On eigenfunctions of the Fourier transform Flavia Lanzara ∗ Vladimir Maz’ya †‡ Abstract. In [6] we considered a nontrivial example of eigenfunction in the sense of distribution for the planar Fourier transform. Here a method to obtain other eigenfunctions is proposed. Moreover we consider positive homogeneous eigenfunctions of order n/2. We show that F (ω) x −n/2, ω = | | | | 1, is an eigenfunction in the sense of distribution of the Fourier transform if and only if F (ω) is an eigenfunction of a certain singular integral operator on n (k) −n/2 the unit sphere of R . Since Ym,n(ω) x are eigenfunctions of the Fourier (k) | | transform, we deduce that Ym,n are eigenfunctions of the above mentioned (k) singular integral operator. Here Ym,n denote the spherical functions of order m in Rn. In the planar case, we give a description of all eigenfunctions of the Fourier transform of the form F (ω) x −1 by means of the Fourier coefficients | | of F (ω). 1 Introduction The Fourier transform of a function f L2(Rn) is defined as ∈ 1 x f(ξ)= f(ξ)= f(x)e−i( ,ξ)dx , F (2π)n/2 Rn Z b (x,ξ)= x1ξ1 + ... + xnξn denoting the standard inner product of x and ξ in Rn. The inverse of the Fourier transform is given by 1 x −1f(ξ)= f(x)ei( ,ξ)dx . F (2π)n/2 Rn Z ∗Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy. email: [email protected] †Department of Mathematics, University of Link¨oping, 581 83 Link¨oping, Sweden ‡Department of Mathematical Sciences, M&O Building, University of Liverpool, Liv- erpool L69 3BX, UK email: [email protected] 1 The Fourier transform is an isomorphism of the space L2(Rn) and, for every f,g L2(Rn), we have Parseval formula ∈ f(ξ)g(ξ)dξ = f(x)g(x)dx . n n ZR ZR In particular, b b f 2 = f 2 . k kL k kL 2 n Here 2 denotes the norm in the space L (R ). As a consequence the k · kL linear map defines a unitaryb operator on L2(Rn), so its spectrum lies F on the unit circle in C. Since 4f = f, if λ C is an eigenvalue then F ∈ λ4 = 1. So λ 1, 1,i, i . Each of these values is an eigenvalue of infinite ∈ { − − } multiplicity. In dimension 1 a complete orthonormal set of eigenfunctions is given by the Hermite functions 1 −x2/2 Φm(x)= Hm(x)e , m 0 (√π2mm!)1/2 ≥ with the Hermite polynomial m 2 d 2 H (x)=( 1)mex e−x . m − dxm They satisfy (Φ )=( i)mΦ (see [9]). In higher dimensions, the eigen- F m − m functions of the Fourier transform can be obtained by taking tensor products of Hermite functions, one in each coordinate variable. That is n Φm(x)= Φmj (xj) jY=1 are eigenfunctions corresponding to the eigenvalues ( i)m1+...+mn . − In this paper we are interested in non standard eigenfunctions i.e. eigen- functions in the sense of distributions. In general such eigenfunctions do not belong to L2(Rn). An interesting example is provided by 1/ x n/2 ([5, | | p.363]). In [6] we have showed that 2 2 x1 + x2 px1x2 is a eigenfunction in the sense of distribution in R2. In section 2 we propose a method to obtain other eigenfunctions (theorems 2.2 and 2.4). We find, for example, that 8x x x2 x2 1 2 x2 + x2 or 2 − 1 x2 + x2 (x2 x2)2 1 2 x2x2 1 2 1 − 2 q 1 2 q 2 are eigenfunctions. In section 3 we consider positive homogeneous distribu- tions of order n/2 that is F (ω) x , ω = , x Rn . (1.1) x n/2 x ∈ | | | | We show that (1.1) is an eigenfunction of the Fourier transform if and only if F (ω) is an eigenfunction of a certain singular integral operator on the unit n (k) −n/2 sphere of R (theorem 3.3). Since Ym,n(ω) x are eigenfunctions of the | | (k) Fourier transform (theorem 3.1), we deduce that Ym,n are eigenfunctions (k) of the above mentioned singular integral operator. Here Ym,n denote the spherical functions of order m in Rn. In section 4 we give a description of all eigenfunctions of the planar Fourier transform of the form (1.1) by means of the Fourier coefficients of F (ω) (theorem 4.1). 2 Fourier transform of distributions Let be the space C∞(Rn) of all real functions with continuous derivative D 0 of all order and with compact support. A sequence Φk converges to n { }∈D Φ if there is a compact K R with suppΦk K, suppΦ K and α∈ D α ⊂ ⊆ ⊆ ∂ Φk converges uniformly to ∂ Φ on K for all α =(α1,...,αn) multi-index. We shall denote by ′ the set of all continuous linear functionals on . The D D elements of ′ are called generalized functions or distributions. We write D the action of a distribution f on a test function Φ as (f, Φ) with the property that if Φ converges to Φ in then k D lim (f, Φk)=(f, Φ) . k→+∞ If f L1 (Rn) the functional ∈ loc (f, Φ) = f(x)Φ(x)dx (2.1) n ZR belongs to the space ′ and distributions of the form (2.1) are called regular D distributions. If f L1 (Rn), if the integral exists in the Cauchy sense, then 6∈ loc (2.1) still defines a distribution called Cauchy principal value distribution (cf., e.g., [5, p.10,p.46]). We denote by the Schwartz space of functions Φ C∞(Rn) rapidly S ∈ decaying at infinity that is, for any multi-indeces α and β, lim xα∂βΦ(x) = 0 . |x|→∞ | | 3 A sequence Φ converges to Φ if, for any multi-indeces α and β, { k}∈S ∈S α β α β lim sup x D Φk(x) x D Φ(x) = 0 . k→∞ x∈Rn | − | We denote by ′ the class of continuous linear functionals f : S S → C. Elements of ′ are called tempered distributions. Obviously ′ ′, S D ⊂ S moreover ′ is dense in ′. Let f ′. Let α =(α ,...,α ) be an n tuple D S ∈S 1 n − of nonnegative integers and α = α + ... + α . The distribution ∂αf is | | 1 n defined by (∂αf, Φ)=( 1)|α|(f,∂αΦ), Φ . − ∈S To introduce some notation, we define the reflection in the origin (r Φ)(x)=Φ( x), Φ , 0 − ∈S and the translation through the vector h Rn ∈ τhΦ(x)=Φ(x h), Φ . − ∈S The reflection of a distribution f ′ is a distribution defined by ∈S (r f, Φ)=(f, (r Φ)), Φ 0 0 ∈S and the translation of f ′ is a distribution defined by duality ∈S (τhf, Φ)=(f,τ hΦ), Φ . − ∈S The Fourier transform is a continuous isomorphism of onto . This S S allows to define the Fourier transform of tempered distributions. Definition 2.1. Let f ′. Its Fourier transform f (or (f)) is the ∈ S F tempered distribution b (f, Φ)=(f, Φ), Φ . ∈S The formulas for theb derivativesb or for the translation of the Fourier transforms are preserved also for distributions. If α is a multi-index and h Rn then the Fourier transform of f ′ has the following properties, ∈ ∈ S in the sense of distribution, (∂αf)=(iξ)α (f), F F (xαf)= i|α|∂α (f), F F −ih·ξ (τhf)(ξ)=e (f)(ξ), F F ih·x (e f)= τh( (f)) . F F 4 Definition 2.2. The distribution f ′ is an eigenfunction of the Fourier ∈S transform corresponding to the eigenvalue λ if (f, Φ) = λ(f, Φ), Φ . (2.2) ∀ ∈S An example of eigenfunctionb understood in the sense of distribution, with eigenvalue 1, is provided by the generalized function 1/ x n/2 (see [3, | | p.71] and [5]). 1/ x n/2 does not define a regular distribution because it has | | a nonsommable singularity at the origin and the integral (2.1) can be defined by analytic continuation (see [5, p.71]). In [6] we proposed an example of non standard eigenfunction of the planar Fourier transform. The distribution x 2 2 f(x1,x2)= | | , x = x1 + x2 (2.3) x1x2 | | q defines a Cauchy principal value distribution in . We define the action on S a test function Φ as ∈S x (f, Φ) = lim | | Φ(x)dx = lim Ψ(x1,x2)dx1dx2 = Ψ(x)dx ǫ→0 |x1|>ǫ x x ǫ→0 x1>ǫ 2 1 2 R+ Z|x2|>ǫ Zx2>ǫ ZZ where Φ(x1,x2) Φ(x1, x2) Φ( x1,x2)+Φ( x1, x2) Ψ(x1,x2)= x − − − − − − | | x1x2 x x2 x1 1 1 = | | Φ (ξ,η)dξdη = x Φ (tx ,sx )dtds. x x ξη | | ξη 1 2 1 2 Z−x2 Z−x1 Z−1 Z−1 The eigenfunction (2.3) gives rise to a one parametric family of eigenfunc- tions. Indeed, denote by cos α sin α R = sin α cos α − the rotation matrix where α is the rotation angle in the counterclockwise direction. Then ( Φ(R ))(ξ)=( Φ( ))(Rξ). If f ′ is an eigenfunction, F · F · ∈S then also the rotation fR defined by (f , Φ)=(f, Φ(R )), Φ R · ∈S is an eigenfunction. As a consequence x2 + x2 f(α,x ,x )= 1 2 (2.4) 1 2 (x cos α + x sin α)( x sin α + x cos α) 1 2 p − 1 2 5 is an eigenfunction for the planar Fourier transform for any value of the parameter α.