On Eigenfunctions of the Fourier Transform

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. eigenfunctions 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 distributions 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

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 eigenfunctions. 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. For example, if we choose α = π/4 we obtain

x2 + x2 1 2 . x2 x2 p1 − 2 With the change of variable a = tan(α), we obtain the family of eigenfunctions x2 + x2 ̥(a,x ,x )= 1 2 . (2.5) 1 2 (x + ax )( x a + x ) 1 p2 − 1 2 If a = 0, putting b =(a2 1)/a and denoting by P (x ,x )= x2 x2 +bx x 6 − b 1 2 1 − 2 1 2 the homogeneous harmonic polynomial of degree 2 we get the following family of eigenfunctions

x2 + x2 x φ(b,x ,x )= 1 2 = | | . (2.6) 1 2 x Ppb(x1,x2) Pb( ) Deﬁnition 2.3. The distribution f ′ is an eigenfunction of the contin- ∈S uous spectrum for if there is a sequence Φ such that the following F { k}∈S conditions are satisﬁed:

lim ( (Φk) λΦk, Φ) = 0, Φ ; k→∞ F − ∀ ∈S

lim (Φk, Φ)=(f, Φ), Φ . k→∞ ∀ ∈S The next theorem shows the relation between eigenfunctions in the sense of distribution and eigenfunctions of the continuous spectrum.

Theorem 2.1. Eigenfunctions in the sense of distribution are eigenfunctions of the continuous spectrum.

Proof. We consider a function ρ(x) C∞(Rn) such that supp(ρ) B (0) ∈ 0 ⊆ 1 and ρ(x)dx = 1. For k 1 we consider the regularizing family of Rn ≥ functions R n ρk(x)= k ρ (k x) . For all Φ , the convolution ρ Φ tends to Φ in as k tends to + . ∈ S k ∗ S ∞ The convolution of ρ and f ′ is given by the formula k ∈S (ρ f)(x)=(f(y), ρ (x y)) . k ∗ k −

6 ρ f belongs to C∞(Rn) and every derivative has at most polynomial k ∗ growth. Moreover ρ f converges to f in ′ when k . Indeed, since k ∗ S → ∞ ρ Φ tends to Φ in , if we replace Φ by the reﬂected Ψ = r Φ we can write k ∗ S 0 lim (ρk f, Φ)= lim (ρk f,r0Ψ)= lim (f,r0(ρk Ψ)) = (f, Φ), Φ . k→∞ ∗ k→∞ ∗ k→∞ ∗ ∀ ∈S Since f satisﬁes (2.2) we have, for any Φ ∈S

lim ( (ρk f) λ(ρk f), Φ)= lim (ρk f, Φ) λ(ρk f), Φ) k→∞ F ∗ − ∗ k→∞ ∗ − ∗ =(f, Φ) λ(f, Φ) = 0 . − b b Let f,g ′ and suppose that at least one has bounded support. We ∈ D deﬁne the convolutional distribution

(f g, Φ)=(f(x) g(y), Φ(x + y)), Φ ∗ × ∀ ∈D (cf. e.g. [5, p.103] or [1, p.89]). If f,g,h ′ and at least two of them ∈ D have bounded support, then f g h ′ and the convolution product is ∗ ∗ ∈ D associative f g h = f (g h)=(f g) h. ∗ ∗ ∗ ∗ ∗ ∗ The next theorem shows that if we convolve (with respect to the parameter) a parametric family of eigenfunctions in the sense of distribution and a distribution we get again an eigenfunction.

Theorem 2.2. Let f(a, x) be a family of distributions in ′ depending on S a parameter a R. Suppose that ∈ i. for fixed a R, f(a, ) ′ is an eigenfunction in the sense of distribution; ∈ · ∈S ii. for fixed x Rn, f( , x) ′ is a distribution on the real line. ∈ · ∈D Let g(a) ′ be a distribution on the real line with bounded support. For ∈ D fixed x Rn consider the convolution g(a) f(a, x) defined as ∈ ∗a (g(a) f(a, x),ψ)=(g(a) f(b, x),ψ(a + b)), ψ C∞(R) . ∗a × ∀ ∈ 0 Then, for fixed a R, g(a) f(a, x) ′ and it is an eigenfunction in the ∈ ∗a ∈S sense of distribution of the Fourier transform.

Proof. g(a) f(a, x) is well deﬁned as an element of ′. By hypothesis ∗a S (f(a, ), Φ) = λ(f(a, ), Φ), Φ . · · ∀ ∈S b 7 If we convolve both terms by g we get

(g(a) f(a, x), Φ(x)) = ((g(a) f(b, x),ψ(a + b)), Φ(x)) ∗a × = ((g(a) (f(b, x), Φ(x)),ψ(a + b)) = λ((g(a) (f(b, x), Φ(x)),ψ(a + b)) × b × b = λ(g(a) f(b, x),ψ(a + b)), Φ(x)) = λ(g(a) a f(a,x), Φ) × b ∗ which proves the theorem.

Example 2.1. The convolution Dδ f(a, x) is well deﬁned, where D is ∗a any diﬀerential operator and δ is the delta function

(δ, ψ)= ψ(0), ψ C∞(R) . ∀ ∈ 0 This follows from the fact that Dδ is concentrated in one point. By deﬁnition

(Dδ f( , x),ψ)=(Dδ(b) f(a, x),ψ(a + b)) ∗a · × =(f(a, x), (Dδ(b),ψ(a + b))=(f(a, x),Dψ(a)) =(D f(a, x),ψ(a))=(D f( , x),ψ) . a a · Thus we have Dδ f(a, x)= D f(a, x) . (2.7) ∗a a Let us apply theorem 2.2 and (2.7) to the family of eigenfunctions (2.4). If we derive with respect to the parameter α we obtain the family of eigenfunctions

2 2 cos(2α) x1 x2 + 2x1x2 sin(2α) 2 2 − 2 2 x1 + x2 . (x2 cos(α) x1 sin(α)) (x1 cos(α)+ x2 sin(α)) − q For α = π/4 we get the eigenfunction 8x x 1 2 x2 + x2 (x2 x2)2 1 2 1 − 2 q whereas, for α = 0, x2 x2 1 − 2 x2 + x2 . x2x2 1 2 1 2 q If we take the second and third derivative at α = 0 we obtain, respectively,

2 x4 + x4 x6 x6 x4x2 x2x4 1 2 x2 + x2, 6 1 − 2 + 2 1 2 − 1 2 x2 + x2 x3x3 1 2 x4x4 x4x4 1 2 1 2 q 1 2 1 2 q

8 and, at α = π/4 ,

4 2 2 4 5 3 3 5 8 x1 + 6x1x2 + x2 2 2 32 5x1x2 + 14x1x2 + 5x1x2 2 2 x1 + x2, x1 + x2. − x2 x2 3 x2 x2 4 1 − 2 q 1 − 2 q In this way we can obtain many eigenfunctions for the planar Fourier Trans- form starting from (2.4). Example 2.2. The integration of an eigenfunction f(a, x) with respect to any Borel measure µ(a) on R gives rise to new eigenfunctions. Indeed

(µ,ψ)= ψ(a)dµ(a) ZR ∞ R deﬁnes a distribution on C0 ( ). If µ(a) has bounded support then (µ f( , x),ψ)=(µ(a) f(b, x),ψ(a + b))=(µ(a), (f(b, x),ψ(a + b))) ∗a · × =(µ(a), (f(c a, x),ψ(c)) = ((µ(a),f(c a, x)),ψ(c)) . − − Then, for any ﬁxed c R ∈ µ f( , x)= f(c a, x)dµ(a) ∗a · − ZR is an eigenfunction. As an example, consider the family of eigenfunctions (2.6). New eigenfunctions are given by

b x2 + x2 µ(b) φ(b, x)= φ(y, x)dy = 1 2 log bx x + x2 x2 , b> 0 . ∗b x x 1 2 1 − 2 Z0 p 1 2

Similarly, if we integrate the family of eigenfunctions (2.5) with respect to the parameter, we get the eigenfunctions a 1 ̥(y,x ,x )dy = (log ax + x log ax x ) . 1 2 x | 2 1|− | 1 − 2| Z0 | | Let us consider now the planar case. We introduce polar coordinates (R,ϕ) and (r, ϑ) such that R = x , eiϕ = x/R and r = y , eiϑ = y/r. We | | | | use the notation f(x) = f(R,ϕ) and g(y) = g(r, ϑ). It is obvious that f and g are 2π periodic functions with respect to the angle. We write the 2D − Fourier transform in polar coordinates

1 ∞ 2π f(R,ϕ)= f(r, ϑ)e−irR cos(ϕ−ϑ)rdrdϑ. (2.8) 2π Z0 Z0 b 9 Let f and g , where is the set of all C∞(R) functions which are ∈ S ∈ P P 2π-periodic. We deﬁne the angular (or circular) convolution as follows ∗ϕ 2π (g f)(r, ϕ)= g(ω)f(r, ϕ ω)dω. ∗ϕ − Z0 We have (g f)= (g f) . ∗ϕ F F ∗ϕ Indeed,

2π (g f)(R,ϕ)= g(ω)( f)(R,ϕ ω)dω ∗ϕ F F − Z0 1 ∞ 2π 2π = rdr g(ω)dω f(r, ϑ)e−irR cos(ϕ−ω−ϑ)dϑ 2π Z0 Z0 Z0 1 ∞ 2π 2π = rdr g(ω)f(r, χ ω)dω e−irR cos(ϕ−χ)dχ 2π − Z0 Z0 Z0 1 ∞ 2π = rdr (g f)(r, χ)e−irR cos(ϕ−χ)dχ = (g f)(R,ϕ) . 2π ∗χ F ∗ϕ Z0 Z0 It follows that, if f is an eigenfunction of the Fourier transform, then ∈ S the angular convolution g f is still an eigenfunction. This is valid also if ∗ϕ g is a distribution supported on the unit circle and f ′. ∈S We denote by ′ the topological dual of . A seguence g converges P P { k}∈P to g in if g(s) tends to g(s) uniformly in R. Elements of ′ are called P k P periodic distributions. The action of g ′ on a test function ψ is ∈ P ∈ P denoted by g,ψ and the translation τ g is defined by τ g,ψ = g,τ ψ h i α h α i h −α i with (τ−αψ)(ϑ)= ψ(ϑ + α). Definition 2.4. If g ′ and f ′, we define the angular convolution ∈P ∈S (g f, Φ) = g,ψ with ψ(ω)=((f(R,ϑ), (τ Φ)(R,ϑ)) Φ ∗ϑ h i −ω ∈S where (τ−ωΦ)(R,ϑ)=Φ(R,ϑ + ω). We can easily check directly that

τ ( (Φ))(R,ϑ)= (τ Φ)(R,ϑ), Φ . (2.9) −ω F F −ω ∈S Proposition 2.3. For f ′ and g ′ we have ∈S ∈P ( (g f), Φ)=(g (f), Φ), Φ . F ∗ϑ ∗ϑ F ∈S

10 Proof. Indeed, by deﬁnition of Fourier transform and angular convolution we have

( (g f), Φ)=(g f, (Φ)) = g(ω), (f(R,ϑ),τ ( (Φ))(R,ϑ) . F ∗ϑ ∗ϑ F h −ω F i Then, keeping in mind (2.9),

( (g f), Φ) = g(ω), (f(R,ϑ), (τ Φ)(R,ϑ)) F ∗ϑ h F −ω i = g(ω), ( (f)(R,ϑ), (τ Φ)(R,ϑ)) =(g (f), Φ) . h F −ω i ∗ϑ F

Theorem 2.4. Let f be an eigenfunction in the sense of distribution for F and g ′. Then the angular convolution g f is an eigenfunction in the ∈P ∗ϑ sense of distribution for . F Proof. Indeed, if f ′ satisfies (2.2), then ∈S ( (g f), Φ)=(g f, (Φ)) = g(ω), (f(R,ϑ), (τ Φ)(R,ϑ)) F ∗ϑ ∗ϑ F h F −ω i = λ g(ω), (f(R,ϑ), (τ Φ)(R,ϑ)) = λ(g f, Φ) . h −ω i ∗ϑ Hence also g f satisfies (2.2). ∗ϑ Corollary 2.5. Suppose that f ′ is an eigenfunction for . Then ∈S F i. the translation of f with respect to the angle, ταf, defined by

(τ f, Φ)=(f,τ Φ), Φ α −α ∈S is an eigenfunction for ; F ii. the derivative in the sense of distribution of f with respect to the angle ∂s s f,s 1 deﬁned by ∂ ϑ ≥ ∂s ∂s ( f, Φ)=( 1)s(f, Φ), Φ ∂sϑ − ∂sϑ ∈S is an eigenfunction for . F

Proof. i. Let us denote by δ(α) the delta function at the point α that is

δ ,ψ = ψ(α), ψ . h (α) i ∈P We have δ ′ and for any f ′ (α) ∈P ∈S τ f = δ f. α (α) ∗ϑ

11 Indeed, by deﬁnition,

(τ f, Φ)=(f(R,ϑ), Φ(R,ϑ + α))=(f(R,ϑ), δ (ω), Φ(R,ϑ + ω) ) α h (α) i = δ (ω), (f(R,ϑ), Φ(R,ϑ + ω)) =(δ f, Φ), Φ h (α) i (α) ∗ϑ ∈S where we have used that

Φ(R,ϑ + α)=(δ(α)(ω), Φ(R,ϑ + ω)) .

Then we can apply theorem 2.4. ii. Let δ(s) be the derivative of the delta function, deﬁned as

δ(s),ψ =( 1)sψ(s)(0), ψ . h i − ∈P Hence δ(s),τ ψ =( 1)sψ(s)(ϑ) . h −ϑ i − Then

(δ(s) f, Φ) = δ(s)(ω), (f(R,ϑ), Φ(R,ϑ + ω)) ∗ϑ h i ds =(f(R,ϑ), δ(s)(ω), Φ(R,ϑ + ω) )=( 1)s(f(R,ϑ), Φ(R,ϑ)) h i − dϑ ds =( f(R,ϑ), Φ(R,ϑ)) . dϑ Thus we have ds δ(s) f = f(R,ϑ) ∗ϑ dϑ and we can apply theorem 2.4.

Remark 2.6. The translation in angle is equivalent to rotation therefore statement i. states that rotated eigenfunctions are still eigenfunctions. The- orem 2.4 can be viewed as a particular case of theorem 2.2. Indeed, if the parameter a in theorem 2.2 is the angle of rotation, that is f(a, x) in polar coordinates is f(R,ϑ a), then the convolution in a gives the same result of − the convolution in the angle.

Example 2.3. Let us write the eigenfunction (2.3) in polar coordinates

Φ(ϑ) 1 f(R,ϑ)= with Φ(ϑ)= . R sin(ϑ) cos(ϑ)

12 By virtue of Corollary 2.5 diﬀerentiation of any order with respect to the angle produces new eigenfunctions. If we consider ﬁrst and second derivatives we get the eigenfunctions

Φ′(ϑ) 1 1 1 x2 x2 = = 2 − 1 x2 + x2 , R R cos2(ϑ) − sin2(ϑ) x2x2 1 2 1 2 q Φ′′(ϑ) 2 cos(ϑ) sin(ϑ) x x = + = 2 1 + 2 x2 + x2 . R R sin3(ϑ) cos3(ϑ) x3 x3 1 2 2 1 q 3 A characterization of eigenfunctions

(k) We denote by Ym,n(ω) the spherical functions of order m in the n dimensional space, ω is a point of the unit sphere S. The upper index k numbers the linearly independent spherical functions of the same order m and it varies between the bounds (m + n 3)! 1 k k = (2m + n 2) − . ≤ ≤ m,n − (n 2)!m! − Theorem 3.1. The functions

Y (k) (ω) x m,n , ω = , k = 1,...,k , m 0 x n/2 x m,n ≥ | | | | are eigenfunctions of the Fourier transform and we have

(k) (k) Ym,n( ) Ym,n(Λ) ξ · (ξ)=( i)m , Λ= . F n/2 − ξ n/2 ξ |·| ! | | | | Proof. We seek for the Fourier transform

1 ∞ (ξ)= Rn/2−1dR Y (k) (θ)e−iR|ξ| cos γd S. F (2π)n/2 m,n θ Z0 ZS

13 In the integral herein we substitute t = R ξ and we obtain | | 1 ∞ (ξ)= tn/2−1dt Y (k) (θ)e−it cos γd S. F (2π)n/2 ξ n/2 m,n θ | | Z0 ZS We use the formula ([7, p.250] ξ Y (k) (θ)eit cos γd S = t1−n/2(2π)n/2imJ (t)Y (k) (Λ), Λ= m,n θ n/2+m−1 m,n ξ ZS | | where Jµ(t) denotes the Bessel function of the ﬁrst kind of order µ (cf.[10]). Hence Y (k) (Λ) ∞ (ξ)= im m,n ( 1)1−n/2 J ( t)dt F ξ n/2 − n/2+m−1 − | | Z0 Y (k) (Λ) ∞ =( i)m m,n J (t)dt. (3.1) − ξ n/2 n/2+m−1 | | Z0 Since (cf. [10, 13.24]) ∞ Jn/2+m−1(t)dt = 1 Z0 the theorem is proved.

Remark 3.2. Theorem 3.1 can be obtained as a particular case of the Bochner formula (cf.[2, Theorem 2]):

1 m (k) x −i(x,ξ) x Ym,n( )ϕ( x )e dx (2π)n/2 Rn | | x | | Z | | m (k) ∞ ( i) Ym,n(Λ) t n +m = − ϕ J n (t)t 2 dt ξ n+m ξ 2 +m−1 | | Z0 | | where ϕ is measurable in (0, ). Indeed, assuming ϕ( x ) = x −m−n/2 we ∞ | | | | get (3.1). Let us consider homogeneous functions of degree n/2 of the form − F (ω) x f(x)= , R = x , ω = . (3.2) Rn/2 | | x | | Here F (ω) is deﬁned on the unit sphere S. Following [5] we use the notation

(σ i0)λ = σλ +e±iλπσλ ± + − where σλ is equal to σλ for σ > 0 and to 0 if σ 0 and σλ is equal to σ λ + ≤ − | | for σ < 0 and to 0 for σ 0. ≥

14 Theorem 3.3. Let be the following singular integral operator on the (n K − 1) dimensional unit sphere − 1 n F (Λ) = Γ e−inπ/4 (ω Λ i0)−n/2 F (ω)dS . n/2 ω (3.3) K (2π) 2 S · − Z The function (3.2) is an eigenfunction of the Fourier transform corresponding to the eigenvalue λ if and only if F is an eigenfunction of (3.3) corresponding to the same eigenvalue, i.e.

F = λF . (3.4) K Proof. The Fourier transform of f in spherical coordinates has the form

∞ 1 −iRρ cos γ n−1 (f)(ξ)= dSω f(Rω)e R dR F (2π)n/2 ZS Z0 where we made use of the notations ρ = ξ ,Λ= ξ/ρ; the angle between x | | and ξ is denoted by γ that is cos γ = ω Λ. · Hence an eigenfunction f(Rω) of the Fourier transform is deﬁned by the equation

∞ 1 −iRρ cos γ n−1 dSω f(Rω)e R dR = λf(ρΛ), ρ> 0, Λ = 1 . (2π)n/2 | | ZS Z0 In the integral above we replace the function f in the form (3.2) and substitute t = Rρ, so that we obtain the following integral equation for F on the unit sphere

∞ 1 n/2−1 −it cos γ F (ω)dSω t e dt = λF (Λ) . (2π)n/2 ZS Z0 Now we make use of the following formula (cf. [5, pp.172-174] )

∞ n n tn/2−1e−itσdt = Γ einπ/4( σ+i0)−n/2 = Γ e−inπ/4(σ i0)−n/2 . 0 2 − 2 − Z (3.4) and (3.3) follow.

Remark 3.4. From theorems 3.1 and 3.3 we obtain that spherical functions (k) Ym,n solve the singular integral equation

Y (k) =( i)mY (k) , 1 k k . K m,n − m,n ≤ ≤ m,n

15 Remark 3.5. If n = 2 then ([5, p.60]) 1 (σ i0)−1 = + iπδ(σ) − σ where δ denotes the delta function. Then (3.3) can be written as i 1 F (Λ) = + πiδ(cos γ) F (ω)dω. K −2π cos γ Z|ω|=1 The homogeneous harmonic polynomials on the unit circle are

Y (1) (ϑ) = cos(mϑ), Y (2) (ϑ) = sin(mϑ) m Z . m,2 m,2 ∈ imϑ If we denote by Ym(ϑ)=e , we obtain that Ym satisﬁes the singular integral equation on the unit circle

Y =( i)mY , m Z . K m − m ∀ ∈ 4 Description of planar eigenfunctions via Fourier series

Every function Φ L2([0, 2π]) admits an expansion into a series with respect ∈ to the spherical functions ∞ ikϑ Φ(ϑ)= ckYk(ϑ), Yk(ϑ)=e (4.1) k=X−∞ 2π 1 −ikϑ with the coefficients ck = Φ(ϑ)e dϑ. A similar result holds for 2π 0 periodic distributions. Z Let be the set of all C∞(R) functions with complex values that are P 2π periodic. Any u can be written as the Fourier series (4.1). Let ′ − ∈ P P be the set of all continuous linear functionals on ′. The action of Φ ′ P ∈ P on a test function ψ is denoted by Φ,ψ . Any Φ ′ can be written as the h i ∈P Fourier series (4.1), which converges in the sense of distributions ∞ c eikϑ,ψ = Φ,ψ , ψ kh i h i ∀ ∈P k=X−∞ where the coefficients are defined by 1 c = Φ(ϑ), e−ikϑ . k 2π h i

16 A complex sequence c Z is said to have polynomial growth if there exists { k}k∈ an integer L and a positive constant C such that

c C k L, k Z . (4.2) | k|≤ | | ∈ ∞ ikϑ Any series of the form cke whose coefficients have polynomial growth k=−∞ converges in the sense ofX distributions to a distribution with the coefficients c as its Fourier coefficients and, conversely, Fourier coefficients of any { k} periodic distribution are a sequence of polynomial growth. (cf. [8, p.225], [4, p.33], [5, p.30]). As a consequence of Theorem 3.1 we prove a characterization of positive homogeneous eigenfunctions of the form Φ(ϑ)r−1 of the planar Fourier transform by means of their Fourier coefficients.

Theorem 4.1. If the distribution Φ(ϑ)r−1, with Φ ′, is an eigenfunction ∈P of the planar Fourier transform corresponding to the eigenvalue λ, then the coeﬃcients ck of the Fourier series (4.1) satisfy the conditions

c ( i)k λ = 0, k Z . (4.3) k − − ∀ ∈ Conversely, let λ C with λ = 1 and c be a sequence with polynomial ∈ | | { k} growth (4.2) satisfying conditions (4.3). Then Φ(ϑ)r−1, with Φ deﬁned in (4.1), is an eigenfunction of the planar Fourier transform. The convergence −ℓ of the series in the sense of distribution is in the space W2 ((0, 2π)), ℓ > L + 1/2.

Proof. Let Φ ′. Φ(ϑ)r−1 is an eigenfunction corresponding to the eigen- ∈P value λ if it satisﬁes Φ(ϕ) Φ(ϑ) ( )(r, ϑ)= λ . F R r If we replace Φ by its Fourier series (4.1), the last equation can be rewritten as ∞ ∞ Y (ϕ) Y (ϑ) c ( k )(r, ϑ)= λ c k . (4.4) kF R k r k=X−∞ k=X−∞ According to theorem 3.1 we have

Y (ϕ) Y (ϑ) ( k )(r, ϑ)=( i)k k , k Z . F R − r ∀ ∈

17 Hence (4.4) implies

∞ c ( i)k λ Y (ϑ) = 0 k − − k k=X−∞ which gives (4.3). Conversely, suppose that c is a sequence of polynomial growth, which { k} satisﬁes (4.3). Then Φ deﬁned in (4.1) belongs to ′ and, due to theorem P 3.1,

∞ ∞ Φ(ϕ) Y (ϕ) Y (ϑ) ( )(r, ϑ)= c ( k )(r, ϑ)= c ( i)k k F R kF R k − r k=−∞ k=−∞ X∞ X Y (ϑ) Φ(ϑ) = λ c k = λ . k r r k=X−∞ We obtain that Φ(ϑ)r−1 is an eigenfunction of the planar Fourier transform. The series (4.4) can be obtained by ℓ term-by-term diﬀerentations of the ∞ ℓ ikϑ 2 series (ck/(ik) )e which converges in L ((0, 2π)) if ℓ>L + 1/2. k=X−∞

Corollary 4.2. Φ(ϑ)r−1 is an eigenfunction of the Fourier transform (2.8) corresponding to the eigenvalue λ if and only if Φ ′, Φ 0, admits the ∈ P 6≡ following Fourier expansion

∞ ∞ 4isϑ −4isϑ Φ(ϑ)= c4se + c−4se if λ = 1; s=0 s=1 X∞ X ∞ Φ(ϑ)= c ei(4s+2)ϑ + c e−i(4s+2)ϑ if λ = 1; 4s+2 −(4s+2) − s=0 s=0 X∞ X∞ (4.5) i(4s+3)ϑ −i(4s+1)ϑ Φ(ϑ)= c4s+3e + c−(4s+1)e if λ = i; s=0 s=0 X∞ X∞ Φ(ϑ)= c ei(4s+1)ϑ + c e−i(4s+3)ϑ if λ = i. 4s+1 −(4s+3) − s=0 s=0 X X Proof. We write the series (4.1) as follows

∞ ∞ ikϑ −ikϑ Φ(ϑ)= cke + c−ke . Xk=0 Xk=1 18 Kepping in mind the obvious relations valid for k 0 ≥ 1 if k 0 mod 4 ≡ i if k 1 mod 4 ( i)k = − ≡ − 1 if k 2 mod 4 − ≡  i if k 3 mod 4 ≡  from theorem 4.1 we deduce that Φ(ϑ)r−1 is an eigenfunction with eigenvalue λ if and only the Fourier series of Φ has the form (4.5)

Example 4.1. The eigenfunction (2.3) in polar coordinates has the form

Φ(ϑ) 2 with Φ(ϑ)= . r sin(2ϑ) Let us compute the Fourier coeﬃcients of Φ. It is clear that

1 π e−ikϕdϕ 2π e−ikϕdϕ 1 π e−ikϕ(1+( 1)k) c = + = − dϕ k π sin(2ϕ) sin(2ϕ) π sin(2ϕ) Z0 Zπ Z0 Hence the coeﬃcients are zero if k is odd. Assume that k = 2s. Then

π/2 −i2sϕ s 0 if s = 2r 2 e (1 ( 1) ) π/2 −i2(2r+1)ϕ c2s = − − dϕ = 4 e π 0 sin(2ϕ)  dϕ if s = 2r + 1 Z π 0 sin(2ϕ) Z It remains to compute 

4 π/2 cos((4r + 2)ϕ) π/2 sin((4r + 2)ϕ) c = dϕ i dϕ 4r+2 π sin(2ϕ) − sin(2ϕ) Z0 Z0 ! The ﬁrst integral is zero. Indeed,

π/2 cos((4r + 2)ϕ) π/4 cos((4r + 2)ϕ) π/2 cos((4r + 2)ϕ) dϕ = dϕ+ dϕ = 0 sin(2ϕ) sin(2ϕ) sin(2ϕ) Z0 Z0 Zπ/4 We prove by induction that

π/2 sin((4r + 2)ϕ) π I = dϕ = , r 0 . r sin(2ϕ) 2 ≥ Z0 Clearly I = π/2. Suppose that I = π/2,r 1. From the relation 0 r ≥ sin((4r + 6)ϕ) = 2 sin(2ϕ) cos((4r + 4)ϕ) + sin((4r + 2)ϕ)

19 we obtain

π/2 sin((4r + 6)ϕ) π/2 π I = dϕ = I + 2 cos((4r + 4)ϕ)dϕ = I = . r+1 sin(2ϕ) r r 2 Z0 Z0 If r< 0 π I = I = . r − −r−1 − 2 Hence 2i r 0 c4r+2 = − ≥ 2i r< 0 and ∞ ∞ 2 = 2i e−i(4r+2)θ ei(4r+2)θ = 4 sin((4r + 2)θ). sin(2θ) − r=0 r=0 X X Example 4.2. Let us compute the Fourier coeﬃcients of

cos(2ϑ) Φ(ϑ) = 2 . sin(2ϑ)

We have 1 2π cos(2ϕ) 1 π cos(2ϕ) c = e−ikϕdϕ = (1+( 1)k)e−ikϕdϕ. k π sin(2ϕ) π sin(2ϕ) − Z0 Z0 Hence the coeﬃcients are zero if k is odd. Assume that k = 2s. Then

π 0 if s = 2r + 1 2 cos(2ϕ) −i2sϕ π/2 c2s = e dϕ = 4 cos(2ϕ) −i4rϕ π 0 sin(2ϕ)  e dϕ if s = 2r Z π 0 sin(2ϕ) Z Let us compute 

4 π/2 cos(2ϕ) π/2 cos(2ϕ) c = cos(4rϕ)dϕ i sin(4rϕ)dϕ . 4r π sin(2ϕ) − sin(2ϕ) Z0 Z0 ! We have

π/2 cos(2ϕ) cos(4rϕ)dϕ sin(2ϕ) Z0 π/4 cos(2ϕ) π/2 cos(2ϕ) = cos(4rϕ)dϕ + cos(4rϕ)dϕ = 0 sin(2ϕ) sin(2ϕ) Z0 Zπ/4 20 where we have made the substitution ϕ = π/2 ϑ in the second integral. − Since 2 cos(2ϕ) sin(4rϕ) = (sin((4r + 2)ϕ) + sin((4r 2)ϕ)) − we get, for r 1, ≥ π/2 cos(2ϕ) 1 π π J := sin(4rϕ)dϕ = (I + I )= , J = J = . r sin(2ϕ) 2 r r−1 2 −r − r − 2 Z0 Hence 2i r 1 − ≥ c = 0 r = 0 4r   2i r< 1 and ∞  ∞ cos(2ϑ) 2 = 2i e4irθ e−4irθ = 4 sin(4rθ) . sin(2ϑ) − − r=1 r=1 X X From Corollary 4.2 we obtain that

cos(2ϑ) 1 x2 x2 1 2 = 1 − 2 sin(2ϑ) r x x 2 2 1 2 x1 + x2 is an eigenfunction of the planar Fourierp transform corresponding to the eigenvalue λ = 1 that is a ﬁxed point of . F References

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