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Funk-Minkowski Transform and Spherical Convolution of Hilbert Type FUNK–MINKOWSKI TRANSFORM AND SPHERICAL CONVOLUTION OF HILBERT TYPE IN RECONSTRUCTING FUNCTIONS ON THE SPHERE S.G. KAZANTSEV Abstract. The Funk–Minkowski transform F associates a function f 2 on the sphere S with its mean values (integrals) along all great circles of the sphere. Thepresented analytical inversion formula reconstruct the unknown function f completely if two Funk–Minkowski transforms, Ff and F∇f, are known. Another result of this article is related to the problem of Helmholtz–Hodge decomposition for tangent vector field on the sphere S2. We proposed solution for this problem which is used the Funk–Minkowski transform F and Hilbert type spherical convolution S. Keywords: Funk–Minkowski transform, Funk–Radon transform, spherical convolution of Hilbert type, Fourier multiplier operators, inverse operator, scalar and vector spherical harmonics, surface gradient, tangential spherical vector fields, Helmholtz–Hodge decomposition. 1. Introduction The paper is devoted to the analytical inverse of the Minkowski–Funk transform (F–M transform). This transform was introduced by P. Funk [9, 10, 11], based on the work [25] of H. Minkowski. In literature Funk–Minkowski transform is known also as the Funk transform, Funk–Radon transform or spherical Radon transform. F–M transform associates a function on the unit sphere S2 in R3 with its mean values (integrals) along all great circles of the sphere. Funk–Minkowski transform is a geodesic transform because the great circles on the sphere are geodesics. In recent time many authors investigate the generalized Funk–Minkowski transforms (or nongeodesic Funk–Minkowski transforms) on the sphere S2, which include nongeodesic paths of integration, such as circles with fixed diameter [46, 37, arXiv:1806.06672v1 [math.DG] 5 Jun 2018 28], circles perpendicular to the equator [54, 19, 15, 28] and circles, which obtained by intersections of the sphere with planes passing through a fixed common point a R3, for example, through the northpole k S2 [43, 33, 2, 6, 18, 30]. ∈Funk–Minkowski transform plays an important∈ role in the study of other integral transforms on the sphere and has various applications, for example, it is used in the convex geometry, harmonic analysis, image processing and in photoacoustic tomography, see [25, 49, 7, 22, 23, 32, 53, 54, 20]. Let B3 and S2 be the unit ball and the unit sphere in R3, respectively, i.e. B3 = x R3 : x < 1 and S2 = ∂B3 = ξ R3 : ξ =1 , where denotes the Euclidean{ ∈ norm.| Throughout| } the paper we{ adopt∈ the| | conventio} n to| denote · | in bold Kazantsev, S. G., Funk–Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere. c 2018 Kazantsev S.G.. 1 2 S.G. KAZANTSEV type the vectors in R3, and in simple type the scalars in R. By the greek letters θ, η, ξ and so on we denote the units vectors S2. We will use for unit vector ξ on the sphere S2 usual angular coordinates (θ, ϕ) ξ = ξ(θ, ϕ)= i sin θ cos ϕ + j sin θ sin ϕ + k cos θ = (sin θ cos ϕ, sin θ sin ϕ, cos θ), where 0 <θ<π (the colatitude), 0 <ϕ< 2π (the longitude) and t = cos θ polar distance. 3 The plane ξ⊥ = x R : x ξ =0 is spanned by the two orthonormal vectors { ∈ } e1, e2 with representations in polar coordinates ∂ξ 1 ∂ξ e (ξ)= = (cos θ cos ϕ, cos θ sin ϕ, sin θ), e (ξ)= = ( sin ϕ, cos ϕ, 0). 1 ∂θ − 2 sin θ ∂ϕ − 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 = f|00 + 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.
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