 (x) := δ(x − na1) n=−∞ 1 1 X x = δ − n ja j a 1 n=−∞ 1 1 x = X ; ja1j a1 and thereby find ^ grida1 (s) = X(a1s) 1 X = δ(a1s − k) k=−∞ 1 k = X s − ja1j a1 (7) = jA1j gridA1 (s): PERIODIC EIGENFUNCTIONS OF THE FT OPERATOR 4 Let g be any univariate distribution with compact support. We can periodize g by writing 1 X f(x) : = g(x − na1) n=−∞ 1 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) = jA1j gridA1 (s) g (s) 1 X ^ = jA1j δ(s − kA1) g (s) k=−∞ 1 X ^ = jA1j g (kA1)δ(s − kA1) ; k=−∞ and thereby obtain the weakly convergent Fourier series 1 X ^ 2πikA1x (9) f(x) = jA1j 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 jA1j 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 distri- butions. 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 eigenfunc- tions 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 21 1 1 ::: 1 3 1 !!2 :::!N−1 1 6 2 4 2N−2 7 F := 61 ! ! :::! 7 ;! := e−2πi=N N N 6. 7 4. 5 1 !N−1 !2N−2 :::!(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 21 0 0 ::: 0 03 0 0 0 ::: 0 1 60 0 0 ::: 1 07 R := 6. 7 N 6. 7 40 0 1 ::: 0 05 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 p p 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 = ±p f; f 6= 0; N then 2 2 1 RN f = (NF )f = N ±p f = f; N N i.e., f is even, and if i FN f = ±p f; f 6= 0; N then 2 2 i RN f = (NF )f = N ±p 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 = ±p f N then 1 CN f = ±p f; N and if i FN f = ±p f N then 1 SN f = ∓p 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 .