Funk-Minkowski Transform and Spherical Convolution of Hilbert Type

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The vectors e1(ξ), e2(ξ) and ξ form the so called local moving triad ξ e1 = 0, 3 ξ e2 =0, e1 e2 =0, where denotes the inner product of two vectors in R . 2 Let denote by feven and fodd the even and odd parts of function f on S , respectively, that is, we have f(ξ)+ f( ξ) f(ξ) f( ξ) f(ξ)= f (ξ)+ f (ξ), f (ξ)= − , f (ξ)= − − . even odd even 2 odd 2 The space of continuous functions on the sphere S2 is denoted by C(S2) and is endowed with the supremum norms

f C(S2) = supξ S2 f(ξ) . || || ∈ | | 2 2 2 2 C(S ), Ceven(S ) and Codd(S ) denote the space of continuous functions on S , the space of even continuous functions on S2 and the space of odd continuous functions 2 2 2 on S , respectively. The subset of Ceven(S ) (Codd(S )) that contains the infinitely S2 S2 differentiable functions will be denoted by Ceven∞ ( ) (Codd∞ ( )). Definition 1. Let f be a continuous function on the sphere S2, f C(S2). Then, for a unit vector ξ S2 the Funk–Minkowski transform of a function∈f is a function f on S2, given by∈ F 1 2π (1) f (ξ) ξf = f e1(ξ)cos ω + e2(ξ) sin ω dω. {F } ≡F 2π 0 Z It is clear that the Funk–Minkowski transform is even, f ( ξ) = f (ξ), and annihilates all odd functions. {F } − {F } TheF inversion of the Funk–Minkowski transform has been treated by many authors and there are exist several inversion formulas in the literature, see [9, 47, 17, 38, 39]. In [9, 11] P. Funk proved that an even function can be recovered from the knowledge of integrals over great circles and presented two different inversion methods: the first method is based on the spherical harmonic decomposition of the functions f, f and the second one utilizes Abel’s integral equation, [28]. The inversionF formula after P. Funk was obtained by V. Semyanisty in [47, formulas (9) and (11)], 1 1 θ η η (2) feven( )= 2 f ( )d , −4π S2 (θ η) {F } Z S2 where the dη is the surface measure on with normalization S2 dη = 4π and integral is understood in the regularized sense. R FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE 3

In [17, p. 99] S. Helgason gives for (1) the inversion formula of ﬁltered back- projection type u 1 d 2 vdv (3) feven(θ)= f (η)δ η θ 1 v dη , 2 2 2π du 0 S2 {F } − − √u v u=1 Z Z − p where δ denotes the the Dirac delta function. Another example of inversion formula is due to B. Rubin [38, 39]

1 ∆θ (4) feven(θ)= f (η)dη + ln η θ f (η) dη , 4π S2 {F } 4π S2 | |{F } Z Z here ∆θ it the Laplace–Beltrami operator (31). In our studies, an important role is played by spherical convolution operator , which is the spherical analogue of Hilbert transform, see [44, 45, 41, 21]. S Deﬁnition 2. Let f C(S2). The spherical convolution operator is deﬁned by, ∈ S 1 v(η) 2 (5) v (θ) θv = p.v. dη, θ S . {S } ≡ S 4π S2 θ η ∈ Z This transform is odd, f ( θ) = f (θ), and annihilates all even functions. {S } − −{S } S The results of this paper are formulated below in Theorems 1 and 2. Theorem 1. For any function f(θ) H1(S2) the following identity take place ∈ 1 1 (η + θ) , f (η) (6) f(θ)= f (η)dη +p.v. F ∇ dη 4π S2 {F } 4π S2 nhη θ i o Z Z =f00

= |f00 + θ{z(η + θ) } , f. S F ∇ η h i Here operators and are the Funk–Minkowski transform (1) and the surface gradient (21), respectively.F ∇ Through the square brackets [., .] we, as usual, denoted the commutator , f = f f, where the F–M transform is applied F ∇ F∇ − ∇F F to vector functionh f byi componentwise. ∇ If we decompose identity (6) on even and odd parts then we can write, 1 1 θ f (η) feven(θ)= f (η)dη p.v. {∇F } dη 4π S2 {F } − 4π S2 θ η Z Z 1 (7) = f (η)dη θ θ f , 4π S2 {F } − S ∇F Z 1 η f (η) (8) fodd(θ)= p.v. {F∇ } dη = θη η f. 4π S2 η θ S F ∇ Z The inversion formulas for feven and fodd follow from these equations and if two F–M transformations g(η)= f (η) and h(η)= f (η) are known, then the unknown function f can be reconstruct{F } completely,{F∇ } 1 1 θ g(η) 1 η h(η) (9) f(θ)= g(η)dη p.v. ∇ dη + p.v. dη . 4π S2 − 4π S2 θ η 4π S2 η θ Z Z Z 4 S.G. KAZANTSEV

The next problem that we will consider is the problem of Helmholtz–Hodge decomposition for a tangential vector ﬁeld on the sphere S2, see [12]. The Helmholtz– Hodge decomposition says that we can write any vector ﬁeld tangent to the surface of the sphere as the sum of a curl-free component and a divergence-free component

(10) f(θ)= θu(θ)+ θ θv(θ), ∇ × ∇ where θ is the surface gradient on the sphere, and rotated gradient θ θ means the cross-product∇ of the surface gradient of v with the unit normal vector× ∇θ to the sphere. Here θu is called also as inrrotational, poloidal, electric or potential field ∇ and θ⊥v is called as incompressible, toroidal, magnetic or stream vector field. Scalar functions∇ u and v are called velocity potential and stream functions, respectively. In the next theorem we show that decomposition (10) is obtained by use of Funk–Minkowski- transform and spherical convolution transform . F S 2 Theorem 2. Any vector field f L2,tan(S ) that is tangent to the sphere can be uniquely decomposed into a sum (10)∈ of a surface curl-free component and a surface divergence-free component with scalar valued functions u, v H1(S2)/R. Functions u and v are velocity potential and stream functions that are∈ calculated unique up to a constant by the formulas u(θ)= , η , f = η f (θ) η f (θ) S F θ S F − F S (11) = h θη ηfi θηn ηf, o n o S F −F S v(θ)= θ , η , f = θ η f (θ) θ η f (θ) S × F θ S ×F − F × S (12) = θ h θη ηfi θ θnη ηf, o n o S ×F − F × S where through [ , , ] we denote the generalized commutator, A B C [ , , ]= . A B C ABC −CBA As a consequence of this theorem, we can obtain formulas for solving two important 2 problems on the sphere S : u = f and ⊥v = g . Answers to solve these problems are ∇ ∇ 2 u(θ) = ( θη η θη η)f for u = f L (S ) S F −F S ∇ ∈ 2,tan and 2 v(θ)= θ ( θη η θη η)g for ⊥v = g L (S ). S ×F −F × S ∇ ∈ 2,tan 2. Basic methods and tools 2.1. Spherical harmonics (SHs). In this section we state some properties of 2 complex spherical harmonics. A spherical harmonic YNℓ of degree N on S is the restriction to S2 of a homogeneous harmonic polynomial of degree N in R3. The Legendre polynomials of the first kind PN of degree N N0 or simply Legendre polynomials are given by the Rodrigues formula ∈ 1 dN P (t)= (t2 1)N . N N!2N dtN − We recall that Legendre polynomials of the first kind PN (t) are the orthogonal (3/2) polynomials on ( 1, 1) with weight function w(t)=1. We define with CN the Gegenbauer polynomial− of degree N with parameter λ =3/2, d C(3/2)(t)= P (t). N dt N+1 FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE 5

The following formulas will be used in our calculations ([1]) Γ(j +1/2) ( 1)j (2j 1)!! (13) P (0) = ( 1)j = − − , 2j − √πj! (2j)!!

(14) (N + 1)PN+1(0) = NPN 1(0), − − j (3/2) ( 1) (2j + 1)!! (3/2) (15) C2j (0) = − or CN 1 (0) = NPN 1(0), N =2j +1. (2j)!! − − The following usefull asymptotics holds as j goes to infinity 1 1 1 1 (16) P2j (0) and (3/2) = if j . ∼ √2j +1 (2j + 1)P2j (0) ∼ √2j +1 → ∞ C2j (0) ℓ The associated Legendre functions of the first kind PN for non negative ℓ 0 are defined as ≥ ℓ ℓ 2 ℓ d P (t)=(1 t ) 2 P (t), N − dtℓ N ℓ where N,ℓ N with ℓ N and for the negative order ℓ, P − are given by ∈ 0 ≤ − N ℓ ℓ (N ℓ)! ℓ P − (t) = ( 1) − P (t), ℓ 0 . N − (N + ℓ)! N ≥ When the order ℓ =0, the associated Legendre function becomes a polynomial in t 0 and instead being written PN (t) it is designated PN (t), the Legendre polynomial. The complex SHs YNℓ are related to the associated Legendre functions as follows Y (ξ) = ( 1)ℓN eiℓϕP ℓ (cos θ), ℓ N, Nℓ − Nℓ N | |≤ where NNℓ is a normalization constant 2N +1 (N ℓ)! NNℓ = − s 4π (N + ℓ)! and the extra factor ( 1)ℓ is called the Condon–Shortley phase. − The YNℓ are complex-valued polynomials of the sines and cosines of θ and ϕ and for complex conjugate functions the following formula fulfil ℓ YNℓ(ξ) = ( 1) YN, ℓ(ξ). − − The parity rule for spherical harmonic is Y ( ξ) = ( 1)N Y (ξ). Nℓ − − Nℓ N It is known that the subspace of all spherical harmonics of degree N, span YNℓ ℓ , is the eigenspace of the Laplace–Beltrami operator (31) corresponding to the{ eigenvalue} λ2 = N(N + 1), − N − ∆ξY (ξ)= N(N + 1)Y (ξ). Nℓ − Nℓ The dimension of this subspace being 2N+1, so one may choose for it an orthonormal basis in different ways. ∞ The collection of all spherical harmonics YNℓ, ℓ N forms an orthonormal | |≤ N=0 2 basis for L2(S ; C) n o

N1 ℓ1 (17) (Y , Y ) S2 = Y (ξ)Y (ξ) dξ = δ δ , N1ℓ1 N2ℓ2 L2( ) N1ℓ1 N2ℓ2 N2 ℓ2 S2 Z 6 S.G. KAZANTSEV where δi is the Kronecker symbol and the space L (S2) L (S2; C) is a Hilbert j 2 ≡ 2 space of square-integrable functions on S2 with the hermitian inner product and the ﬁnite norm,

2 S2 ξ ξ ξ 2 S2 (u, v)L2( ) = u( )v( ) d , u L2(S ) = (u,u)L2( ). S2 || || Z S2 The Fourier coeﬃcients for u L2( ) are uNℓ = (u, YNℓ)L2 . Then, every function u L (S2) admits a spherical∈ harmonics series expansion in L –sense ∈ 2 2 ∞ (18) u(ξ)= uNℓYNℓ(ξ), NX=0 Xℓ ∞ 2 2 (19) u S2 = uNℓ . || ||L2( ) | | NX=0 Xℓ We close this section with Funk–Hecke formula. It was ﬁrst published by Funk (1916) and a little later by Hecke (1918).

Theorem 3. [The Funk–Hecke Theorem] Suppose f(t) L1( 1, 1) is an integrable function. Then for every spherical harmonics of degree ∈N we− have 1 (20) f(ξ η)YNℓ(ξ) dξ =2πYNℓ(η) f(t)PN (t) dt, S2 1 Z Z− where ξ η denotes the inner product of unit vectors ξ and η, PN denotes the Nth order Legendre polynomial. The Funk–Hecke formula is useful in simplifying calculations of certain integrals over S2 and plays an important role in the theory of spherical harmonics. For more details on the Funk–Hecke formula see [3, 45], for example. A general overview on spherical harmonics and the relevant problems can be found in the monographs [1, 3, 27, 51, 27, 5, 12, 13]. 2.2. Surface differential operators on the sphere S2. Here we briefly recall the definitions and some properties of surface differential operators. 2 2 The space L2(S ) L2(S ; C) is a Hilbert space of square-integrable vector functions on S2 with the≡ inner product and the finite norm,

2 u v S2 u ξ v ξ ξ u 2 u u S2 ( , )L2 ( ) = ( ) ( ) d , L2(S ) = ( , )L2( ). S2 || || Z Definition 3. The tangential gradient or the surface gradient, denoted by ξ ∇ ≡ ∇ and the tangential rotated gradient (the surface curl-gradient), denoted by ⊥ ∇ ≡ ⊥, are defined accordingly as ∇ξ ∂u 1 ∂u (21) ξu = e (ξ)+ e (ξ), ∇ ∂θ 1 sin θ ∂ϕ 2 1 ∂u ∂u (22) ⊥u = ξ ξu = e (ξ)+ e (ξ), ∇ξ × ∇ −sin θ ∂ϕ 1 ∂θ 2 where ξ = i sin θ cos ϕ + j sin θ sin ϕ + k cos θ. Obviously, we have ξ ξu(ξ)=0, ξ ξ⊥u(ξ)=0 and u ⊥u =0, thus u ∇ ∇ ∇2 ∇ ∇ and ⊥u are will be tangential vector fields on the sphere S with ⊥ is rotation by π/∇2 in the tangent plane. ∇ FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE 7

We must note here that integration by parts formulas on the sphere for operators (21) and (22) are diﬀer. Namely, for u, v C1(S2), we have ∈

(23) u(ξ) ξv(ξ) dξ = v(ξ) ξu(ξ)dξ +2 ξu(ξ)v(ξ) dξ, S2 ∇ − S2 ∇ S2 Z Z Z

(24) u(ξ) ξ⊥v(ξ) dξ = v(ξ) ξ⊥u(ξ) dξ. S2 ∇ − S2 ∇ Z Z Definition 4. In canonical coordinates, the surface divergence divξ of vector-valued 1 2 3 2 function v(ξ)= v e1(ξ)+ v e2(ξ)+ v ξ on the sphere S is written as, 1 ∂ ∂ (25) div v = (v1 sin θ)+ v2 +2v3 . ξ sin θ ∂θ ∂ϕ For tangent vector field v we define the scalar surface rotation (or scalar curl operator) curlξ by

1 ∂ 2 ∂ 1 (26) curlξv = divξ(ξ v)= (v sin θ) v . − × sin θ ∂θ − ∂ϕ If u C1(S2) and tangential vector ﬁeld v C1(S2), then we have integral formulas,∈ which are also understood as inner products∈

(27) v(ξ) ξu(ξ) dξ = u(ξ)divξv(ξ) dξ S2 ∇ − S2 Z Z (28) or (v, u) S2 = (div v,u) S2 , ∇ L2( ) − L2( ) (29) v(ξ) ξ⊥u(ξ) dξ = u(ξ)curlξv(ξ) dξ S2 ∇ − S2 Z Z (30) or (v, ⊥u) S2 = (curl v,u) S2 . ∇ L2( ) − L2( ) Deﬁnition 5. Finally, we deﬁne the Beltrami operator, which is also called the Laplace–Beltrami operator ∆ ∆ξ as ≡ (31) ∆ξu(ξ) = divξ ξu(ξ) , ∇ i.e. the divergence of a gradient is the Laplacian. One easily checks that

(32) ∆ξu(ξ) = curlξ ⊥u(ξ) ∇ξ and also

curlξ ξu(ξ)=0, divξ ⊥u(ξ)=0, ∇ ∇ξ thus we say that ξu is the curl-free, but ⊥u is the divergence-free vector ﬁelds. ∇ ∇ξ The next formula is Green–Beltrami identity or Green’s ﬁrst surface identity, see [3, Proposition 3.3], [24, Theorem 4.12]: for any u C1(S2) and any v C2(S2) we have, ∈ ∈

(33) ξu(ξ) ξv(ξ) dξ = u(ξ)∆ξv(ξ) dξ S2 ∇ ∇ − S2 Z Z (34) or ( u, v) S2 = (u, ∆v) S2 . ∇ ∇ L2( ) − L2( ) 8 S.G. KAZANTSEV

For example, if we take u = YN1ℓ1 and v = YN2ℓ2 , then

(35) ( Y , Y ) S2 ∇ N1ℓ1 ∇ N2ℓ2 L2( ) = ξYN1ℓ1 (ξ) ξYN2ℓ2 (ξ) dξ = YN1ℓ1 (ξ)∆ξYN2ℓ2 (ξ)dξ S2 ∇ ∇ − S2 Z Z = N (N + 1) Y (ξ)Y (ξ) dξ = N (N + 1)δN2 δℓ2 . 2 2 N1ℓ1 N2ℓ2 2 2 N1 ℓ1 S2 Z For more deﬁnitions and properties of these diﬀerential operators see e.g. [51, 3, 29, 12, 13].

2.3. Two systems of vector spherical harmonics (VSHs). There are vectorial analogues of scalar spherical harmonics called vector spherical harmonics. VSHs can be deﬁned in several ways. In this section we give deﬁnitions and properties of the vector spherical harmonics, which are needed in our work. We refer to [26, 51, 29, 8, 12] for more details in this theme.

2.3.1. Pure–spin vector spherical harmonics. Let us now deﬁne a complete orthogonal 2 set of vectors in L2(S ). Deﬁnition 6. The vector spherical harmonics (or pure–spin VSHs) are arranged in three families: y(1)(ξ), y(2)(ξ) and y(3)(ξ). For ξ S2 and given a scalar spherical Nℓ Nℓ Nℓ ∈ harmonic YNℓ(ξ) the unnormalized vector spherical harmonics are the set

(36) y(1)(ξ)= ξY (ξ), N 0 N, Nℓ Nℓ ∈ ∪ (2) (37) y (ξ)= ξY (ξ), N N, Nℓ ∇ Nℓ ∈ (3) (2) (38) y (ξ)= ξ y (ξ)= ⊥Y (ξ), N N. Nℓ × Nℓ ∇ξ Nℓ ∈ The pure–spin VSHs form a complete set of orthogonal vector functions on the 2 2 surface of a sphere S with the inner product of the L2(S ) space, see [13, Theorem (1) (2) 2 5.2.7]. Clearly, yNℓ L2( ) =1. To calculate the norms of vector functions yNℓ and (3) || || S yNℓ, we can use (35). Therefore, the normalizing vector harmonics or orthonormal system of VSHs are

(1) (2) (2) (3) (3) yNℓ, yNℓ = yNℓ/ N(N + 1) , yNℓ = yNℓ/ N(N + 1) . Each vector function f L (S2p) has the Fourier expansionp e ∈ 2 e

(1) ∞ (1) (2) (3) f(ξ)= f1,00y00 (ξ)+ f1,NℓyNℓ(ξ)+ f2,NℓyNℓ(ξ)+ f3,NℓyNℓ(ξ), NX=1 Xℓ ∞ 2 2 2 2 e 2 e f S2 = f1,00 + f1,Nℓ + f2,Nℓ + f3,Nℓ . || ||L2( ) | | | | | | | | NX=1 Xℓ The hermitian inner products are then given by

∞ 2 (f, h)L2 (S ) = f1,00h1,00 + f1,Nℓh1,Nℓ + f2,Nℓh2,Nℓ + f3,Nℓh3,Nℓ. NX=1 Xℓ FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE 9

2.3.2. Pure–orbit vector spherical harmonics. An alternative orthogonal basis in the S2 (e) (e) (i) (3) space L2( ) is the system of pure–orbit VSHs h00 , hNℓ, hNℓ, yNℓ, ℓ N N∞=1, (e) (i) { | |≤ } where vector functions hNℓ and hNℓ deﬁned by (39) h(e) = y(1), 00 − 00 (40) h(e) = (N + 1)y(1) + y(2), N N, Nℓ − Nℓ Nℓ ∈ (41) h(i) = Ny(1) + y(2), N N. Nℓ Nℓ Nℓ ∈ The pure-orbit vector spherical harmonics also has a nice properties, in particular, they are eigenfunctions for the vectorial Funk–Minkowski operator in the space 2 2F L2,even(S ) and for vectorial Hilbert operator in the space L2,odd(S ), see Lemmas 1 and 2 in the section Proofs. S

2.3.3. Tangent vector ﬁelds and Helmholtz–Hodge decomposition. Consider the tangent vector ﬁeld f L (S2), it can be written uniquely as ∈ 2,tan ∞ (2) ∞ (3) f(θ)= f2,NℓyNℓ(θ) + f3,NℓyNℓ(θ) NX=1 Xℓ NX=1 Xℓ the curl-free componente the divergence-freee component |∞ {z1 } | {z } = f2,Nℓ YNℓ(θ)+ f3,Nℓθ YNℓ(θ). N(N + 1) ∇ × ∇ NX=1 Xℓ Then formally wep have

∞ 1 ∞ 1 f(θ)= f2,NℓYNℓ(θ)+ ⊥ f3,NℓYNℓ(θ), ∇ N(N + 1) ∇ N(N + 1) NX=1 Xℓ NX=1 Xℓ where accordingp to (10) the velocity potential and streamp functions are

∞ 1 u(θ)= f2,NℓYNℓ(θ), N(N + 1) NX=1 Xℓ ∞ p 1 v(θ)= f3,NℓYNℓ(θ). N(N + 1) NX=1 Xℓ Another evident approach consistsp in solving the Laplace–Beltrami equations on the sphere

∆θu(θ) = divθ f(θ),

∆θv(θ) = curlθ f(θ). They can be solved in integral form, for example, involving GreenпїЅs function with respect to the Laplace–Beltrami ∆θ, see [13, Theorem 4.6.9]. 2.4. Hilbertian Sobolev spaces on the sphere.

2.4.1. Sobolev scalar functions on S2. The Sobolev space Hs(S2) with a smoothness index s 0 is deﬁned by ([3, 24, 41, 29, 32]) ≥ ∞ Hs(S2) := u L (S2; C): (1 + N(N + 1))s u 2 < . { ∈ 2 | Nℓ| ∞} NX=0 Xℓ 10 S.G. KAZANTSEV

s 2 s/2 2 s 2 In other words u H (S ) if and only if (I ) u L2(S ). The space H (S ) is a Hilbert space∈ with the hermitian inner product−△ ∈

∞ s (u, v)Hs(S2) = (1 + N(N + 1)) uNℓvNℓ NX=0 Xℓ and the induced norm

2 ∞ s 2 s/2 2 u s S2 = (1 + N(N + 1)) uNℓ = (I ) u S2 . || ||H ( ) | | || −△ ||L2( ) NX=0 Xℓ 0 2 2 Putting s = 0 we obtain H (S ) = L2(S ). If s = 1 then in addition to (18), (19) we have

∞ ∞ (2) ξu(ξ)= u ξY (ξ), = N(N + 1) (u, Y ) S2 y , ∇ Nℓ∇ Nℓ Nℓ L2( ) Nℓ N=0 ℓ N=1 ℓ X X X p X e

2 ∞ 2 u S2 = N(N + 1) uNℓ . ||∇ ||L2( ) | | NX=0 Xℓ Thus we can deﬁne the Sobolev space H1(S2) as (see [29, p. 14])

H1(S2)= u L (S2): u L (S2) { ∈ 2 ∇ ∈ 2 } with its inner product and the ﬁnite Sobolev norm

2 2 2 (u, v) 1 S2 = (u, v) S2 + ( u, v)L S2 , u 1 S2 = u S2 + u S2 , H ( ) L2( ) ∇ ∇ 2( ) || ||H ( ) || ||L2( ) ||∇ ||L2( ) where is the surface gradient on the sphere. Generally, if s = m which is a positive integer,∇ we can deﬁne the Sobolev norm via the following formula ( -deﬁnition of Sobolev spaces) ∇

m 2 k k u s S2 = (u, v) S2 + ( u, v) S2 . || ||H ( ) L2( ) ∇ ∇ L2( ) kX=1 If we s consider a closed linear subspace H1(S2)/R) H1(S2), ⊂ H1(S2)/R = u H1(S2): u(ξ) dξ =0 , { ∈ S2 } Z then due to a Poincar´einequality for all u H1(S2)/R we can deﬁne an equivalent norm for H1(S2)/R ∈ 2 u 1 S2 R = u S2 , || ||H ( )/ ||∇ ||L2( ) such that H1(S2)/R becomes a Hilbert space with the inner product

(u, v) 1 S2 R = ( u, v) S2 . H ( )/ ∇ ∇ L2( ) For more details on these spaces, we refer the reader to [3], [24, Theorems 4.12 and 6.12], [29, p. 41] . FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE11

2.4.2. Sobolev tangent vector fields on S2. For tangential vector fields we have the vectorial Sobolev space Hs (S2), which is the set of all f L (S2) such that tan ∈ 2,tan 2 ∞ s 2 2 f Hs (S2) = (1 + N(N + 1)) f2,Nℓ + f3,Nℓ . || || tan | | | | NX=1 Xℓ s S2 For the scale of Sobolev spaces Htan( ) there is a Helmholtz–Hodge decomposition ([4, Theorem 4.1]) Hs (S2)= Hs+1(S2)/R ker(div) = Hs (S2)+ Hs (S2), s 0. tan ∇ ⊕ tan,curl tan,div ≥ s S2 s S2 Here we denote by Htan,div( ) and Htan,curl( ) the divergence-free and curl-free s S2 subspaces of Htan( ), respectively. s S2 Another words vector field tangent to the sphere f Htan( ) can be uniquely decomposed into surface curl-free and surface divergence-∈ free components:

f = u + ⊥v, u dξ = v dξ =0, ∇ ∇ S2 S2 Z Z where functions u, v Hs+1(S2)/R. We can deﬁne its Hs norm, among other equivalent versions, as∈ 2 2 2 f s S2 = u s+1 S2 + v s+1 S2 . || ||H ( ) || ||H ( ) || ||H ( ) 2.5. Fourier multiplier and spherical convolution operators.

2.5.1. Fourier multiplier operators. Here we define Fourier multiplication operators. Definition 7. The operator Λ: L (S2) L (S2) is called the Fourier multiplier 2 → 2 operator with corresponding sequence of multipliers λN N∞=0 if operator Λ acts on a function u L (S2) by the formula { } ∈ 2 Λu (ξ) Λξu = λ u Y (ξ), { } ≡ N Nℓ Nℓ NX=0 Xℓ where uNℓ denote the Fourier coefficients of u with respect to the spherical harmonics,

u(ξ)= uNℓYNℓ(ξ). NX=0 Xℓ The sequence of multipliers λN N∞=0 gives complete information about properties of operator Λ, especially the behavior{ } and asymptotics of multipliers at inﬁnity. It 2 is not hard to see that a multiplier operator on L2(S ) is bounded if and only if its sequence of multipliers is bounded. The works of many authors are devoted to the study of such operators, see [44, 39, 5].

2.5.2. Spherical convolution operators. An important example of the multiplier operator will be a spherical convolution operator.

Deﬁnition 8. The spherical convolution K u of K L2( 1, 1) with a function u L (S2) is deﬁned as ∗ ∈ − ∈ 2 (K u)(ξ)= K(ξ η)u(η) dη, ξ S2, ∗ S2 ∈ Z dη is the rotation invariant measure, normalized so that S2 dη =4π the surface area of S2. We recall that η ξ is the usual pointwise inner product. R 12 S.G. KAZANTSEV

By the FunkпїЅ-Hecke formula in Theorem 3 we have the sequence of multipliers λ ∞ { N }N=0 1 K YNℓ (ξ)=2πYNℓ(ξ) K(x)PN (x) dx = λN YNℓ(ξ). { ∗ } 1 Z− 2.5.3. Funk’s inversion formula for the F–M transform. In [9] Funk showed that Funk–Minkowski- transform (1) is the Fourier multiplier operator with multiplicators λ2j = P2j (0), Y (ξ)= P (0)Y (ξ), {F Nℓ} 2j 2j,ℓ 1/2 and asymptotics λ2j = P2j (0) (2j + 1)− if j ([1]). Hence any even 2 ∼ → ∞ function feven C∞(S ) can be reconstructed explicitly from its Funk–Minkowski transform by the∈ formula

∞ ∞ ( feven, Y2j,ℓ) 2 f (ξ)= f Y (ξ)= F L2(S ) Y (ξ) , even 2j,ℓ 2j,ℓ P (0) 2j,ℓ j=0 j=0 2j X Xℓ X Xℓ where

( f , Y ) S2 = P (0)f . F even 2j,ℓ L2( ) 2j 2j,ℓ The following mapping property of the Funk–Minkowski transform between Sobolev spaces was shown by R. S. Strichartz in [50, Lemma 4.3] : operator : Hs (S2) Hs+1/2(S2), s 0 F even → even ≥ is continuous and bijective, see also [16, 32]. 2.5.4. The spherical convolution operator . Now consider the spherical convolution operator , which deﬁned by formula (5),S we repeat it S 1 1 1 v(η) 2 v (ξ) ξv = x− u (ξ)= dη, ξ S . {S } ≡ S 4π { ∗ } 4π S2 ξ η ∈ Z The operator does not exist as an absolutely convergent integral and should be understood inS the principal value sense, see [45, 41], 1 v(η) 1 v(η) v (ξ) = lim dη = p.v. dη. {S } ε 0 4π ξη >ε ξ η 4π S2 ξ η → Z| | Z 2 2 The operator is considered as operator from L2(S ) into L2(S ) and can be regarded as the sphericalS analogue of the Hilbert transform, [41]. Evidently, that for even spherical harmonics Y2j,ℓ (ξ)=0, so we can consider this operator only {S }2 on the subspace of odd SHs, L2,odd(S ). Proposition 1 ([41, 21]). The spherical analogue of the Hilbert transform (5) : L (S2) L (S2) S 2,odd → 2,odd 2 is a compact operator and a multiplier operator on L2,odd(S ) with corresponding 1 1 ∞ sequence of Fourier-Laplace multipliers (3/2) = , N =2j +1 , NPN−1(0) CN−1 (0) j=0 1 n 1 o (42) YNℓ (ξ) = (3/2) YNℓ(ξ)= YNℓ(ξ), N =2j +1 NPN−1(0) {S } CN−1 (0) and asymptotics 1 1 1 (43) (3/2) = if j . (2j + 1)P2j (0) ∼ √2j +1 → ∞ C2j (0) FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE13

The operator , as well as the operator , S F : Hs (S2) Hs+1/2(S2), s 0 S odd → odd ≥ s S2 is continuous and bijective in the scale of Sobolev spaces Hodd( ), see [41, Proposition 3.2].

2.5.5. Analytic family of fractional integrals and Funk–Minkowski transform. We can write the F–M operator (1) in the form of spherical convolution operator as follows 1 2π u (ξ)= u e1(ξ)cos ω + e2(ξ) sin ω dω {F } 2π 0 1 Z 2π 1 2 2 = δ(t) u e1(ξ)cos ω 1 t + e2(ξ) 1 t sin ω dω dt 2π 1 0 − − Z− Z 1 1 p p = δ(ξ θ)u(θ) dθ = δ u (ξ) , 2π S2 2π { ∗ } Z where δ is the Dirac delta function. The papers [23, 32] give a deﬁnition of the generalized Funk–Radon transform (j) 2 S for u C∞(S ) by ∈ 1 S(j)u (ξ)= δ(j)(ξ θ)u(θ) dθ, j 0 N. { } 2π S2 ∈ ∪ Z Here use the notation from [23, 32] and δ(j) denotes the j-th derivative of the Dirac delta function and operator S(0) is the Funk–Minkowski transform . The spherical Hilbert type operator in (5) as well as operatorsF (j) are the S S members of analytic family of fractional integrals λ, λ deﬁned by {C C } Γ λ λ 2 λ e (44) f (θ)= −1+λ f(σ) θ σ dσ, {C } 2πΓ S2 | | 2 Z Γ 1 λ λ −2 λ (45) f (θ)= λ f(σ) θ σ sgn(θ σ) dσ, {C } 2πΓ 1+ S2 | | 2 Z see [35, 41]. Thee operators λ and λare called the λ-cosine transforms of f with C C 2 even and odd kernel, respectively. If f C∞(S ), they extend analytically to all ∈ λ C with the only poles λ =0, 2, 4e, ... for λ and λ =1, 3, 5, ... for Cλ . ∈The limit case λ = 1 corresponds to theC Funk–Minkowski transform and − F Hilbert spherical transform , [41, Lemma 3.4.] e S 1 1 − f (θ)= f(η)δ(θ η) dη, F ∼{C } 2√π S2 Z 1 f(η) 1 θ η − f ( )= 3/2 d . S∼{C } 2π S2 θ η Z The integral operator in thee inverse formula (2) by V. Semyanisty also belongs to this family with λ = 2, − 1 1 2 θ η η − f ( )= −3/2 f( ) 2 d . {C } 4π S2 (θ η) Z 14 S.G. KAZANTSEV

2 2 The corresponding operator − for − is the generalized Funk–Radon transform C C

(1) 2 1 S −ef (θ)= − f(η)δ′(θ η) dη. ∼{C } 4√π S2 Z If for an analytic continuatione we use formulas, see for example [14],

x λ ( 1)mm! (46) | | = − δ(2m)(x), m =0, 1, 2, ... , Γ 1+λ λ= (2m+1) (2m)! 2 − λ m x sgn( x) ( 1) (m 1)! (2m 1) (47) | | = − − δ − (x), m =1, 2, 3, ... , Γ 1+ λ λ= 2m (2m 1)! 2 − − then as the result, the following connection between (2m), (2m+1) and analytic S S family λ, ˜λ take place {C C } (48) ( 1)m√π (2m) 2m 1 θ σ (2m) θ σ σ − − f ( )= − 2m f( )δ ( ) d , m =0, 1, 2, ... , S ∼ {C } 2π2 S2 Z (49) m (2m+1) 2m ( 1) √π (2m 1) − f (θ)= − 2m 1 f(σ)δ − (θ σ) dσ, m =1, 2, 3, ... . S ∼{C } 2π2 S2 − Z According to thee general theory of analytic family λ, λ on the sphere S2, we {C C } can ﬁnd inverse operators of λ, λ by the formulas (see [41, Proposition 3.1]) C C e λ λ 3 λ 3 λ 2 − − f = − − f = f, where λ, λ 3 =0, 2, 4, ... f C∞ (S ), C C C C e − − 6 ∈ even and

λ λ 3 λ 3 λ 2 − − f = − − f = f, where λ, λ 3 =1, 3, 5, ... , f C∞ (S ). C C C C − − 6 ∈ odd 1 1 1 2 Ine thee particulare casee λ = 1 we have − − − = − and it is appropriate to formula (2) by V.− Semyanisty,F see also∼ C [40, CorollaryC 3.3]. If we apply (formally) the integration by parts formula (23) to (7), then we get

θ f (η) θ 1 θ η f (η) {∇F } dη = f (η)dη + {F } dη 4π S2 θ η −4π S2 ∇θ η {F } 2π S2 η θ Z Z Z θ θ (θ η)η 1 η η η η = − 2 f ( )d + f ( )d 4π S2 (θ η) {F } 2π S2 {F } Z Z 1 1 (θ η)2 1 η η η η = − 2 f ( )d + f ( )d 4π S2 (θ η) {F } 2π S2 {F } Z Z 1 1 1 η η η η = 2 f ( )d + f ( )d . 4π S2 (θ η) {F } 4π S2 {F } Z Z Thus, this formal calculations show that formula (7) corresponds to formula (2) and serves as its regularization

1 1 1 1 θ f (η) η η η η η 2 f ( )d = f ( )d p.v. {∇F } d . −4π S2 (θ η) {F } 4π S2 {F } − 4π S2 θ η Z Z Z FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE15

3. Proofs In this section we present the proofs of Theorems 1, 2, which will be based on Lemmas 1 and 2. In vector case, as in the scalar case, the vectorial Funk–Minkowski 2 2 transform : L2,even(S ) L2,even(S ) and vectorial Hilbert type spherical F 2 → 2 transform : L2,odd(S ) L2,odd(S ) are multiplier operators and relevant mapping propertiesS between Sobolev→ spaces are valid. The accurate formulations are given below.

2 2 Lemma 1. Vectorial Funk–Minkowski transform : L2,even(S ) L2,even(S ) is a multiplier operator F →

(i) (i) (50) h = PN 1(0)h , N =2j +1, F Nℓ − Nℓ (51) y(3) = P (0)y(3), N =2j, F Nℓ N Nℓ (52) h(e) = P (0)h(e) , N =2j +1, F Nℓ N+1 Nℓ

(i) (3) (e) where hNℓ, yNℓ, hNℓ are pure–orbit vector spherical harmonics (39)–(41). We have s S2 s+1/2 S2 that in the scale of Sobolev spaces operator : Heven( ) Heven ( ), s 0 is continuous and bijective. F → ≥ If we choose as a basis pure–spin vector spherical harmonics, then following formulas take place

(2) (1) yNℓ (53) y = PN 1(0) , N =2j +1, F Nℓ − N +1 (2) (2) (1) yNℓ (54) y = PN 1(0) Ny + , N =2j +1 , F Nℓ − Nℓ N +1 (55) y(3) = P (0)y(3) , N =2j . F Nℓ N Nℓ Similar statements are valid for the operator S 2 2 Lemma 2. Vectorial spherical convolution transform : L2,odd(S ) L2,odd(S ) is a multiplier operator S →

(56) h(e) = h(e), S 00 00 (e) (e) hNℓ (57) hNℓ = , N =2j, S (N + 1)PN (0) (i) 1 N +1 (i) (58) hNℓ = hNℓ, N =2j, S −(N + 1)PN (0) N (3) (3) yNℓ (59) yNℓ = , N =2j +1 , S NPN 1(0) −

(i) (3) (e) where hNℓ, yNℓ, hNℓ are pure–orbit vector spherical harmonics (39)–(41). In the s S2 s+1/2 S2 scale of Sobolev spaces operator : Hodd( ) Hodd ( ), s 0 is continuous and bijective. S → ≥ 16 S.G. KAZANTSEV

The images of pure–spin spherical harmonics under the action of operator are listed below S

(60) y(1) = y(1), S 00 00 (2) (1) 1 yNℓ (61) yNℓ = − , N =2j, S PN (0) N(N + 1) (2) (2) 1 (1) yNℓ (62) yNℓ = − yNℓ + , N =2j, S PN (0) N(N + 1) (3) (3) 1 yNℓ (63) yNℓ = , N =2j +1 . S PN 1(0) N − Proof of Lemma 1. The pure–orbit VSHs are expressed through scalar spherical harmonics with the help of three term relations, see for example [8],

(i) (1) (2) (64) hNℓ = NyNℓ(ξ)+ yNℓ(ξ) 1 0 1 = α1YN 1,ℓ 1(ξ) i + β1YN 1,ℓ(ξ) 0 + γ1YN 1,ℓ+1(ξ) i , − −  0  −  1  −  −0  (65) h(e) = (N + 1)y(1)(ξ)+ y(2)(ξ)     Nℓ − Nℓ Nℓ 1 0 1 = α2YN+1,ℓ 1(ξ) i + β2YN+1,ℓ(ξ) 0 + γ2YN+1,ℓ+1(ξ) i , −  0   1   −0   1  0  1  (3) (66) yNℓ(ξ)= α3YN,ℓ 1(ξ) i + β3YNℓ(ξ) 0 + γ3YN,ℓ+1(ξ) i , −  0   1   −0        where αi,βi,γi (i = 1, 2, 3) some coeﬃcients. The values of this coeﬃcients are unimportant here, but their accurate expressions can be found in [8]. By applying the operator to these three term relations we immediately obtain: for N =2j +1 F

(i) (i) (e) (e) (67) h = PN 1(0)h , h = PN+1(0)h F Nℓ − Nℓ F Nℓ Nℓ and for N =2j

y(3) = P (0)y(3) . F Nℓ N Nℓ Because the multipliers have asymptotics P (0) 1 as j goes to inﬁnity, 2j ∼ √2j+1 s S2 s+1/2 S2 we have that : Heven( ) Heven ( ) is a continuous operator in the scale of Sobolev spaces,F as in the scalar→ case. The two equations (67) can be written as

(1) (2) (1) (2) N yNℓ + yNℓ = NPN 1(0)yNℓ + PN 1(0)yNℓ F F (1) (2) − (1)− (2) ( (N + 1) y + y = (N + 1)PN+1(0)y + PN+1(0)y . − F Nℓ F Nℓ − Nℓ Nℓ FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE17

(1) (2) We need to solve this system with respect to yNℓ and yNℓ. Subtracting the second from the ﬁrst equation, we obtain F F

(1) (1) (2) (2N + 1) y = (NPN 1(0) + (N + 1)PN+1(0))y + (PN 1(0) PN+1(0))y F Nℓ − Nℓ − − Nℓ N (1) N (2) = PN 1(0) N (N + 1) y + PN 1(0) 1+ y − − N +1 Nℓ − N +1 Nℓ 2N +1 (2) = PN 1(0) y . − N +1 Nℓ

Here we used the formula (14), (N + 1)PN+1(0) = NPN 1(0), thus we have − − (2) (2) (1) yNℓ (2) (1) yNℓ y = PN 1(0) , y = PN 1(0) Ny + , N =2j +1. F Nℓ − N +1 F Nℓ − Nℓ N +1 Proof of Lemma 2. By applying operator to three term relations, as well as in the previous case, we obtain: for N =2j S

(i) (i) hNℓ 1 N +1 (i) hNℓ = = hNℓ, S (N 1)PN 2(0) −(N + 1)PN (0) N − − (e) (e) hNℓ hNℓ = S (N + 1)PN (0) and for N =2j +1 (3) (3) yNℓ yNℓ = . S NPN 1(0) −

In the ﬁrst formula we used equality (N 1)PN 2(0) = NPN (0). Continuity of the operator in the− scale H−s (S2)−follows from asymptotic S odd behavior 1 1 if j . (2j+1)P2j (0) ∼ √2j+1 → ∞ The ﬁrst two equations above are equivalent to the system

(1) (2) y(1) y(2) N y + y = Nℓ Nℓ S Nℓ S Nℓ − PN (0) − NPN (0) (1) (2) y(1) y(2)  (N + 1) y + y = Nℓ + Nℓ .  − S Nℓ S Nℓ − PN (0) (N+1)PN (0) Solving this system, we obtain the desired

(2) (2) (1) yNℓ (2) 1 (1) yNℓ yNℓ = − , yNℓ = − yNℓ + . S PN (0)N(N + 1) S PN (0) N(N + 1) 3.1. Proof Theorem 1. We recall some of the basic properties that are implied in our proof. The Funk–Minkowski transform is even, f ( ξ) = f (ξ), and {F } − {F } fodd =0, but spherical transform is odd, f ( ξ)= f (ξ), and feven = 0F. It is obviously that the surface gradientS {S, scalar} − (dot)−{S product} η andS vector (cross) product η change the parity. We also∇ recall the parity rules for scalar and × vector spherical harmonics : Y ( ξ) = ( 1)N Y (ξ), y(1)( ξ) = ( 1)N+1y(1)(ξ), Nℓ − − Nℓ Nℓ − − Nℓ y(2)( ξ) = ( 1)N+1y(2)(ξ), y(3)( ξ) = ( 1)N y(3)(ξ). Nℓ − − Nℓ Nℓ − − Nℓ 18 S.G. KAZANTSEV

Now we can proceed to our formula (6) and without loss of generality, we assume that f(θ) H1(S2)/R, then we have ∈ 1 (η + θ) , f (η) f(θ)= F ∇ dη 4π S2 nhη θ i o Z 1 η , f (η)+ θ , f (η) = F ∇ F ∇ dη 4π S2 nh i o η θ nh i o Z =0 eveen

1 η η f η ηf θ ηf θ η f = F ∇ − ∇F − ∇F dη + F ∇ dη 4π S2 η θ 4π S2 η θ Z z }| { Z z }| { 1 η η f θ ηf = F ∇ − ∇F dη = θη η f θθ ηf. 4π S2 η θ S F ∇ − S ∇F Z s S2 s S2 It is clear that ker( θη η ) = Heven( ) and ker(θ θ η) = Hodd( ). S F ∇ s 2 s 2S ∇F From the Lemmas 1, 2 we have η η : H (S ) H (S ) if s 1, that looks on the diagram S F ∇ → ≥ s 2 s 1 2 η η s 1/2 2 s 2 H (S ) ∇ H − (S ) F H − (S ) S H (S ). −−−−→ tan −−−−→ −−−−→ s 2 s 2 Similarly, θ θ : H (S ) H (S ) if s 1/2, which is also conﬁrmed by the diagram S ∇F → ≥

s 1/2 θ θ Hs(S2) F Hs+1/2(S2) ∇ H − (S2) S Hs(S2). −−−−→ −−−−→ tan −−−−→ For further calculations we take speciﬁcally f = YNℓ, then from (54) and (55) we have (2) for N =2j : η Y =0, ηY = P (0)y (η), F ∇ Nℓ ∇F Nℓ N Nℓ (2) (1) yNℓ(η) for N =2j + 1 : η YNℓ = PN 1(0) Ny (η)+ , ηYNℓ =0. F ∇ − Nℓ N +1 ∇F Consequently (2) PN (0)θ yNℓ(η), N =2j η η YNℓ θ ηYNℓ = − F ∇ − ∇F PN 1(0)NYNℓ(η), N =2j +1 − and ﬁnally, using formulas (42) and (62), we get

1 η η Y θ ηY Y (θ), N =2j F ∇ Nℓ − ∇F Nℓ dη = Nℓ 4π S2 η θ YNℓ(θ), N =2j +1 . Z s 2 s 2 So we proved that, if s 1 then operator θη η θθ η : H (S ) H (S ) is identical operator. ≥ S F ∇−S ∇F →

3.2. Proof Theorem 2. We have already mentioned two approaches to the solution 2 of Helmholtz–Hodge decomposition problem for f L2,tan(S ). Now we proof formulas (11) and (12) in Theorem 2 for the velocity∈ potential u and stream functions v of the Helmholtz–Hodge decomposition (10),

u(θ)= θη ηf θη ηf, S F −F S v(θ)= θ θη ηf θ θη ηf. S ×F − F × S FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE19

(2) For the proof it suﬃces to verify these formulas on the basis elements f = yNℓ (3) and f = yNℓ. Applying the scalar and cross products to the the formulas (54)–(55) and (62)–(63), we obtain

(3) (3) (2) for N =2j : η ηy =0, η ηy = P (0)y (η), F Nℓ ×F Nℓ − N Nℓ (3) (2) (2) yNℓ(η) for N =2j + 1 : η ηy = PN 1(0)NYNℓ(η), η ηy = PN 1(0) , F Nℓ − ×F Nℓ − N +1

y(3) η (2) 1 (2) 1 Nℓ( ) for N =2j : η ηyNℓ = − YNℓ(η), η ηyNℓ = − , S PN (0) × S PN (0) N(N + 1) y(2) η (3) (3) 1 Nℓ( ) for N =2j + 1 : η ηyNℓ =0, η ηyNℓ = − . S × S PN 1(0) N − Based on the above, we get

(2) (2) YNℓ(θ), N =2j θη ηyNℓ θη ηyNℓ = S F −F S YNℓ(θ), N =2j +1, (3) (3) 0, N =2j θη ηy θη ηy = S F Nℓ −F S Nℓ 0, N =2j +1, (2) (2) 0, N =2j θ θη ηy θ θη ηy = S ×F Nℓ − F × S Nℓ 0 N =2j +1, (3) (3) YNℓ(θ), N =2j θ θη ηyNℓ θ θη ηyNℓ = S ×F − F × S YNℓ(θ), N =2j +1. 4. Conclusion This paper is devoted to the study of Funk–Minkowski transform and Hilbert type spherical convolution . We provide inversion formulas for two F–MF transforms f and f. In this case bothS even and odd parts of the function f are determined. FAlso, theF∇ formulas for decomposition of a tangent vector ﬁeld on the sphere into divergence-free and curl-free parts with the participation of operators and are derived. In the process of obtaining and proving all formulas, theF sphericalS multipliers approach is used.

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Sergei Gavrilovich Kazantsev Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia E-mail address: [email protected]