text:
\begin{document}
\renewcommand{\refname}{References}

\thispagestyle{empty}

\title[Funk--Minkowski  transform and  spherical convolution of Hilbert type

]
       {Funk--Minkowski  transform and  spherical convolution of Hilbert type in reconstructing functions on the sphere}
\author{{S.\,G. Kazantsev}}
\address{Sergei Gavrilovich  Kazantsev
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}
\email{kazan@math.nsc.ru}

\thanks{\sc Kazantsev, S.\,G., Funk--Minkowski  transform and  spherical convolution of Hilbert type    in reconstructing functions on the sphere}
\thanks{\copyright \ 2018 Kazantsev S.G.}

 \vspace{1cm}
\maketitle
 {\small
\begin{quote}
\noindent{\sc Abstract. } The Funk--Minkowski transform
 ${\mathcal F}$  associates a function $f$ on the sphere ${\mathbb S}^2$
  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, ${\mathcal F}f$ and ${\mathcal F} \nabla f$, are known. Another result  of this
 article is related to the problem of  Helmholtz--Hodge decomposition for
 tangent vector field on the sphere ${\mathbb S}^2$. We proposed solution for this problem which is used  the Funk--Minkowski  transform ${\mathcal F}$ and  Hilbert type spherical convolution ${\mathcal S}$.
\medskip

\noindent{\bf 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.
 \end{quote}
}

\section{Introduction}

 The paper is devoted to the  analytical  inverse of the
 Minkowski--Funk  transform (F--M transform). This transform  was introduced by
 P. Funk ~\cite{Funk:1913, Funk:1915, Funk:1916},
 based on the work ~\cite{Minkowski:1904} 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 ${\mathbb S}^2$ in
 ${\mathbb R}^3$ 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 ${\mathbb S}^2$, which
  include nongeodesic paths of integration, such as
  circles with fixed diameter ~\cite{Schneider:1969, Rubin:2000, NattererWubelling:2001},
  circles perpendicular to the equator
  ~\cite{ZangerlScherzer:2010, HielscherQuellmalz:2016, GindikinReedsShepp:1994, NattererWubelling:2001}
  and  circles, which   obtained
  by intersections of  the sphere with planes passing through
  a fixed common point ${\bf a} \in {\mathbb R}^3$, for example,  through the  northpole
  ${\bf k} \in {\mathbb S}^2$
  ~\cite{Salman:2016, Quellmalz:2017, AbouelazDaher:1993,
   Daher:2001, Helgason:2011, Palamodov:2017}.

 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 ~\cite{Minkowski:1904,
  Stepanov:2017, Dann:2010,
  LouisRiplingerSpiessSpodarev:2011, Louis:2016,
  QuellmalzHielscherLouis:2018, YarmanYazici:2011,
  ZangerlScherzer:2010, HristovaaMoonSteinhauer:2016}.

  Let ${\mathbb B}^3$ and  ${\mathbb S}^2$ be the unit ball and
  the unit sphere in $\mathbb R^3$, respectively,
  i.e.
  ${\mathbb B}^3 = \{{\bf x} \in \mathbb R^3 : |{\bf x}|<1 \}$ and
  ${\mathbb S}^2 = \partial \mathbb B^3 =\{{\bm \xi} \in \mathbb R^3 : |{\bm \xi}|=1 \},$
  where $| \cdot|$ denotes the Euclidean norm.
  Throughout the paper we adopt the convention to denote in bold type the vectors in
  ${\mathbb R}^3,$ and in simple type the scalars in ${\mathbb R}.$ By the  greek
   letters ${\bm \theta}$, ${\bm \eta}$, ${\bm \xi}$ and so on we denote the units
   vectors ${\mathbb S}^2$. We will use for unit vector ${\bm \xi}$ on the sphere $\mathbb{S}^2$
  usual angular coordinates $(\theta, \varphi)$
  \[
  {\bm\xi}={\bm\xi}({\theta},{\varphi})={\bf i}\sin{\theta}\cos{\varphi}+
  {\bf  j}\sin{\theta}\sin{\varphi}+ {\bf  k}\cos{\theta}=
  ({\sin\theta} \cos {\varphi},  {\sin\theta} \sin {\varphi},\cos {\theta}),
  \]
  where   $0 < \theta < \pi$ (the colatitude),
  $0 < \varphi < 2 \pi$  (the longitude) and

  $t=\cos \theta$ --- polar
  distance.

  The  plane ${\bm\xi}^\perp=\{{\bf x}\in\mathbb{R}^3 : {\bf x}\centerdot {\bm\xi}=0\}$
  is spanned by the two orthonormal
  vectors ${\bf e}_{1},$ ${\bf e}_{2}$  with
  representations in polar coordinates
  \[
   {\bf e}_{1}(\bm\xi)=\frac{\partial{\bm\xi}}{\partial {\theta}}=
  ({\cos\theta}  \cos {\varphi}, {\cos\theta}  \sin {\varphi},-\sin {\theta}),
  \,
  {\bf e}_2(\bm\xi)=\frac{1}{\sin{\theta}}\frac{\partial{\bm\xi}}{\partial {\varphi}}=
  ( -\sin {\varphi}, \cos {\varphi}, 0).
   \]
   The vectors ${\bf e}_{1}({\bm\xi}),$
  ${\bf e}_{2}({\bm\xi})$ and   ${\bm\xi}$
  form the so called local moving
  triad ${\bm\xi} \centerdot  {\bf e}_{1}=0$,
  ${\bm\xi} \centerdot  {\bf e}_{2}=0$,
  ${\bf e}_{1}\centerdot  {\bf e}_{2}=0$,
  where  $ \centerdot $ denotes the inner product
  of two vectors in ${\mathbb R}^3$.

 Let  denote by $f_{even}$ and
 $f_{odd}$ the even and odd parts of function $f$ on ${\mathbb S}^2$, respectively, that is,
 we have
 \[
 f({\bm \xi})=f_{even}({\bm \xi})+f_{odd}({\bm
  \xi}), \
 f_{even}({\bm \xi})=\frac{f({\bm \xi})+f(-{\bm \xi})}{2}, \ \
 f_{odd}({\bm \xi})=
 \frac{f({\bm \xi})-f(-{\bm \xi})}{2}.
 \]

 The space of continuous functions on the sphere ${\mathbb S}^2$ is
 denoted by $C({\mathbb S}^2)$  and is  endowed with the supremum
 norms
 \[
 ||f||_{C({\mathbb S}^2)}=
 {\rm sup}_{{\bm \xi}\in{\mathbb S}^2 }|f({\bm \xi})|.
 \]
 $C({\mathbb S}^2)$, $C_{even}({\mathbb S}^2)$ and $C_{odd}({\mathbb S}^2)$
 denote the space of continuous functions on ${\mathbb S}^2$,
  the space of even  continuous functions on ${\mathbb S}^2$
  and the space of odd  continuous functions on ${\mathbb S}^2$,
 respectively. The subset of  $C_{even}({\mathbb S}^2)$ ($C_{odd}({\mathbb S}^2)$) that
 contains the infinitely differentiable functions will be denoted
 by $C^{\infty}_{even}({\mathbb S}^2)$ ($C^{\infty}_{odd}({\mathbb S}^2)$).

 \begin{definition} Let $f$ be a
 continuous function on the sphere ${\mathbb S}^2$,
 $f \in  C({\mathbb S}^2)$.
 Then, for a unit vector  ${\bm \xi} \in {\mathbb S}^2$
 the Funk--Minkowski transform  of a function  $f$ is a
  function ${\mathcal F}f$ on ${\mathbb S}^2$,
  given by
    \begin{align}
   \label{SEMR:Funk--Minkowski-scalar}
  \{ {\mathcal F}f \}({\bm \xi})\equiv{\mathcal F}_{\bm \xi}f=
\frac{1}{2\pi}
  \int_{0}^{2\pi}f \Big ({\bf e}_{1}({\bm \xi})\cos \omega
 +{\bf e}_{2}({\bm \xi})\sin \omega \Big ) \,{\rm d}{\omega} .
  \end{align}
 \end{definition}
It is clear that the Funk--Minkowski transform is even,
 $\{ {\mathcal F}f\}(-{\bm \xi})=\{ {\mathcal F}f\}({\bm \xi})$, and
${\mathcal F}$ annihilates all odd functions.

  The inversion of the Funk--Minkowski transform has been treated by
  many authors and there are exist several inversion formulas
  in the literature, see
  ~\cite{Funk:1913, Semyanistyi:1961, Helgason:1999, Rubin:2002,  Rubin:2003}.
  In ~\cite{Funk:1913, Funk:1916} 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
  $f$,  ${\mathcal F}f$ and the second one utilizes Abel's integral equation,
  ~\cite{NattererWubelling:2001}.

The inversion formula after P. Funk was obtained by V. Semyanisty
  in \cite[formulas (9) and (11)]{Semyanistyi:1961},
 \begin{align}\label{SEMR:Semyanistyi}
  f_{even}({\bm \theta})=-\frac{ 1}{4\pi} \int_{{\mathbb S}^2}
   \frac{ 1}
      {({\bm \theta} \centerdot{\bm \eta})^2}
      \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta},
 \end{align}
 where  the ${\rm d}{\bm \eta}$  is  the surface measure
  on $\mathbb S^2$ with normalization   $\int_{\mathbb S^2}{\rm d}{\bm
  \eta}=4\pi$ and
  integral is understood in the regularized sense.

  In  \cite[p. 99]{Helgason:1999} S. Helgason gives
  for (\ref{SEMR:Funk--Minkowski-scalar}) the inversion formula of filtered
  back-projection type
  \begin{align} \label{SEMR:Helgason}
  f_{even}({\bm \theta}) = \frac{1}{2 \pi }
  \frac{d}{du}\int_0^u
  \int_{{\mathbb S}^2}
  \{ {\mathcal F}f \}({\bm \eta})
  \delta \Big ({\bm \eta} \centerdot{\bm \theta} -\sqrt{1-v^2} \Big )
   {\rm d}{\bm \eta} \frac{v d v}{\sqrt{u^2-v^2}}  \Big |_{u=1}
   \, ,
  \end{align}
  where  $\delta$ denotes the the Dirac delta function.

 Another example of inversion formula is due to
 B. Rubin ~\cite{Rubin:2002, Rubin:2003}
 \begin{align} \label{SEMR:Rubin}
 f_{even}({\bm \theta})=
 \frac{1}{4\pi} \int_{{\mathbb S}^2}\{ {\mathcal F}f \}({\bm \eta})
 {\rm d}{\bm \eta}
 +  \frac{\Delta_{\bm \theta}}{4\pi} \int_{{\mathbb S}^2}
 \ln|{\bm \eta} \centerdot{\bm \theta}|\, \{ {\mathcal F}f \}({\bm \eta}) \,{\rm d}{\bm \eta} \, ,
 \end{align}
 here  ${\Delta}_{{\bm \theta}}$ it  the Laplace--Beltrami operator
(\ref {SEMR:Laplace--Beltrami}).

 In our studies, an important role is played by spherical convolution
 operator ${\mathcal S}$,
 which is the spherical   analogue  of Hilbert transform,
 see  ~\cite{Samko:1983, Samko:2002, Rubin:2014, Kazantsev:2015}.
    \begin{definition}Let  $f \in  C({\mathbb S}^2)$.
    The  spherical convolution operator ${\mathcal S}$ is defined by,
   \begin{eqnarray}\label{SEMR:OperatorS}
      \{ {\mathcal S}v \}({\bm  \theta}) \equiv
     {\mathcal S}_{{\bm  \theta}}v=p.v.
     \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{v({\bm  \eta})}{{{\bm  \theta}} \centerdot{\bm  \eta}}
   \, {\rm d}{\bm  \eta}, \ {{\bm  \theta}}\in {\mathbb S}^2.
   \end{eqnarray}
 \end{definition}
 This transform is odd,
 $\{ {\mathcal S}f\}(-{\bm \theta})=-\{ {\mathcal S}f\}({\bm \theta})$, and
${\mathcal S}$ annihilates all even  functions.

The results of this paper are  formulated below in Theorems
\ref{th:main1} and \ref{th:main2}.
\begin{theorem}\label{th:main1}
For any function $f({\bm \theta}) \in H^{1}({\mathbb S}^2)$ the
following identity  take place
\begin{align}\label{SEMR:representation}
f(\bm \theta)
&=\underbrace{\frac{1}{4\pi}  \int_{{\mathbb S}^2}
  \{ {\mathcal F}f \}(\bm \eta) {\rm d}{\bm \eta}}_{=f_{00}}+
  p.v.
       \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
    ({\bm \eta} + {\bm \theta}) \, \centerdot
    \Big \{
    \Big [ {\mathcal F}, \nabla   \Big ] f  \Big   \}   (\bm \eta)
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}
   \\ \nonumber
   &=f_{00} +
 {\mathcal S}_{\bm \theta}
 ({\bm \eta} + {\bm \theta} ) \centerdot \Big [ {\mathcal F},
\nabla \Big ]_{\bm \eta}f .
\end{align}
 Here operators  ${\mathcal F}$ and  $\nabla$

are  the  Funk--Minkowski transform
(\ref{SEMR:Funk--Minkowski-scalar}) and the surface gradient
(\ref{SEMR:grad}),

respectively. Through the square brackets $ [. , . ]$
we, as usual, denoted the commutator
$ \Big [ {\mathcal F}, \nabla \Big ]f ={\mathcal F} \nabla f
  -\nabla{\mathcal F} f$, where the F--M transform  ${\mathcal F} $ is applied to vector function $\nabla f$ by componentwise.
\end{theorem}

 If  we decompose  identity (\ref {SEMR:representation}) on even and odd parts
  then we can write,
  \begin{align} \nonumber
  f_{even}({\bm \theta})&=
   \frac{1}{4\pi}  \int_{{\mathbb S}^2}
  \{ {\mathcal F}f \}(\bm \eta) {\rm d}{\bm \eta}
   - p.v.
    \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   {\bm \theta} \, \centerdot \{ \nabla{\mathcal F} f \}(\bm \eta)}
   {{\bm \theta} \centerdot{\bm \eta}}
   {\rm d}{\bm \eta}
   \\ \label{SEMR:inverseEVEN}
    &=
  \frac{1}{4\pi}  \int_{{\mathbb S}^2}
  \{ {\mathcal F}f \}(\bm \eta) {\rm d}{\bm \eta}
  -{{\bm \theta} \, \centerdot}
    {\mathcal S}_{\bm \theta} \nabla{\mathcal F} f  \, ,
  \\ \label{SEMR:inverseODD}
    f_{odd}({\bm \theta})&= p.v.
       \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   {\bm \eta}   \centerdot \{ {\mathcal F} \nabla  f \}(\bm \eta)
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta} = {\mathcal S}_{\bm \theta} {\bm \eta}  \centerdot
   {\mathcal F}_{\bm \eta} \nabla f.
  \end{align}

The inversion formulas  for $f_{even}$ and  $f_{odd}$ follow from these equations
and if  two F--M transformations
 $g({\bm \eta  })= \{{\mathcal F} f \}({\bm \eta})$ and
 ${\bf h}({\bm \eta  }) =\{ {\mathcal F} \nabla f \}({\bm \eta})$
 are known, then  the unknown function $f$   can be  reconstruct
 completely,
\begin{align}\label{SEMR:fful-f}

  f({\bm \theta})=\frac{1}{4\pi}  \int_{{\mathbb S}^2}
  g(\bm \eta) {\rm d}{\bm \eta}
  - p.v.
    \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   {\bm \theta} \, \centerdot  \nabla g(\bm \eta)}
   {{\bm \theta} \centerdot{\bm \eta}}\,
   {\rm d}{\bm \eta}+
    p.v.
       \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   {\bm \eta}   \centerdot {\bf h}(\bm \eta)
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta} \, .
    \end{align}

The next problem that we will consider
 is   the problem of Helmholtz--Hodge
 decomposition for  a tangential vector field on the sphere ${\mathbb
 S}^2$, see \cite{FreedenSchreiner:2009}.
 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
 \begin{align}\label{SEMR:Helmholtz--Hodge}
 {\bf f}({\bm \theta})=\nabla_{\bm \theta}u({\bm \theta})+{\bm \theta} \times
 \nabla_{{\bm \theta}}v({\bm \theta}),
 \end{align}
  where $\nabla_{{\bm  \theta}} $ is the surface gradient on the sphere, and
 rotated gradient
 ${\bm \theta} \times \nabla_{{\bm  \theta}} $
 means the cross-product of the surface gradient of $v$ with the
 unit normal vector ${\bm \theta}$ to the sphere.
  Here   $\nabla_{\bm \theta} {u}$ is called also as inrrotational,
  poloidal, electric or potential field
  and
  $\nabla^{\bot}_{\bm \theta} {v}$ 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 (\ref{SEMR:Helmholtz--Hodge})
 is obtained
 by use of  Funk--Minkowski- transform ${\mathcal F}$ and  spherical convolution
 transform  ${\mathcal S}$.
  \begin{theorem}\label{th:main2}
  Any vector field ${\bf f} \in {\bf L}_{2,tan}({\mathbb S}^2) $ that is
  tangent to the sphere
  can be uniquely decomposed into a sum
 (\ref{SEMR:Helmholtz--Hodge})
  of a surface curl-free component and  a surface
  divergence-free
  component
  with scalar valued functions $u,v \in H^{1}({\mathbb S}^2)/{\mathbb R}$.
 Functions
 $u$ and $v$ are  velocity potential  and stream
 functions  that are calculated unique up to a constant by the formulas
\begin{align}\nonumber
u({\bm \theta})&=\Big [{\mathcal S, {\bm \eta} \,  \centerdot \,  ,
{\mathcal F}} \Big ]_{\bm \theta}  {\bf f}=
 \Big \{  {\mathcal S } {\bm \eta}
\centerdot {\mathcal F} {\bf f}
 \Big \} ({\bm \theta})-
 \Big \{{\mathcal F} {\bm \eta} \centerdot {\mathcal S}{\bf f}
 \Big \}({\bm \theta})
  \\ \label{SEMR:u}
  &=
   {\mathcal S}_{\bm \theta}
    {\bm \eta}
\centerdot {\mathcal F}_{\bm \eta} {\bf f}
 -
 {\mathcal F}_{\bm \theta} {\bm \eta} \centerdot {\mathcal S}_{\bm \eta}{\bf f}
  ,
\\ \nonumber
v({\bm \theta})&= {\bm \theta} \centerdot
 \Big [{\mathcal S, {\bm \eta} \times , {\mathcal F}} \Big ]_{\bm \theta}{\bf  f}
  ={\bm \theta} \centerdot
 \Big \{
 {\mathcal S } {\bm \eta} \times {\mathcal F} {\bf f}
 \Big \}({\bm \theta}) - {\bm \theta} \centerdot
   \Big \{ {\mathcal F}{ \bm \eta} \times {\mathcal S} {\bf f }
   \Big \} ({\bm \theta})
   \\\label{SEMR:v}
   &={\bm \theta} \centerdot
  {\mathcal S }_{\bm \theta} {\bm \eta} \times {\mathcal F}_{ \bm \eta} {\bf f}
  - {\bm \theta} \centerdot
   {\mathcal F}_{\bm \theta}{ \bm \eta} \times {\mathcal S}_{ \bm \eta} {\bf f }
   ,
\end{align}
 where  through $[{\mathcal A},{\mathcal B},{\mathcal C}]$ we denote the generalized commutator,
\[
 [{\mathcal A},{\mathcal B},{\mathcal C}]={\mathcal A}{\mathcal B}{\mathcal C}
  -{\mathcal C}{\mathcal B}{\mathcal A}.
  \]
\end{theorem}
As a consequence of this  theorem, we can obtain formulas for solving two important problems
on the sphere ${\mathbb S}^2$:
$\nabla u={\bf f}$ and $\nabla^{\bot} v={\bf g}$ .
Answers  to solve these problems are
\[
u({\bm \theta})=({\mathcal S }_{\bm \theta} {\bm \eta} \centerdot
  {\mathcal F}_{\bm \eta} -
  {\mathcal F}_{\bm \theta} {\bm \eta} \centerdot {\mathcal S}_{\bm \eta}
     ){\bf f} \  \ \text{for} \ \   \nabla u={\bf f} \in {\bf L}_{2,tan}({\mathbb S}^2)
\]
and
\[v({\bm \theta})={\bm \theta} \centerdot(
  {\mathcal S }_{\bm \theta} {\bm \eta} \times {\mathcal F}_{\bm \eta}
   - {\mathcal F}_{\bm \theta} {\bm \eta} \times {\mathcal S}_{\bm \eta} ){\bf g}
    \  \ \text{for} \  \ \nabla^{\bot} v={\bf g}\in {\bf L}_{2,tan}({\mathbb S}^2).
\]

\section{Basic methods and tools}

\subsection{Spherical harmonics (SHs)}

  In this section  we state some properties of  complex spherical harmonics.
  A spherical harmonic  $Y_{N\ell}$ of degree  $N$ on ${\mathbb S}^2$
  is the
  restriction to ${\mathbb S}^2$ of a homogeneous harmonic polynomial
  of degree $N$ in ${\mathbb R}^3$.

  The Legendre polynomials of the first kind  $P_{N}$ of degree $N\in {\mathbb N}_0$
  or simply Legendre polynomials   are given by the Rodrigues formula
  \[
  P_{N}(t)=\frac{1}{N!2^N}\frac{{\rm d}^N}{{\rm  d}t^N}(t^2-1)^N.
  \]
  We recall that  Legendre polynomials of the first kind $P_N(t)$
  are the orthogonal polynomials  on $(-1,1)$
  with weight function $w(t)=1.$
  We define with  $C_N^{(3/2)}$ the Gegenbauer polynomial of degree $N$ with  parameter
  ${\lambda}=3/2$,
  \[
  C_N^{(3/2)}(t)=\frac{d}{dt}P_{N+1}(t).
  \]

 The following formulas will be used in our calculations
 (\cite{AbramowitzStegun:1972})
  \begin{align}\label{SEMR:P0}
  P_{2j}(0)=(-1)^j\frac{\Gamma(j+1/2)}{\sqrt{\pi} j!}
  =\frac{(-1)^j(2j-1)!!}{(2j)!!},
 \end{align}
 \begin{align}\label{SEMR:PN}
 (N+1)P_{N+1}(0)=-N P_{N-1}(0),
  \end{align}
  \begin{align}\label{SEMR:P0C0}
      C^{(3/2)}_{2j}(0)=\frac{(-1)^j(2j+1)!!}{(2j)!!}  \  \text{or}  \
      C^{(3/2)}_{N-1}(0)={NP_{N-1}(0)}, \ N=2j+1 .
  \end{align}
The following usefull asymptotics holds as $j$ goes to infinity
  \begin{align}\label{SEMR:asymptotic}
    P_{2j}(0) \sim \frac{1}{\sqrt{2j+1}} \ \text{and} \
     \frac{1}{C^{(3/2)}_{2j}(0)}=\frac{1}{(2j+1)P_{2j}(0)}
     \sim \frac{1}{\sqrt{2j+1}} \ \text{if } \
    j \to \infty .
        \end{align}

The associated Legendre functions of the first kind $P^{\ell}_{N}$  for non negative $\ell\geq  0$  are defined as
  \[
  P^{\ell}_{N}(t)=(1-t^2)^{\frac{\ell}{2}}\frac{{\rm d}^\ell}{{\rm  d}t^\ell}P_N(t),
  \]
  where $N,\ell \in {\mathbb N}_0$ with $\ell \leq N$
  and for the  negative order $-\ell,$ $P^{-\ell}_{N}$ are given by
  \[
  P^{-\ell}_{N}(t)=
  (-1)^{\ell}\frac{(N-\ell)!}{(N+\ell)!}P^{\ell}_{N}(t), \  \ell \geq
  0 \, .
  \]
  When the order $\ell=0$, the associated Legendre function becomes a
  polynomial in $t$ and instead being written $P^{0}_N(t)$ it is designated
  $P_N(t)$, the Legendre polynomial.

  The complex SHs $Y_{N\ell}$
  are related to  the associated Legendre functions as follows
  \[
  Y_{N\ell}({\bm \xi})=(-1)^\ell N_{N\ell}{e}^{{\rm i}\ell\varphi}P^{\ell}_{N}(\cos \theta), \  |\ell| \leq  N,
  \]
  where  $N_{N\ell}$ is a normalization constant
  \[
  N_{N\ell}=\sqrt{\frac{2N+1}{4\pi}\frac{(N-\ell)!}{(N+\ell)!}}
  \]
  and the extra factor $(-1)^\ell$ is called the Condon--Shortley phase.

  The $Y_{N\ell}$ are complex-valued polynomials  of the sines and cosines
  of $\theta$ and $\varphi$
  and for complex conjugate functions  the following formula  fulfil
  \[
  \overline {Y_{N \ell}({\bm \xi})}=(-1)^{\ell} Y_{N,-\ell}({\bm \xi}).
  \]

  The  parity rule for spherical harmonic is
  \[
  Y_{N\ell}(-{\bm \xi})=(-1)^NY_{N\ell}({\bm \xi}).
  \]
  It is known that the subspace of all spherical  harmonics of degree $N$,
    ${\rm span} \{ Y_{N\ell} \}_{\ell}^{N}$,
    is the eigenspace of the Laplace--Beltrami
  operator
  (\ref{SEMR:Laplace--Beltrami})
   corresponding to the eigenvalue $-\lambda^2_N=-N(N+1),$
  \[
  \Delta_{\bm \xi}Y_{N\ell}({\bm \xi})=-N(N+1)Y_{N \ell}({\bm \xi}).
  \]
  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
 $\Big \{ Y_{N\ell}, |\ell| \leq N  \Big \}_{N=0}^{\infty}$
 forms an orthonormal basis
 for ${L}_2(\mathbb S^2; {\mathbb C})$
 \begin{equation}
 \label{SEMR:orthSH}
 ({Y}_{N_1\ell_1},  Y_{N_2\ell_2})_{L_2(\mathbb S^2)}=
 \int_{{\mathbb S}^2}{Y}_{N_1\ell_1}({\bm \xi})\overline{Y_{N_2\ell_2}({\bm
 \xi})}
     \, {\rm d} {\bm \xi}=
   \delta^{N_1}_{N_2}\delta^{\ell_1}_{\ell_2},
 \end{equation}
 where  $\delta^{i}_{j}$ is the Kronecker symbol and the space  ${L}_2(\mathbb S^2) \equiv {L}_2(\mathbb S^2; {\mathbb C})$ is a
 Hilbert space of square-integrable functions on ${\mathbb S}^2$
 with the hermitian inner product and the  finite norm,
 \begin{equation*}

  (u,v)_{L_2(\mathbb S^2)}=
  \int_{\mathbb S^2}  {u}({\bm \xi})  \overline{v({\bm \xi})} \,
  {\rm d}{\bm \xi},
 \

 ||u||^2_{L_2(\mathbb S^2)}= (u,u)_{L_2(\mathbb S^2)}.
 \nonumber
 \end{equation*}

 The Fourier coefficients for $u \in L_2({\mathbb S}^2)$ are
 $ u_{N\ell}=(u,Y_{N\ell})_{L_2}$.
 Then, every  function $u \in {L}_2(\mathbb S^2)$ admits a
 spherical harmonics series expansion  in $L_2$--sense
  \begin{align} \label{SEMR:L2}
  u({\bm \xi}) &=\sum^{\infty}_{N=0} \sum_{\ell} u_{N\ell}
  Y_{N\ell}({\bm \xi}),
  \\
  \label{SEMR:L2norm}
 ||u||^2_{L_2(\mathbb S^2)}
  &=\sum^{\infty}_{N=0} \sum_{\ell} |u_{N\ell}|^2  \, .
   \end{align}

   We close this section with Funk--Hecke formula.

  It was first published
  by Funk (1916) and a little later by Hecke (1918).

  \begin{theorem}\label{th:Funk}[The Funk--Hecke Theorem]
  Suppose $f(t)$ $\in L_1(-1,1)$ is an integrable function.
  Then for
  every spherical harmonics of degree $N$ we have
  \begin{equation}
  \label{SEMR:Funk-Heckeformula}
  \int_{{\mathbb S}^2}f({\bm \xi}\centerdot{\bm \eta})Y_{N \ell}({\bm \xi})
  \, {\rm d}{\bm \xi}= 2\pi Y_{N \ell}({\bm \eta})
  \int_{-1}^{1}f(t)P_N(t)\, {\rm d}t,
  \end{equation}
  where  ${\bm \xi}\centerdot{\bm \eta}$ denotes the inner product
  of  unit vectors ${\bm \xi}$ and ${\bm \eta}$,
  $P_N$ denotes the
  $N$th order Legendre polynomial.
   \end{theorem}

   The Funk--Hecke formula is useful in simplifying calculations of certain integrals
  over ${\mathbb S}^2$ and plays an important role in the theory of  spherical harmonics.
  For more details on the Funk--Hecke formula see ~\cite{AtkinsonHan:2012, Samko:2002},
  for example.
  A general overview on spherical harmonics and the  relevant
  problems can be found in the monographs
  ~\cite{AbramowitzStegun:1972, AtkinsonHan:2012, Muller:1980, VarMosKher:1988,
  Muller:1980, DaiXu:2013, FreedenSchreiner:2009, FreedenGutting:2013}.

\subsection{Surface differential operators on the sphere ${\mathbb S}^2$}
  Here  we briefly recall the definitions and some properties of surface
  differential operators.

  The space  ${\bf L}_2(\mathbb S^2)\equiv {{\bf L}}_2(\mathbb S^2; {\mathbb C})$
 is a Hilbert space of square-integrable vector functions on
 ${\mathbb S}^2$
 with the inner product and the  finite norm,
 \begin{equation*}
 ({\bf u},{\bf v})_{{\bf L}_2(\mathbb S^2)}=
 \int_{\mathbb S^2}  {{\bf u}}({\bm \xi})  \centerdot  \overline{{\bf v}({\bm \xi})}
 \, {\rm d}{\bm \xi},
 \
 ||{\bf u}||^2_{{\bf L}_2(\mathbb S^2)}= ({\bf u},{\bf u})_{{\bf L}_2(\mathbb S^2)}.
 \nonumber
 \end{equation*}

  \begin{definition}
   {\it The tangential gradient or the surface gradient},
  denoted   by $\nabla \equiv \nabla_{\bm \xi}$
  and
  {\it  the tangential rotated gradient (the surface curl-gradient)}, denoted by
  $\nabla^{\bot} \equiv \nabla^{\bot}_{\bm \xi}$,
  are  defined  accordingly  as
  \begin{align}\label{SEMR:grad}
  \nabla_{\bm \xi}u &=
  \frac{\partial u}{\partial \theta} {\bf e}_1(\bm \xi) +
  \frac{1}{\sin \theta} \frac{\partial u}{\partial \varphi} {\bf  e}_2(\bm \xi),
  \\
  \label{SEMR:grad-bot}
  \nabla^{\bot}_{\bm \xi}u &={\bm \xi} \times \nabla_{\bm \xi}u=
 -\frac{1}{\sin \theta} \frac{\partial u}{\partial \varphi} {\bf e}_{1}(\bm \xi)+
   \frac{\partial u}{\partial \theta} {\bf e}_{2}(\bm \xi),
 \end{align}
 where
  ${\bm \xi} ={\bf  i}\sin{ \,\theta}\cos{ \,\varphi}+
  {\bf  j}\sin{ \,\theta}\sin{ \,\varphi}+
  {\bf  k}\cos{ \,\theta}$.
 \end{definition}

 Obviously, we have  ${\bm \xi} \centerdot \nabla_{\bm \xi}u({\bm
 \xi})=0$, ${\bm \xi} \centerdot \nabla^{\bot}_{\bm \xi}u({\bm
 \xi})=0$ and $\nabla u \centerdot \nabla^{\bot}u =0 $, thus $\nabla u$ and  $\nabla^{\bot} u$
 are will be  tangential vector fields on the sphere ${\mathbb S}^2$
 with $\nabla^{\bot}$
 is rotation by $\pi /2 $ in the tangent plane.

 We  must note here that integration by parts formulas on the
 sphere for  operators (\ref{SEMR:grad}) and  (\ref{SEMR:grad-bot}) are differ.
 Namely, for $u, v \in C^1({\mathbb S}^2),$ we have
  \begin{align}\label{SEMR:intgrad1}
     \int_{{\mathbb S}^2}
     u({\bm \xi})\nabla_{\bm \xi}v(\bm \xi)\,
    {\rm d}{\bm \xi}
  &=-\int_{{\mathbb S}^2}
     v({\bm \xi})\nabla_{\bm \xi}u(\bm \xi)
    {\rm d}{\bm \xi}+2  \int_{{\mathbb S}^2}{\bm \xi}
     u({\bm \xi})v(\bm \xi)\,
    {\rm d}{\bm \xi},
  \\
  \label{SEMR:intgradbot}
     \int_{{\mathbb S}^2}
     u({\bm \xi})\nabla^{\bot}_{\bm \xi}v(\bm \xi)\,
    {\rm d}{\bm \xi}
  &=-\int_{{\mathbb S}^2}
     v({\bm \xi})\nabla^{\bot}_{\bm \xi}u(\bm \xi)\,
    {\rm d}{\bm \xi}.
 \end{align}
 \begin{definition}
  In canonical coordinates, the surface divergence
  ${\rm div}_{\bm \xi}$
  of   vector-valued function
  ${\bf v}({\bm \xi})=v^1{\bf  e}_1({\bm \xi})+v^2{\bf  e}_2({\bm \xi})+v^3{\bm \xi}$
  on the sphere ${\mathbb S}^2$  is written as,
 \begin{align}\label{SEMR:div}
 {\rm div}_{\bm \xi}{\bf v}& =\frac{1}{\sin \theta}\left (
 \frac{\partial}{\partial \theta}(v^{1}\sin \theta )+
 \frac{\partial}{\partial \varphi} v^{2} \right )+2v^3 \, .
  \end{align}
For  tangent vector field ${\bf v}$  we define   the scalar surface rotation (or scalar curl operator) ${\rm curl}_{\bm \xi}$
  by
 \begin{align}\label{SEMR:curl}
 {\rm curl}_{\bm \xi}{\bf v}& =-{\rm div}_{\bm \xi}({\bm \xi }\times {\bf v})=
  \frac{1}{\sin \theta}\left (
 \frac{\partial}{\partial \theta}(v^{2}\sin \theta )-
 \frac{\partial}{\partial \varphi} v^{1} \right ) .
  \end{align}
\end{definition}

 If $u \in C^1({\mathbb S}^2)$ and tangential vector field
   ${\bf v} \in {\bf C}^1({\mathbb S}^2)$, then  we have
   integral formulas, which are also understood as inner  products
  \begin{align}
    \label{SEMR:intdiv}
     \int_{{\mathbb S}^2}
     {\bf v}({\bm \xi}) \centerdot \nabla_{\bm \xi}u(\bm \xi)
    \,{\rm d}{\bm \xi}
  &=-\int_{{\mathbb S}^2}
     u({\bm \xi}){\rm div}_{\bm \xi}{\bf v}(\bm \xi)
    \,{\rm d}{\bm \xi}
    \\ \text{or} \ \
    ({\bf v},\nabla u)_{{\bf L}_2({\mathbb S}^2)}&=
    -({\rm div}\,{\bf v},u)_{L_2({\mathbb S}^2)},
\\    \label{SEMR:intcurl}
     \int_{{\mathbb S}^2}
     {\bf v}({\bm \xi}) \centerdot \nabla^{\bot}_{\bm \xi}u(\bm \xi)
    \,{\rm d}{\bm \xi}
  &=-\int_{{\mathbb S}^2}
     u({\bm \xi}){\rm curl}_{\bm \xi}{\bf v}(\bm \xi)
    \,{\rm d}{\bm \xi}
    \\ \text{or} \ \
   ({\bf v},\nabla^{\bot} u)_{{\bf L}_2({\mathbb S}^2)}&=
    -({\rm curl}\,{\bf v},u)_{L_2({\mathbb S}^2)}.
\end{align}

 \begin{definition}
 Finally, we define  the Beltrami operator, which is also  called the
 Laplace--Beltrami operator
 ${\Delta}\equiv {\Delta}_{\bm \xi}$ as
 \begin{align}\label{SEMR:Laplace--Beltrami}
 \Delta_{\bm \xi}u({\bm \xi})={\rm div}_{\bm \xi}\nabla_{\bm \xi}u({\bm
 \xi})\, ,
 \end{align}
 i.e. the divergence of a gradient is the Laplacian.
 \end{definition}
 One easily checks that

\begin{align}\label{SEMR:Laplace--Beltrami2}
 \Delta_{\bm \xi}u({\bm \xi})={\rm curl}_{\bm \xi}\nabla^{\bot}_{\bm \xi}u({\bm
 \xi})
 \end{align}
 and also
\begin{align*}
  {\rm curl}_{\bm \xi}\nabla_{\bm \xi}u({\bm  \xi})=0,
\ \ {\rm div}_{\bm \xi}\nabla^{\bot}_{\bm \xi}u({\bm  \xi})=0,
 \end{align*}
 thus we say that $\nabla_{{\bm \xi}} u$ is the curl-free,
  but $ \nabla^{\bot}_{{\bm \xi}}u$ is the divergence-free vector
 fields.

 The next formula is Green--Beltrami identity
 or Green's first surface identity, see \cite[Proposition 3.3]{AtkinsonHan:2012},
 \cite[Theorem 4.12]{Michel:2013}: for
 any $u \in C^1({\mathbb S}^2)$ and any $v \in C^2({\mathbb S}^2)$ we have,
 \begin{align}\label{SEMR:L-B}
     \int_{{\mathbb S}^2}
     \nabla_{\bm \xi}{u}({\bm \xi}) \centerdot \nabla_{\bm \xi}v(\bm \xi)
    \,{\rm d}{\bm \xi}
  &=-\int_{{\mathbb S}^2}
     u({\bm \xi}){\Delta}_{\bm \xi}{v}(\bm \xi)\,
    {\rm d}{\bm \xi}
    \\
    \text{or} \ \
     ( \nabla u,\nabla v)_{{\bf L}_2(\mathbb  S^2)}
    &= -( u, \Delta v)_{L_2(\mathbb  S^2)}
     \, .
        \end{align}
 For example, if we take $u=Y_{N_1\ell_1}$  and  $v=Y_{N_2\ell_2}$, then
\begin{align}\label{SEMR:OrtGrad}
     &( \nabla Y_{N_1\ell_1},\nabla Y_{N_2\ell_2})_{{\bf L}_2(\mathbb
     S^2)}
     \\ \nonumber
    &=
     \int_{{\mathbb S}^2}
     \nabla_{\bm \xi}{Y}_{N_1\ell_1}({\bm \xi})
     \centerdot \nabla_{\bm \xi} \overline {Y_{N_2\ell_2}(\bm
     \xi)}\,
    {\rm d}{\bm \xi}
    =-\int_{{\mathbb S}^2}
     Y_{N_1\ell_1}({\bm \xi}){\Delta}_{\bm \xi}\overline{{Y}_{N_2\ell_2}(\bm
     \xi)}
    {\rm d}{\bm \xi}
    \\ \nonumber
&=N_2(N_2+1)\int_{{\mathbb S}^2}
     Y_{N_1\ell_1}({\bm \xi})\overline{{Y}_{N_2\ell_2}(\bm
     \xi)}\,
    {\rm d}{\bm \xi}=N_2(N_2+1)\delta_{N_1}^{N_2}\delta_{\ell_1}^{\ell_2}
    \, .
        \end{align}

 For more definitions and properties of
 these differential operators see e.g. ~\cite{ VarMosKher:1988, AtkinsonHan:2012, Nedelec:2001,
 FreedenSchreiner:2009, FreedenGutting:2013}.

\subsection{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 ~\cite{MorseFeshbach:1953,  VarMosKher:1988, Nedelec:2001,
DerevtsovKazantsev:2009,   FreedenSchreiner:2009} for more details in this theme.

\subsubsection{Pure--spin vector spherical harmonics}

Let us now define a complete  orthogonal set of vectors in ${\bf L}_2({\mathbb S}^2)$.
 \begin{definition}
 The vector spherical harmonics (or pure--spin VSHs)
 are arranged in three families:
 ${\bf y}^{(1)}_{N \ell}({\bm \xi}), $
 ${\bf y}^{(2)}_{N \ell}({\bm \xi}) $
 and ${\bf y}^{(3)}_{N \ell}({\bm \xi}).$
 For ${\bm \xi} \in {\mathbb S}^2$ and given a scalar spherical harmonic
 $Y_{N \ell}({\bm \xi})$
 the unnormalized vector spherical harmonics  are the set
 \begin{align}
  {\bf y}^{(1)}_{N \ell}({\bm \xi})&={\bm \xi}Y_{N \ell}({\bm \xi}), \ \
  N \in 0 \cup {\mathbb N},  \label{SEMR:VSH1}
  \\

 {\bf y}^{(2)}_{N \ell}({\bm \xi})&=\nabla_{{\bm \xi}}Y_{N \ell}({\bm \xi}),
 \ \ N \in {\mathbb N},

  \label{SEMR:VSH2}
  \\

  {\bf y}^{(3)}_{N \ell}({\bm \xi})
  &={\bm \xi}\times {\bf y}^{(2)}_{N \ell}({\bm \xi})
   =\nabla^{\bot}_{\bm \xi}Y_{N \ell}({\bm \xi}),
    \ \ N \in {\mathbb N}.

  \label{SEMR:VSH3}
 \end{align}
\end{definition}
The pure--spin VSHs form a complete set of orthogonal vector functions on the
surface
 of a sphere ${{\mathbb S}^2}$ with the inner product of the ${\bf L}_2({{\mathbb S}^2})$ space, see \cite [Theorem 5.2.7]{FreedenGutting:2013}.

 Clearly, $||{\bf y}^{(1)}_{N\ell}||_{{\bf L}_2({\mathcal S}^2)}=1.$
  To calculate the  norms  of vector functions ${\bf y}^{(2)}_{N \ell}$ and   ${\bf y}^{(3)}_{N \ell}$, we can use  (\ref{SEMR:OrtGrad}). Therefore, the normalizing vector  harmonics
  or orthonormal  system of  VSHs are
 \begin{align*}{\bf y}^{(1)}_{N\ell}, \
 \widetilde{\bf y}^{(2)}_{N \ell}={\bf y}^{(2)}_{N \ell}/
\sqrt{N(N+1)}\, , \
 \widetilde{\bf y}^{(3)}_{N \ell}={\bf y}^{(3)}_{N
 \ell}/\sqrt{N(N+1)}\, .
 \end{align*}
 Each vector function ${\bf f} \in {\bf L}_2({\mathbb S}^2)$ has
 the Fourier expansion
 \begin{align*}
   {\bf f }({\bm \xi})&=f_{1,00} {\bf y}^{(1)}_{00}({\bm \xi})
   +\sum^{\infty}_{N=1} \sum_{\ell}
  f_{1,N\ell} {\bf y}^{(1)}_{N \ell}({\bm \xi})+
 f_{2,N\ell} \widetilde{\bf y}^{(2)}_{N \ell}({\bm \xi})
 + f_{3,N\ell}\widetilde{\bf y}^{(3)}_{N \ell}({\bm \xi}),
 \\
 ||{\bf f }||^2_{{\bf L}_2({\mathbb S}^2)}&=|f_{1,00}|^2+\sum^{\infty}_{N=1}
 \sum_{\ell} |f_{1,N\ell}|^2 +|f_{2,N\ell}|^2 + |f_{3,N\ell}|^2 .
  \end{align*}

  The hermitian inner products are then given by
\begin{align*}
 ({\bf f },  {\bf h })_{{\bf L}_2({\mathbb S}^2)}
 = f_{1,00} \overline { h}_{1,00}+ \sum^{\infty}_{N=1}
  \sum_{\ell}f_{1,N \ell} \overline { h}_{1,N\ell}+ {f_{2,N \ell} \overline { h}_{2,N\ell}}+ f_{3,N\ell}\overline{h}_{3,N\ell} .
  \end{align*}

\subsubsection{Pure--orbit vector spherical harmonics}

An alternative orthogonal basis in the space ${\bf L}_2({\mathbb S}^2)$
 is the system of pure--orbit VSHs
 $\{{\bf h}^{(e)}_{00}, {\bf h}^{(e)}_{N \ell}, {\bf h}^{(i)}_{N \ell}, {\bf y}^{(3)}_{N \ell},
 \ |\ell| \leq N \}_{N=1}^{\infty}$,
where vector functions ${\bf h}^{(e)}_{N \ell}$ and ${\bf h}^{(i)}_{N \ell}$
defined by
\begin{align}
 \label{SEMR:he0}
  {\bf h}^{(e)}_{00}
  &= -{\bf y}^{(1)}_{00},
\\
 \label{SEMR:he}
  {\bf h}^{(e)}_{N \ell}
  &= -(N+1){\bf y}^{(1)}_{ N \ell}
   + {\bf y}^{(2)}_{N\ell}, \ \ N \in {\mathbb N},
   \\
      \label{SEMR:hi}
    {\bf h}^{(i)}_{N \ell}
  &=N{\bf y}^{(1)}_{N \ell}
  + {\bf y}^{(2)}_{N \ell}, \ N \in {\mathbb N}.
 \end{align}

 The pure-orbit vector spherical harmonics also has a nice  properties,
 in particular, they are  eigenfunctions for the vectorial
 Funk--Minkowski operator ${\mathcal F}$  in the space ${\bf L}_{2,even}({\mathbb S}^2)$ and for
 vectorial Hilbert  operator ${\mathcal S}$ in the space ${\bf L}_{2,odd}({\mathbb S}^2)$,
  see Lemmas \ref{l:F} and \ref{l:S} in the section Proofs.

\subsubsection{Tangent vector fields and  Helmholtz--Hodge decomposition}
 Consider the tangent vector field ${\bf f} \in {\bf L}_{2,tan}({\mathbb S}^2)$,
 it  can be written uniquely as
\begin{align*}
   {\bf f }({\bm \theta})&=
    \underbrace{\sum^{\infty}_{N=1} \sum_{\ell}
 f_{2,N\ell} \widetilde{\bf y}^{(2)}_{N \ell}({\bm \theta})}_
 {\text{the curl-free component}}
 +
 \underbrace{\sum^{\infty}_{N=1} \sum_{\ell}
 f_{3,N\ell}\widetilde{\bf y}^{(3)}_{N \ell}({\bm \theta})}_{\text{the divergence-free component}}

\\
&=\sum^{\infty}_{N=1} \frac{1}{\sqrt{N(N+1)}}\sum_{\ell}
 f_{2,N\ell}\nabla Y_{N\ell}({\bm \theta})
 + f_{3,N\ell}{\bm \theta} \times \nabla Y_{N\ell}({\bm \theta}).

 \end{align*}

Then formally we have
\begin{align*}
 {\bf f }({\bm \theta})=\nabla\sum^{\infty}_{N=1} \frac{1}{\sqrt{N(N+1)}}\sum_{\ell}
 f_{2,N\ell} {Y}_{N \ell}({\bm \theta})
 +\nabla^{{\bot}}
\sum^{\infty}_{N=1} \frac{1}{\sqrt{N(N+1)}}\sum_{\ell}
 f_{3,N\ell}{Y}_{N \ell}({\bm \theta}),
  \end{align*}
  where according to (\ref{SEMR:Helmholtz--Hodge})
     the velocity
potential and stream functions are
\begin{align*}
u({\bm \theta})&=\sum^{\infty}_{N=1}
\frac{1}{\sqrt{N(N+1)}}\sum_{\ell}
 f_{2,N\ell} {Y}_{N \ell}({\bm \theta}),
\\
v({\bm \theta})&=\sum^{\infty}_{N=1}
\frac{1}{\sqrt{N(N+1)}}\sum_{\ell}
 f_{3,N\ell}{Y}_{N \ell}({\bm \theta}).
 \end{align*}

Another evident approach  consists in solving the Laplace--Beltrami equations
on the sphere
  \begin{align*}
  \Delta_{\bm \theta}{u}(\bm \theta)&={\rm div}_{\bm \theta} \, {\bf f}(\bm \theta),
  \\
  \Delta_{\bm \theta}{v}(\bm \theta)&={\rm curl}_{\bm \theta} \, {\bf f}(\bm \theta).
  \end{align*}
 They  can be solved
  in integral form, for example, involving
 Green�s function with respect to the Laplace--Beltrami
 $\Delta_{\bm \theta}$, see ~\cite[Theorem 4.6.9]{FreedenGutting:2013}.

\subsection{Hilbertian Sobolev spaces on the sphere}

\subsubsection{ Sobolev scalar functions  on ${\mathbb S}^2$}

The Sobolev space $H^s({\mathbb S}^2)$
 with a smoothness index
 $s \geq 0$ is defined by (\cite{AtkinsonHan:2012, Michel:2013,
 Rubin:2014, Nedelec:2001, QuellmalzHielscherLouis:2018})
\begin{align*}
H^s({\mathbb S}^2):= \{ u \in {L}_2(\mathbb S^2; {\mathbb C})
  :\sum^{\infty}_{N=0}(1+N(N+1))^s \sum_{\ell} |u_{N\ell}|^2
  < \infty \} .
  \end{align*}
In other words $u \in H^s({\mathbb S}^2)$ if and only if
$(I-\triangle)^{s/2}u \in L_2({\mathbb S}^2)$. The space
$H^s({\mathbb S}^2)$ is a Hilbert space with the hermitian inner
product
\begin{align*}
 (u,v)_{{H}^s({\mathbb S}^2)}
 =\sum^{\infty}_{N=0} (1+N(N+1))^s
 \sum_{\ell} u_{N\ell} \overline{v_{N\ell}}
  \end{align*}
and the induced norm
\[
||u||^2_{H^s({\mathbb S}^2)}
  =\sum^{\infty}_{N=0}(1+N(N+1))^s \sum_{\ell} |u_{N\ell}|^2=
  ||(I-\triangle)^{s/2}u||^2_{L_2({\mathbb S}^2)}\, .
\]
Putting $s=0$ we obtain $H^0({\mathbb S}^2)=L_2({\mathbb S}^2)$.
If $s=1$ then in addition to (\ref{SEMR:L2}), (\ref{SEMR:L2norm})
we have
\[
 \nabla_{{\bm \xi}}u({\bm \xi})=\sum^{\infty}_{N=0}
\sum_{\ell}
 u_{N\ell}\nabla_{{\bm \xi}}
  Y_{N\ell}({\bm \xi}) ,
=\sum_{N=1}^{\infty}\sqrt{N(N+1)}
 \sum_{\ell}(u,Y_{N\ell})_{L_{2}({\mathbb S}^2)}\widetilde{\bf
 y}^{(2)}_{N\ell}\, ,
 \]
 \begin{align*}
 ||\nabla u||^2_{{\bf L}_2({\mathbb S}^2)}=\sum^{\infty}_{N=0}N(N+1) \sum_{\ell}
 |u_{N\ell}|^2  .
  \end{align*}
Thus we can define the Sobolev space $H^1({\mathbb S}^2)$ as (see
 \cite[p. 14]{Nedelec:2001})
\[H^1({\mathbb S}^2)=\{u \in L_{2} ({\mathbb S}^2) :
\nabla u \in {\bf L}_{2} ({\mathbb S}^2) \}
\]
with its  inner product and the  finite  Sobolev norm
\begin{align*}
 (u,v)_{H^1({\mathbb S}^2)}=(u,v)_{L_2({\mathbb S}^2)}+
(\nabla u,\nabla v)_{{\bf L}_2({\mathbb S}^2)},
 \ \
 ||u||^2_{H^1({\mathbb S}^2)}= ||u||^2_{L_2({\mathbb S}^2)}+
  ||\nabla u||^2_{{\bf L}_{2}({\mathbb S}^2)} \, ,
\end{align*}
where $\nabla$ 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
($\nabla$-definition of Sobolev spaces)
\[||u||^2_{H^s({\mathbb S}^2)}=(u, v)_{L_2({\mathbb S}^2)}+
\sum_{k=1}^{m}(\nabla^k u, \nabla^k v)_{{\bf L}_2({\mathbb S}^2)}.
\]
If we s  consider  a closed linear subspace $H^1({\mathbb S}^2)/{\mathbb R})
 \subset H^1({\mathbb S}^2)$,
\[
 H^1({\mathbb S}^2)/{\mathbb R}=\{ u\in H^1({\mathbb S}^2) : \int_{{\mathbb S}^2}
  {u}({\bm \xi}) \,{\rm d}{\bm \xi}=0  \} ,
\]
 then due to a Poincar\'{e} inequality for all
 $u \in H^1({\mathbb S}^2)/{\mathbb R} $ we can  define  an equivalent norm
 for $H^1({\mathbb S}^2)/{\mathbb R}$
\[
||u||^2_{H^1({\mathbb S}^2)/{\mathbb R}}=|| \nabla u||_{{\bf L}_{2}({\mathbb S}^2)} \,,
\]
 such that $H^1({\mathbb S}^2)/{\mathbb R}$ becomes a Hilbert space with
 the inner product
\[
(u,v)_{H^1({\mathbb S}^2)/{\mathbb R}}=(\nabla u,\nabla v)_{{\bf L}_{2}({\mathbb S}^2)} \,.
\]

For more details on these spaces, we refer the reader to
     ~\cite{AtkinsonHan:2012},
     ~\cite[Theorems 4.12 and 6.12]{Michel:2013},
     ~\cite[p. 41]{Nedelec:2001} \,.

\subsubsection{Sobolev tangent vector fields on ${\mathbb S}^2$}
 For tangential vector fields
 we have the vectorial Sobolev space ${\bf H}^{s}_{tan}({\mathbb S}^2)$,
 which is the set of all $f\in {\bf  L}_{2,tan}({\mathbb S}^2)$ such that
\[
||{\bf f }||^2_{{\bf H}^s_{tan}({\mathbb S}^2)}=
\sum^{\infty}_{N=1}(1 + N(N + 1))^s
 \sum_{\ell}|f_{2,N\ell}|^2 + |f_{3,N\ell}|^2.
\]

For the scale of Sobolev spaces ${\bf H}_{tan}^{s}({\mathbb S}^2)$
there is a  Helmholtz--Hodge  decomposition (\cite[Theorem 4.1]{Cantor:1981})
\[{\bf H}_{tan}^{s}({\mathbb S}^2)=\nabla \left (H^{s+1}({\mathbb S}^2)/{\mathbb R} \right )
 \oplus {\rm ker}({\rm div})
  ={\bf H}^{s}_{tan,curl}({\mathbb S}^2)+{\bf H}^{s}_{tan,div}({\mathbb S}^2), \ s \geq 0 .
\]
Here we denote by ${\bf H}^{s}_{tan,div}({\mathbb S}^2)$
  and  ${\bf H}^{s}_{tan,curl}({\mathbb S}^2)$
   the divergence-free and curl-free subspaces of ${\bf H}^{s}_{tan}({\mathbb S}^2)$, respectively.

 Another words vector field tangent to the sphere
 ${\bf f} \in {\bf H}^{s}_{tan}({\mathbb S}^2)$ can be uniquely decomposed
 into surface curl-free and surface divergence-free
 components:
\[
{\bf f}=\nabla{u}+ \nabla^{\bot}{v}, \
 \int_{{\mathbb S}^2}{u} \, {\rm d}{\bm \xi}=
 \int_{{\mathbb S}^2}{v} \, {\rm d}{\bm \xi}=0,
\]
where functions $u,v \in H^{s+1}({\mathbb S}^2)/{\mathbb R}$.
We can
define its ${\bf H}^s$ norm, among other equivalent versions, as
\[
 ||{\bf f}||^2_{{\bf H}^s({\mathbb S}^2)}=||u||^2_{H^{s+1}({\mathbb S}^2)} +||v||^2_{H^{s+1}({\mathbb S}^2)}\, .
 \]
\subsection{Fourier multiplier  and spherical convolution operators}

\subsubsection{Fourier multiplier operators}
 Here we define Fourier multiplication operators.
\begin{definition}
The operator  $\Lambda: L_2({\mathbb S}^2) \to L_2({\mathbb S}^2)$
is called the Fourier  multiplier operator with corresponding
sequence of multipliers $\{ \lambda_N \}_{N=0}^{\infty}$
 if operator $\Lambda$ acts  on a function $u \in
L_2({\mathbb S}^2)$ by the formula
 \[
 \{\Lambda u \}({\bm \xi})\equiv \Lambda_{{\bm \xi}} u
 = \sum_{N=0}\lambda_{N}\sum_{\ell}u_{N\ell}Y_{N\ell}({\bm \xi}),
 \]

 where $u_{N\ell}$ denote the Fourier coefficients of $u$ with respect to
 the spherical harmonics,
 \[
 u({\bm \xi})=\sum_{N=0}\sum_{\ell}u_{N\ell}Y_{N\ell}({\bm \xi}).
 \]
\end{definition}
 The sequence of multipliers

 $\{ \lambda_N \}_{N=0}^{\infty}$
 gives complete information about properties of
 operator $\Lambda$, especially  the behavior and asymptotics  of multipliers at
 infinity. It is not hard to see that a multiplier operator on
 $L_2({\mathbb S}^2)$ 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 \cite{Samko:1983, Rubin:2003, DaiXu:2013}.

\subsubsection{Spherical convolution operators}
An important example of the multiplier operator will be a spherical
convolution operator.
\begin{definition}
 The spherical convolution
 $K *u$  of $K \in
 L_2(-1,1)$ with a function $u \in L_2({\mathbb S}^2)$ is
 defined as
 \[
 (K *u) ({\bm \xi})= \int_{{\mathbb S}^2}
 K({\bm \xi} \centerdot{\bm  \eta})u({{\bm  \eta}})
   \, {\rm d}{\bm  \eta}, \, {\bm \xi}\in S^2,
\]
${\rm d}{\bm \eta}$ is the
rotation invariant measure, normalized so that
 $\int_{\mathbb S^2}{\rm d}{\bm \eta}=4\pi$ ---
 the surface area of ${\mathbb S}^2$.
  We recall that  ${\bm \eta} \centerdot {\bm \xi}$ is the usual pointwise  inner
  product.
\end{definition}

By  the Funk�-Hecke formula in Theorem \ref{th:Funk} we have
the sequence of multipliers $\{ \lambda_N \}_{N=0}^{\infty}$
\[
 \{K * Y_{N \ell}\}
({\bm \xi})=2{\pi}
 Y_{N \ell}({\bm \xi})
\int_{-1}^{1} K(x) P_{N}(x) \,dx =\lambda_{N} Y_{N \ell}({\bm
\xi}).
\]

\subsubsection{Funk's inversion formula for the F--M transform}

In \cite{Funk:1913} Funk   showed that Funk--Minkowski-
  transform (\ref{SEMR:Funk--Minkowski-scalar})  is the Fourier multiplier
 operator with multiplicators
 $\lambda_{2j}=P_{2j}(0)$,
 \[
 \{ {\mathcal F} Y_{N \ell} \}({\bm \xi})=P_{2j}(0)Y_{2j,\ell}({\bm\xi}),
\]
 and  asymptotics $\lambda_{2j}=P_{2j}(0) \sim (2j+1)^{-1/2}$  if  $j \to \infty$
 (\cite{AbramowitzStegun:1972}).
 Hence any even function $f_{even} \in C^{\infty}({\mathbb S}^2)$
 can be reconstructed explicitly from
 its Funk--Minkowski transform by the formula
 \begin{align*}
  f_{even}({\bm \xi}) =\sum^{\infty}_{j=0} \sum_{\ell} f_{2j,\ell}
  Y_{2j,\ell}({\bm \xi})=
  \sum^{\infty}_{j=0} \sum_{\ell} \frac{( {\mathcal F} f_{even},
  Y_{2j,\ell})_{L_2(S^2)}}{P_{2j}(0)}
  Y_{2j,\ell}({\bm \xi})\, ,
 \end{align*}
where
\begin{align*}
 ( {\mathcal F} f_{even},  Y_{2j,\ell})_{L_2({\mathbb S}^2)}
 =P_{2j}(0)f_{2j,\ell} \, .
  \end{align*}
 The following mapping property of the Funk--Minkowski transform
 between Sobolev spaces was shown by R.\,S. Strichartz in
 ~\cite[Lemma 4.3]{Strichartz:1981}
 : operator
 \[
  {\mathcal F}:H_{even}^s({\mathbb S}^2) \to H_{even}^{s+1/2}({\mathbb S}^2), \ s\geq 0
 \]
  is continuous and bijective, see also ~\cite{GoodeyWeil:1992, QuellmalzHielscherLouis:2018}.

\subsubsection{The spherical convolution operator ${\mathcal S}$}
  Now consider the spherical convolution operator ${\mathcal S}$,
  which defined by formula  (\ref{SEMR:OperatorS}), we repeat it
     \begin{eqnarray*}
      \{ {\mathcal S}v \}({\bm  \xi}) \equiv
     {\mathcal S}_{{\bm  \xi}}v=
    \frac{1}{4\pi} \{ x^{-1} *u\} ({\bm \xi})
=   \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{v({\bm  \eta})}{{{\bm  \xi}} \centerdot{\bm  \eta}}
   \, {\rm d}{\bm  \eta}, \ {{\bm  \xi}}\in {\mathbb S}^2.
   \end{eqnarray*}

      The operator ${\mathcal S}$ does not
   exist as an absolutely convergent integral and
   should be understood in the principal value sense,
   see  ~\cite{Samko:2002, Rubin:2014},
   \[
   \{ {\mathcal S}v \}({{\bm  \xi}})=\lim_{\varepsilon \to 0}\frac{1}{4\pi}
   \int_{|{{\bm  \xi} } \centerdot{\bm  \eta}|> \varepsilon}
   \frac{v({\bm  \eta})}
   {{{\bm  \xi} } \centerdot{\bm  \eta}}
   \, {\rm d}{\bm  \eta}=p.v. \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{v({\bm  \eta})}{{{\bm  \xi}} \centerdot{\bm  \eta}}
   \, {\rm d} {\bm  \eta}.
   \]

  The operator ${\mathcal S}$ is  considered as  operator from
  $L_2({\mathbb S}^2)$ into $L_2({\mathbb S}^2)$ and
  can be regarded as the spherical analogue
  of the Hilbert transform, \cite{Rubin:2014}.
    Evidently, that for even spherical harmonics
  $ \{{\mathcal S}Y_{2j,\ell} \}({\bm  \xi})=0$,
  so we can  consider this operator  only
  on  the subspace of odd SHs, $L_{2,odd}({\mathbb S^2})$.

 \begin{proposition} [\cite {Rubin:2014, Kazantsev:2015}]
  The spherical analogue
  of the Hilbert transform  (\ref{SEMR:OperatorS})
 \[
 {\mathcal S}: L_{2,odd}({\mathbb S^2}) \to L_{2,odd}({\mathbb S^2})
 \]
 is a compact operator and a multiplier operator on $L_{2,odd}({\mathbb S^2})$ with
 corresponding sequence of Fourier-Laplace multipliers
 $ \Big \{\frac{1}{C^{(3/2)}_{N-1}(0)}=\frac{1}{NP_{N-1}(0)}, \ N=2j+1
 \Big \}_{j=0}^{\infty}$,
 \begin{eqnarray}
  \label{SEMR:SY}
    \{ {\mathcal S}Y_{N\ell} \}({\bm  \xi})
    &=\frac{1}{C^{(3/2)}_{N-1}(0)}Y_{N\ell}({\bm  \xi}) =\frac{1}{NP_{N-1}(0)}Y_{N\ell}({\bm  \xi}), \ N=2j+1 \
  \end{eqnarray}
and asymptotics
\begin{equation}
    \frac{1}{C^{(3/2)}_{2j}(0)}=\frac{1}{(2j+1)P_{2j}(0)}
    \sim \frac{1}{\sqrt{2j+1}} \ \text{if } \
    j \to \infty .
        \end{equation}
   \end{proposition}

 The operator ${\mathcal S}$, as well as the operator
  ${\mathcal F}$,
  \[
  {\mathcal S}: H_{odd}^s({\mathbb S}^2) \to H_{odd}^{s+1/2}({\mathbb
  S}^2), \ s \geq  0
 \]
  is continuous and bijective in the scale of Sobolev spaces
$H_{odd}^s({\mathbb S}^2)$, see  \cite[Proposition 3.2]{Rubin:2014}.

\subsubsection{Analytic family of  fractional integrals and  Funk--Minkowski transform}

 We can write the F--M operator (\ref {SEMR:Funk--Minkowski-scalar}) in the form
 of spherical convolution operator  as follows
\begin{align*}
 & \{ {\mathcal F}{u} \}(\bm \xi)=\frac{1}{2\pi}
  \int_{0}^{2\pi}u \Big ({\bf e}_{1}({\bm \xi})\cos \omega
 +{\bf e}_{2}({\bm \xi})\sin \omega \Big ) \,{\rm d}{\omega}
\\
&=
  \frac{1}{2\pi}\int_{-1}^{1}\delta(t)
  \int_{0}^{2\pi}u
  \Big({\bf e}_{1}({\bm \xi})\cos \omega \sqrt{1-t^2}
   +{\bf e}_{2}({\bm \xi})\sqrt{1-t^2}\sin \omega \Big )\,
    {\rm d}{\omega}\,{\rm d}{t}
 \\
 &= \frac{1}{2\pi}
  \int_{{\mathbb S}^2}\delta({\bm \xi} \centerdot{\bm \theta})
  u({\bm \theta}) \, {\rm d}{\bm \theta}=
\frac{1}{2\pi} \{\delta *u\} ({\bm \xi})\, ,
   \end{align*}
   where $\delta$ is the Dirac delta function.

The papers \cite{Louis:2016, QuellmalzHielscherLouis:2018}
give a definition
 of   the generalized Funk--Radon transform
 $S^{(j)}$ for $u\in C^{\infty}({\mathbb S}^2)$  by
  \[
   \{S^{(j)}u\}({\bm \xi}) =\frac{1}{2\pi}
  \int_{{\mathbb S}^2}\delta^{(j)}({\bm \xi} \centerdot{\bm \theta})
  u({\bm \theta}) \, {\rm d}{\bm \theta}, \ \
  j \in 0 \cup {\mathbb N}
  .
 \]
 Here use the notation from ~\cite{Louis:2016, QuellmalzHielscherLouis:2018}
  and $\delta^{(j)}$  denotes the $j$-th derivative of the Dirac delta
 function and  operator $S^{(0)}$ is the Funk--Minkowski
 transform ${\mathcal F}$.

 The spherical Hilbert type operator ${\mathcal S}$ in (\ref{SEMR:OperatorS}) as well as operators
 ${\mathcal S}^{(j)}$ are the members of analytic family
 of fractional integrals $\{ {\mathcal C}^{\lambda}, \widetilde{\mathcal C}^{\lambda} \}$ defined by \begin{align}
 \{{\mathcal C}^{\lambda} f \}({\bm \theta})& =
 \frac { \Gamma \left ( -\frac{\lambda}{2} \right )}
  {2\pi\Gamma \left (  \frac{1+\lambda}{2} \right )}
    \int_{{\mathbb S}^2}
  f({\bm \sigma})
  |{\bm \theta} \centerdot  {\bm \sigma} |^{\lambda}\,  d{\bm \sigma},
  \\
  \{\widetilde{\mathcal C}^{\lambda} f \}({\bm \theta}) &=
 \frac { \Gamma \left ( \frac{1-\lambda}{2} \right )}
  {2\pi \Gamma \left (1+ \frac{\lambda}{2} \right )}
  \int_{{\mathbb S}^2}
 f({\bm \sigma}) |{\bm \theta} \centerdot  {\bm \sigma} |^{\lambda}
  {\rm sgn}({\bm \theta} \centerdot  {\bm \sigma})\,  d{\bm \sigma},
   \end{align}
   see ~\cite{Rubin2:1998, Rubin:2014}.
 The operators  ${\mathcal C}^{\lambda}$ and $\widetilde{{\mathcal C}}^{\lambda}$
 are  called the  $\lambda$-cosine transforms
 of $f$ with even and odd kernel, respectively. If $f \in C^{\infty}({\mathbb S}^2)$, they extend analytically to all $\lambda \in {\mathbb C}$
 with the only poles $\lambda =0,2,4,...$ for ${\mathcal C}^{\lambda}$
 and $\lambda=1,3,5,...$ for $\widetilde{C}^{\lambda}$ .

The limit case $\lambda =-1$ corresponds to the Funk--Minkowski
  transform ${\mathcal F}$ and Hilbert spherical transform ${\mathcal S}$,
  ~\cite[Lemma 3.4.]{Rubin:2014}
  \begin{align*}{\mathcal F} \sim
\{ {\mathcal C}^{-1} f \} ({\bm \theta} )& =\frac{1} {2\sqrt{\pi}}
 \int_{{\mathbb S}^2}
 f({\bm \eta}) \delta({\bm \theta} \centerdot  {\bm \eta} )\, d{\bm \eta},
 \\
 {\mathcal S} \sim
 \{\widetilde{\mathcal C}^{-1} f \}({\bm \theta}) &= \frac {1}{2 \pi^{3/2}}
 \int_{{\mathbb S}^2}\frac{f({\bm \eta})}{{\bm \theta} \centerdot  {\bm \eta} }\,
   d{\bm \eta}.
  \end{align*}
The integral operator in the inverse formula
(\ref{SEMR:Semyanistyi}) by  V. Semyanisty also belongs to this family with
$\lambda=-2$,
\begin{align*}
 \{ {\mathcal C}^{-2} f \}({\bm \theta})& = \frac {-1}{4\pi^{3/2}}
 \int_{{\mathbb S}^2}
 f({\bm \eta}) \frac{1}{({\bm \theta} \centerdot  {\bm \eta})^2 }
   d{\bm \eta}.
  \end{align*}
The corresponding operator $\widetilde{\mathcal C }^{-2}$ for
 ${\mathcal C}^{-2}$ is the generalized
Funk--Radon transform
\begin{align*}S^{(1)} \sim
 \{ \widetilde{\mathcal C }^{-2} f \}({\bm \theta}) &= \frac {-1}{4 \sqrt{\pi}}
 \int_{{\mathbb S}^2}f({\bm \eta})\delta^{\prime}({\bm \theta} \centerdot  {\bm \eta} )\,
 d{\bm \eta}.
 \end{align*}

If for an analytic continuation we use formulas,
  see for example \cite{GelfandShilov:1964},
\begin{align}
 \frac{ |x|^{\lambda}}
 {\Gamma \left (\frac{1+\lambda}{2}\right )}\Big |_{\lambda=-(2m+1)}
 &=\frac{(-1)^m m!}{(2m)!} \delta^{(2m)}(x), \ m=0,1,2,... \ ,
 \\
 \frac{|x|^{\lambda} {\rm sgn}(x)}
 {\Gamma \left (1+ \frac{\lambda}{2} \right)}\Big |_{\lambda=-2m}&=
 \frac{(-1)^m (m-1)!}{(2m-1)!}\delta ^{(2m-1)}(x)
  , \ m=1,2,3,... \ ,
   \end{align}

   then as the  result,  the following connection between  ${\mathcal S}^{(2m)}$,
       ${\mathcal S}^{(2m+1)}$ and  analytic family $\{ {\mathcal C}^{\lambda}, \tilde{\mathcal C}^{\lambda} \}$
   take place
  \begin{align}{\mathcal S}^{(2m)} \sim
  \{{\mathcal C}^{-2m-1} f \}({\bm \theta})& =
  \frac { (-1)^m \sqrt{\pi}}
  {2\pi 2^{2m}}
    \int_{{\mathbb S}^2}
  f({\bm \sigma})
  \delta^{(2m)}(
  {\bm \theta} \centerdot  {\bm \sigma})\,  d{\bm \sigma}, \ m=0,1,2,... \,  ,
  \\
   {\mathcal S}^{(2m+1)} \sim  \{ \widetilde{\mathcal C}^{-2m} f \}({\bm \theta}) &=
 \frac {(-1)^m \sqrt{\pi}}
  {2\pi2^{2m-1}}
  \int_{{\mathbb S}^2}
 f({\bm \sigma})\delta ^{(2m-1)}({\bm \theta} \centerdot  {\bm \sigma})
    \,  d{\bm \sigma}, \ m=1,2,3,... \, .
   \end{align}
 According to the general theory of
  analytic family $\{
 {\mathcal C}^{\lambda},  \widetilde{\mathcal C}^{\lambda} \}$ on the sphere ${\mathbb S}^2$,
 we can  find  inverse operators of
 ${\mathcal C}^{\lambda}$, $\widetilde{\mathcal C}^{\lambda}$
  by the formulas (see \cite[Proposition 3.1]{Rubin:2014})
\[  {\mathcal C}^{\lambda} {\mathcal C}^{-\lambda-3}f
={\mathcal C}^{-\lambda-3}{\mathcal C}^{\lambda}f=f, \ \text{where} \
\lambda, -\lambda-3 \neq 0,2,4,... \ f\in C^{\infty}_{even}({\mathbb S}^2),
\]
and
\[  \widetilde{\mathcal C}^{\lambda}\widetilde {\mathcal C}^{-\lambda-3}f
=\widetilde{\mathcal C}^{-\lambda-3}\widetilde{\mathcal C}^{\lambda}f=f, \
\text {where} \ \lambda, -\lambda-3 \neq 1,3,5,... \ ,\  f\in C^{\infty}_{odd}({\mathbb S}^2).
\]

In the particular case  $\lambda =-1$  we have
${\mathcal F}^{-1} \sim  \left ({\mathcal C}^{-1} \right )^{-1}={\mathcal C}^{-2}$
and it is appropriate to formula (\ref{SEMR:Semyanistyi}) by V. Semyanisty,
see also \cite [Corollary 3.3]{Rubin:2008}.
If we  apply (formally) the integration by parts formula
(\ref{SEMR:intgrad1}) to  (\ref{SEMR:inverseEVEN}), then we  get
\begin{align*}
 \frac{ {\bm \theta} \, \centerdot}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   \{ \nabla{\mathcal F} f \}(\bm \eta)}
   {{\bm \theta} \centerdot{\bm \eta}}
   {\rm d}{\bm \eta}&=-
 \frac{ {\bm \theta} \, \centerdot}{4\pi} \int_{{\mathbb S}^2}
   \nabla \frac{1}
      {{\bm \theta} \centerdot{\bm \eta}}
      \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}
   +\frac{ {\bm \theta} \, \centerdot}{2\pi}
   \int_{{\mathbb S}^2}
   \frac{{\bm \eta}
   \{{\mathcal F} f \}(\bm \eta)}
   {{\bm \eta} \centerdot{\bm \theta}}
   {\rm d}{\bm \eta}
\\&=
 \frac{ {\bm \theta} \, \centerdot}{4\pi} \int_{{\mathbb S}^2}
   \frac{{\bm \theta}-({\bm \theta} \centerdot{\bm \eta}){\bm \eta}}
      {({\bm \theta} \centerdot{\bm \eta})^2}
      \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}
   +\frac{ 1}{2\pi}
   \int_{{\mathbb S}^2}
     \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}
 \\
 &=
 \frac{ 1}{4\pi} \int_{{\mathbb S}^2}
   \frac{ 1-({\bm \theta} \centerdot{\bm \eta})^2}
      {({\bm \theta} \centerdot{\bm \eta})^2}
      \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}
   +\frac{ 1}{2\pi}
   \int_{{\mathbb S}^2}
     \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}
 \\
&=
 \frac{ 1}{4\pi} \int_{{\mathbb S}^2}
   \frac{ 1}
      {({\bm \theta} \centerdot{\bm \eta})^2}
      \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}
   +\frac{ 1}{4\pi}
   \int_{{\mathbb S}^2}
     \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}.
   \end{align*}
  Thus, this  formal calculations show that formula (\ref
 {SEMR:inverseEVEN}) corresponds to formula (\ref{SEMR:Semyanistyi}) and
  serves  as  its  regularization
\begin{align*}
-\frac{ 1}{4\pi} \int_{{\mathbb S}^2}
   \frac{ 1}
      {({\bm \theta} \centerdot{\bm \eta})^2}
      \{ {\mathcal F} f \}(\bm \eta)
   {\rm d}{\bm \eta}=
   \frac{1}{4\pi}  \int_{{\mathbb S}^2}
  \{ {\mathcal F}f \}(\bm \eta) {\rm d}{\bm \eta}
   - p.v.
    \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   {\bm \theta} \, \centerdot \{ \nabla{\mathcal F} f \}(\bm \eta)}
   {{\bm \theta} \centerdot{\bm \eta}}
   {\rm d}{\bm \eta}   .
\end{align*}

\section{Proofs}
In this section we present the proofs of Theorems \ref{th:main1},
 \ref{th:main2}, which will be based on Lemmas
 \ref{l:F} and \ref{l:S}.
In vector case, as in the scalar case,
 the vectorial   Funk--Minkowski transform
 $
 {\mathcal F}: {\bf L}_{2,even}({\mathbb S^2}) \to {\bf L}_{2,even}({\mathbb S^2})
 $
 and
  vectorial    Hilbert type  spherical transform
 $
 {\mathcal S}: {\bf L}_{2,odd}({\mathbb S^2}) \to {\bf L}_{2,odd}({\mathbb S^2})
 $
 are multiplier operators and relevant mapping properties
 between Sobolev spaces are valid.
 The accurate formulations  are given below.

 \begin{lemma}\label{l:F}
  Vectorial   Funk--Minkowski transform
 $
 {\mathcal F}: {\bf L}_{2,even}({\mathbb S^2}) \to {\bf L}_{2,even}({\mathbb S^2})
 $
  is  a multiplier operator
  \begin{align}\label{SEMR:Fhi}
  {\mathcal F}{\bf h}^{(i)}_{N{\ell}}&=P_{N-1}(0) {\bf h}^{(i)}_{N\ell},  \  N=2j+1,
  \\\label{SEMR:Fy}
  {\mathcal F}{\bf y}^{(3)}_{N{\ell}}&=  P_{N}(0){\bf y}^{(3)}_{N\ell}, \ \ N=2j,
  \\\label{SEMR:Fhe}
  {\mathcal F}{\bf h}^{(e)}_{N\ell} &= P_{N+1}(0) {\bf h}^{(e)}_{N\ell}, \ N=2j+1,
  \end{align}
     where ${\bf h}^{(i)}_{N{\ell}}, {\bf y}^{(3)}_{N{\ell}}, {\bf h}^{(e)}_{N\ell}$
  are  pure--orbit vector spherical harmonics
 (\ref{SEMR:he0})--(\ref{SEMR:hi}).
   We have that in the scale of Sobolev spaces operator
   ${\mathcal F}:{\bf H}_{even}^s({\mathbb S}^2) \to {\bf H}_{even}^{s+1/2}({\mathbb S}^2)$, $s\geq 0$ is continuous and bijective.

  If we choose as a basis  pure--spin vector spherical harmonics,
  then following formulas take place
 \begin{align}\label{SEMR:Fy1}
{\mathcal F}{\bf y}^{(1)}_{N\ell}
&=P_{N-1}(0) {\frac{{\bf y}^{(2)}_{N\ell} }{N+1}}\,
 , \ N=2j+1,
  \\\label{SEMR:Fy2}
{\mathcal F}{\bf y}^{(2)}_{N\ell}
 &=  P_{N-1}(0)\Big ( N{\bf y}^{(1)}_{N\ell} +
 \frac{
  {\bf y}^{(2)}_{N\ell} }{N+1} \Big  ), \ N=2j+1
 \, ,
\\\label{SEMR:Fy3}
    {\mathcal F} {\bf y}^{(3)}_{N{\ell}} &=
    P_{N}(0) {\bf   y}^{(3)}_{N\ell} \, , \  N=2j
    \, .
 \end{align}
    \end{lemma}

Similar statements are valid for the operator
${\mathcal S}$
 \begin{lemma}\label{l:S}
 Vectorial spherical convolution   transform
  $
  {\mathcal S}: {\bf L}_{2,odd}({\mathbb S^2}) \to {\bf L}_{2,odd}({\mathbb S^2})
  $
  is  a multiplier operator
  \begin{align}
  \label{SEMR:She0}
  &{\mathcal S} {\bf h}^{(e)}_{00}
  = {\bf h}^{(e)}_{00},
  \\
  \label{SEMR:She}
  &{\mathcal S} {\bf h}^{(e)}_{N \ell}
  =
  \frac{{\bf h}^{(e)}_{N \ell}}{(N+1)P_{N}(0)} ,  \ N=2j,
  \\
  \label{SEMR:Syi}
   &{\mathcal S} {\bf h}^{(i)}_{N \ell}
    =-\frac{1}{(N+1)P_{N}(0)}\frac{N+1}{N}{\bf h}^{(i)}_{N\ell}, \ N=2j,
  \\
  \label{SEMR:Shy3}
  &{\mathcal S}{\bf y}^{(3)}_{N\ell}=\frac{{\bf y}^{(3)}_{N\ell}}{NP_{N-1}(0)}
  , \ N=2j+1  \, ,
  \end{align}
  where ${\bf h}^{(i)}_{N{\ell}}, {\bf y}^{(3)}_{N{\ell}}, {\bf h}^{(e)}_{N\ell}$
  are  pure--orbit vector spherical harmonics
 (\ref{SEMR:he0})--(\ref{SEMR:hi}).
  In the scale of Sobolev spaces operator
  ${\mathcal S}:{\bf H}_{odd}^s({\mathbb S}^2) \to {\bf H}_{odd}^{s+1/2}({\mathbb S}^2),
  \ s\geq 0$  is continuous and bijective.

The images of  pure--spin spherical harmonics under the action of operator $ {\mathcal S}$
are listed below
   \begin{align}\label{SEMR:Sy0}
  {\mathcal S} {\bf y}^{(1)}_{00} &={\bf y}^{(1)}_{00},
    \\
    \label{SEMR:Sy1}
  {\mathcal S} {\bf y}^{(1)}_{N\ell} &=
 \frac{-1}{P_{N}(0)}\frac{{\bf y}^{(2)}_{N \ell}
 }
 {{N(N+1)}},  \ N=2j,
\\ \label{SEMR:Sy2}
  {\mathcal S} {\bf y}^{(2)}_{N\ell}&=
    \frac{-1}{P_{N}(0)} \Big (
     {\bf y}^{(1)}_{N \ell}
    +\frac{ {\bf y}^{(2)}_{N \ell}}  {N(N+1)}
    \Big ), \    N=2j,
    \\ \label{SEMR:Sy3}
  {\mathcal S} {\bf y}^{(3)}_{N\ell}
   &= \frac{1}{P_{N-1}(0)} \frac{{\bf y}^{(3)}_{N\ell}}{N}
  , \ N=2j+1  \, .
    \end{align}
  \end{lemma}

    {\bf Proof} of  Lemma \ref{l:F}. The
     pure--orbit VSHs are expressed through scalar spherical
 harmonics with the help of three term relations, see for example
 ~\cite{DerevtsovKazantsev:2009},
 \begin{align}\label{SEMR:ThreeTerm1}
  &  {\bf h}^{(i)}_{N \ell}=N{\bf y}^{(1)}_{N \ell}({\bm \xi})+{\bf y}^{(2)}_{N \ell}({\bm \xi})
  \\ \nonumber
  &=
  \alpha_1 Y_{N-1,\ell-1}({\bm \xi})
 \left (
 \begin{array}{c}
   1 \\
   {\rm i} \\
   0 \\
 \end{array}
 \right )
 +\beta_1 Y_{N-1, \ell}({\bm \xi})
 \left (
 \begin{array}{c}
   0 \\
   0 \\
   1 \\
 \end{array}
 \right )
 +\gamma_1 Y_{N-1,\ell+1}({\bm \xi})
 \left (
 \begin{array}{r}
    1 \\
   -{\rm i} \\
    0 \\
 \end{array}
 \right ),
\\
\label{SEMR:ThreeTerm1}
 & {\bf h}^{(e)}_{N \ell}=-(N+1){\bf y}^{(1)}_{N \ell}({\bm \xi})+{\bf y}^{(2)}_{N \ell}({\bm \xi})
 \\ \nonumber
 &=
 \alpha_2 Y_{N+1,\ell-1}({\bm \xi})
\left (
 \begin{array}{c}
   1 \\
   {\rm i} \\
   0 \\
 \end{array}
 \right )
 +\beta_2 Y_{N+1, \ell}({\bm \xi})
 \left (
 \begin{array}{c}
   0 \\
   0 \\
   1 \\
 \end{array}
 \right )
 +\gamma_2 Y_{N+1,\ell+1}({\bm \xi})
 \left (
 \begin{array}{r}
    1 \\
   -{\rm i} \\
     0 \\
 \end{array}
 \right ),
\\%3
& {\bf y}^{(3)}_{N \ell}({\bm \xi})=
 \alpha_{3} Y_{N,\ell-1}({\bm \xi})
\left (
 \begin{array}{c}
   1 \\
   {\rm i} \\
   0 \\
 \end{array}
 \right )
 +\beta_{3} Y_{N \ell}({\bm \xi})
 \left (
 \begin{array}{c}
   0 \\
   0 \\
   1 \\
 \end{array}
 \right )
 +\gamma_{3} Y_{N,\ell+1}({\bm \xi})
 \left (
 \begin{array}{r}
    1 \\
   -{\rm i} \\
    0 \\
 \end{array}
 \right ),
\end{align}
 where $\alpha_{i}, \beta_{i}, \gamma_{i} \ (i=1,2,3)$ some
 coefficients. The values of this coefficients  are unimportant
 here, but their accurate expressions can be found in
 ~\cite{DerevtsovKazantsev:2009}.

By applying the operator ${\mathcal F}$ to these three  term
relations  we immediately obtain:
for  $N=2j+1$
\begin{align}\label{SEMR:Fie}
 {\mathcal F} {\bf h}^{(i)}_{N \ell}= P_{N-1}(0){\bf h}^{(i)}_{N \ell},
\ \
{\mathcal F} {\bf h}^{(e)}_{N \ell}=P_{N+1}(0){\bf h}^{(e)}_{N \ell}
\end{align}
and for  $N=2j$
\[
  {\mathcal F}{\bf y}^{(3)}_{N \ell}=P_{N}(0){\bf y}^{(3)}_{N \ell}\, .
\]

Because the multipliers  have asymptotics $P_{2j}(0) \sim \frac{1}{\sqrt{2j+1}}$
 as $j$ goes to infinity,  we
have that ${\mathcal F}:{\bf H}^{s}_{even}({\mathbb S}^2) \to {\bf H}^{s+1/2}_{even}({\mathbb S}^2)$
 is a continuous operator  in the scale of Sobolev spaces,
as in the scalar case.

The two equations (\ref{SEMR:Fie})  can  be written  as
\begin{align*}
 \left \{
 \begin{array}{lll}
   N{\mathcal F}{\bf y}^{(1)}_{N \ell}+{\mathcal F}{\bf y}^{(2)}_{N
 \ell}&=
  NP_{N-1}(0){\bf y}^{(1)}_{N \ell}+P_{N-1}(0){\bf y}^{(2)}_{N\ell}
\\
 -(N+1){\mathcal F}{\bf y}^{(1)}_{N \ell}+{\mathcal F}{\bf y}^{(2)}_{N \ell}
 &=-(N+1)P_{N+1}(0){\bf y}^{(1)}_{N \ell}+P_{N+1}(0){\bf y}^{(2)}_{N \ell} \, .
\\
 \end{array}
\right .
\end{align*}

We need to  solve this system  with respect to
 ${\mathcal  F}{\bf y}^{(1)}_{N \ell}$ and
 ${\mathcal F}{\bf y}^{(2)}_{N \ell}$.
 Subtracting the second from the first equation, we obtain
\begin{align*}
&(2N+1){\mathcal F}{\bf y}^{(1)}_{N \ell}=
  (NP_{N-1}(0)+(N+1)P_{N+1}(0)){\bf y}^{(1)}_{N\ell}
  +(P_{N-1}(0)- P_{N+1}(0)) {\bf y}^{(2)}_{N \ell}
\\
&=P_{N-1}(0)
  \left ( N-(N+1)\frac{N}{N+1} \right ){\bf y}^{(1)}_{N\ell}
  +P_{N-1}(0) \left (1   +  \frac{N}{N+1} \right ) {\bf y}^{(2)}_{N \ell}
  \\
  &=P_{N-1}(0)
  \frac{2N+1}{N+1}{\bf y}^{(2)}_{N \ell}.
\end{align*}
 Here  we used the formula (\ref{SEMR:PN}),  $(N+1)P_{N+1}(0)=-N P_{N-1}(0)$,
 thus   we have
\[{\mathcal F}{\bf y}^{(1)}_{N \ell}=P_{N-1}(0)\frac{{\bf y}^{(2)}_{N \ell}}{N+1}, \  \
{\mathcal F}{\bf y}^{(2)}_{N\ell}
 =  P_{N-1}(0)\Big ( N{\bf y}^{(1)}_{N\ell} +
 \frac{
  {\bf y}^{(2)}_{N\ell} }{N+1} \Big ), \ N=2j+1.
\]

{\bf Proof} of  Lemma \ref{l:S}.
By applying  operator ${\mathcal S}$ to three  term relations,
as well as in the previous case, we obtain:
for  $N=2j$
\begin{align*}
    {\mathcal S} {\bf h}^{(i)}_{N \ell}
   & = \frac{{\bf h}^{(i)}_{N \ell}}{(N-1)P_{N-2}(0)}
   =
   -\frac{1}{(N+1)P_{N}(0)}\frac{N+1}{N}{\bf h}^{(i)}_{N\ell},
   \\
   {\mathcal S} {\bf h}^{(e)}_{N \ell}
  &=
  \frac{{\bf h}^{(e)}_{N \ell}}{(N+1)P_{N}(0)}
  \end{align*}
and for  $N=2j+1$
\[
{\mathcal S}{\bf y}^{(3)}_{N \ell}=\frac{{\bf y}^{(3)}_{N \ell}}{NP_{N-1}(0)}\, .
\]
In the first formula we used equality
$(N-1)P_{N-2}(0)=-NP_{N}(0)$.

Continuity of the operator ${\mathcal S}$ in the scale
${\bf H}_{odd}^s({\mathbb S}^2)$
 follows from  asymptotic behavior
$\frac{1}{(2j+1)P_{2j}(0)} \sim \frac{1}{\sqrt{2j+1}}$ if  $j \to \infty$.

The first two equations above are equivalent to the system
\begin{align*}
 \left \{
 \begin{array}{lll}
 N{\mathcal S}{\bf y}^{(1)}_{N \ell}+{\mathcal S}{\bf y}^{(2)}_{N \ell}
  &=
  -\frac{{\bf y}^{(1)}_{N \ell}}{P_{N}(0)}-\frac{{\bf y}^{(2)}_{N \ell}}{N P_{N}(0)}
 \\
  -(N+1){\mathcal S}{\bf y}^{(1)}_{N \ell}+{\mathcal S}{\bf y}^{(2)}_{N \ell}
 &=
  -\frac{{\bf y}^{(1)}_{N \ell}}{P_{N}(0)}+ \frac{{\bf y}^{(2)}_{N  \ell}}{(N+1)P_{N}(0)}\,.
  \\
 \end{array}
\right .
\end{align*}

 Solving this system, we obtain the desired
\begin{align*}
  {\mathcal S}{\bf y}^{(1)}_{N \ell}=
         \frac{-{\bf y}^{(2)}_{N \ell}}{P_{N}(0)N(N+1)} ,
\ \
 {\mathcal S}{\bf y}^{(2)}_{N \ell}=
    \frac{-1}{P_{N}(0)}
   \Big  ( {\bf y}^{(1)}_{N \ell}
       + \frac{{\bf y}^{(2)}_{N \ell}}{N(N+1)} \Big  )  .
\end{align*}
\subsection{Proof Theorem \ref{th:main1}}

We recall some of the basic properties that are implied in our proof.
The Funk--Minkowski transform is even, $\{ {\mathcal F}f\}(-{\bm
\xi})=\{ {\mathcal F}f\}({\bm \xi})$, and ${\mathcal F}
f_{odd} =0$,
 but spherical transform  ${\mathcal S}$ is odd,
   $\{{\mathcal S} f\}(-{\bm \xi})=- \{ {\mathcal S} f\}({\bm
   \xi})$,
and ${\mathcal S} f_{even} =0$. It is obviously that the surface
gradient  $\nabla$, scalar (dot) product ${\bm \eta} \, \centerdot
$ and vector (cross) product ${\bm \eta} \times $ change the
parity.
 We also recall the  parity rules for scalar and vector spherical harmonics :
  $  Y_{N\ell}(-{\bm \xi})=(-1)^NY_{N\ell}({\bm \xi})$,
  ${\bf y}^{(1)}_{N\ell}(-{\bm \xi})= (-1)^{N+1}{\bf y}^{(1)}_{N\ell}({\bm \xi})$,
  ${\bf y}^{(2)}_{N\ell}(-{\bm \xi})= (-1)^{N+1}{\bf y}^{(2)}_{N\ell}({\bm \xi})$,
  ${\bf y}^{(3)}_{N\ell}(-{\bm \xi})= (-1)^{N}{\bf y}^{(3)}_{N\ell}({\bm \xi})$.

 Now we can proceed to our
formula (\ref{SEMR:representation}) and without loss of
generality, we assume that
 $f({\bm \theta}) \in H^{1}({\mathbb S}^2)/{\mathbb R}$, then we have
\begin{align*}
 f({\bm \theta})&=
        \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
    ({\bm \eta} + {\bm \theta}) \, \centerdot
    \Big \{
    \Big [ {\mathcal F}, \nabla   \Big ] f  \Big   \}   (\bm \eta)
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}
\\ &=
        \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
    {\bm \eta}     \centerdot \Big \{  \Big [ {\mathcal F}, \nabla   \Big ]
 f \Big \}
    (\bm   \eta)+
   {\bm \theta} \, \centerdot  \Big \{ \Big [ {\mathcal F}, \nabla   \Big ]
   f \Big \}(\bm \eta)
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}
   \\
  &=  \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
    {\bm \eta}   \centerdot  {\mathcal F}_{\bm   \eta} \nabla  f -
\overbrace{ {\bm \eta}   \centerdot  \nabla{\mathcal F}_{\bm   \eta} f }^{=0}
    -
   {\bm \theta}  \centerdot \nabla{\mathcal F}_{\bm   \eta} f
     }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}+
       \frac{ {\bm \theta}  \centerdot}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   \overbrace{  {\mathcal F}_{\bm   \eta} \nabla  f}^{eveen}
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}
      \\
 &=       \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
    {\bm \eta}   \centerdot {\mathcal F}_{\bm   \eta} \nabla  f-
   {\bm \theta} \centerdot  \nabla {\mathcal F}_{\bm   \eta} f
      }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}
   = {\mathcal S}_{\bm \theta} {\bm \eta}   \centerdot {\mathcal F}_{\bm \eta} \nabla  f -
    {\mathcal S}_{\bm \theta}{\bm \theta}  \centerdot \nabla {\mathcal F}_{\bm \eta} f .
   \end{align*}

   It is clear that
   ${\rm ker} ({\mathcal S}_{\bm \theta} {\bm \eta}
    \centerdot {\mathcal F}_{\bm \eta} \nabla)=H^{s}_{even}({\mathbb S }^2)$
   and
     ${\rm ker} ({{\bm \theta}  \centerdot}\,
   {\mathcal S}_{\bm \theta} \nabla{\mathcal F}_{\bm \eta})
 =H^{s}_{odd}({\mathbb S }^2)$.
 From the  Lemmas  \ref{l:F},  \ref{l:S} we have
  ${\mathcal S} {\bm \eta}  \centerdot
   {\mathcal F}_{\bm \eta}  \nabla :H^{s}({\mathbb S }^2)\to H^{s}({\mathbb S }^2)$
   if $s\geq 1$, that looks on the diagram
\[
   \begin{CD} H^{s}({\mathbb S }^2)  @>  \nabla     >>
{\bf H}^{s-1}_{tan}({\mathbb S }^2)
                         @>  {\bm \eta }  \centerdot {\mathcal F}_{\bm \eta}          >>
           H^{s-1/2}({\mathbb S }^2)  @>  {\mathcal S}       >>
             H^{s}({\mathbb S }^2).
   \\
   \end{CD}
\]
Similarly, $ {\bm \theta} \centerdot
  {\mathcal S}_{\bm \theta} \nabla {\mathcal F}:
  H^{s}({\mathbb S }^2)  \to   H^{s}({\mathbb S }^2)$ if $s\geq 1/2$,
  which is also confirmed by the diagram
  \[
  \begin{CD} H^{s}({\mathbb S }^2)  @>  {\mathcal F}      >>
  H^{s+1/2}({\mathbb S }^2)          @>   \nabla        >>
 {\bf H}^{s-1/2}_{tan}({\mathbb S }^2)
           @>  {\bm \theta}  \centerdot {\mathcal S}_{\bm \theta}       >>
     H^{s}({\mathbb S }^2).
   \\
   \end{CD}
 \]

 For further calculations we take specifically $f={Y}_{N\ell}$, then
 from (\ref{SEMR:Fy2}) and (\ref{SEMR:Fy3}) we have
 \begin{align*} \text{for} \ N=2j &:
 {\mathcal F}_{\bm \eta} \nabla {Y}_{N\ell}= 0,
 \ \
  \nabla {\mathcal F}_{\bm \eta} {Y}_{N\ell}
 =  P_{N}(0) {\bf y}^{(2)}_{N\ell} ({\bm \eta}),
  \\ \text{for} \ N=2j+1 &:
  {\mathcal F}_{\bm \eta} \nabla {Y}_{N\ell}
 =  P_{N-1}(0)\Big ( N{\bf y}^{(1)}_{N\ell}({\bm \eta}) +
 \frac{
  {\bf y}^{(2)}_{N\ell} ({\bm \eta})}{N+1}
 \Big  ), \ \
  \nabla
 {\mathcal F}_{\bm \eta} {Y}_{N\ell}
 =0.
 \end{align*}
Consequently
\begin{align*}
{\bm \eta}   \centerdot  {\mathcal F}_{{\bm \eta}} \nabla  Y_{N\ell} -
   {\bm \theta} \centerdot \nabla{\mathcal F}_{\bm \eta} Y_{N\ell}
 = \left \{
\begin{array}{ll}
   -P_{N}(0)
   {\bm \theta} \centerdot  {\bf y}^{(2)}_{N\ell} ({\bm \eta}), & N=2j \\
   P_{N-1}(0) N{Y}_{N\ell}({\bm \eta}),
  & N=2j+1
  \\
\end{array}
\right.
\end{align*}
and finally,   using formulas (\ref{SEMR:SY}) and
 (\ref{SEMR:Sy2}), we get
\begin{align*}
       \frac{1}{4\pi} \int_{{\mathbb S}^2}
   \frac{
   {\bm \eta}   \centerdot  {\mathcal F}_{{\bm \eta}} \nabla  Y_{N\ell} -
   {\bm \theta} \, \centerdot \nabla{\mathcal F}_{\bm \eta} Y_{N\ell}
         }
   {{\bm \eta} \centerdot{\bm \theta}}\,
   {\rm d}{\bm \eta}=
   \left \{
\begin{array}{ll}
   {Y}_{N\ell}({\bm \theta}), & N=2j \\
   {Y}_{N\ell}({\bm \theta}),
  & N=2j+1 \,  .
  \\
\end{array}
\right.
\end{align*}
So we proved that, if
    $s\geq 1$ then
operator
${\mathcal S}_{\bm \theta} {\bm \eta}   \centerdot {\mathcal F}_{\bm \eta} \nabla -
    {\mathcal S}_{\bm \theta}{\bm \theta}  \centerdot \nabla {\mathcal F}_{\bm \eta}
    :H^{s}({\mathbb S }^2) \to H^{s}({\mathbb S }^2)$
    is identical operator.

\subsection{Proof Theorem \ref{th:main2}}
  We have already mentioned two approaches to the solution of
  Helmholtz--Hodge  decomposition problem for
  ${\bf f} \in {\bf L}_{2, tan}({\mathbb S}^2)$.
 Now we proof  formulas (\ref{SEMR:u}) and
(\ref{SEMR:v}) in Theorem  \ref{th:main2}
 for the velocity potential  $u$ and stream functions $v$ of  the Helmholtz--Hodge
  decomposition (\ref{SEMR:Helmholtz--Hodge}),
\begin{align*}
u({\bm \theta})&=
   {\mathcal S }_{\bm \theta} {\bm \eta}
 \centerdot {\mathcal F}_{\bm \eta} {\bf f}
 -
 {\mathcal F}_{\bm \theta} {\bm \eta} \centerdot {\mathcal S}_{\bm \eta}{\bf f},
\\ v({\bm \theta})&= {\bm \theta} \centerdot
 {\mathcal S }_{\bm \theta} {\bm \eta} \times {\mathcal F}_{\bm \eta} {\bf f}
 - {\bm \theta} \centerdot
   {\mathcal F}_{\bm \theta}{ \bm \eta} \times {\mathcal S}_{\bm \eta} {\bf f }.
  \end{align*}

  For the proof it suffices to verify these formulas on the basis elements
 $f={\bf y}^{(2)}_{N\ell}$ and
$f={\bf y}^{(3)}_{N\ell}$. Applying the scalar and cross products
to the the formulas (\ref{SEMR:Fy2})--(\ref{SEMR:Fy3}) and
(\ref{SEMR:Sy2})--(\ref{SEMR:Sy3}), we obtain
\begin{align*}
\text{for} \  N=2j &:
 {\bm \eta} \centerdot  {\mathcal F}_{\bm \eta} {\bf y}^{(3)}_{N{\ell}}= 0,
 \ {\bm \eta} \times {\mathcal F}_{\bm \eta} {\bf y}^{(3)}_{N{\ell}}=
   - P_{N}(0) {\bf   y}^{(2)}_{N\ell}({\bm \eta}),
 \\
 \text{for} \   N=2j+1 &:
 {\bm \eta} \centerdot {\mathcal F}_{\bm \eta}{\bf y}^{(2)}_{N\ell}
 =  P_{N-1}(0) N {Y}_{N\ell}({\bm \eta}), \
 {\bm \eta} \times {\mathcal F}_{\bm \eta}{\bf y}^{(2)}_{N\ell}
 =  P_{N-1}(0)\frac{
  {\bf y}^{(3)}_{N\ell} ({\bm \eta})}{N+1},
       \end{align*}

\begin{align*}
 \text{for} \    N=2j &:
        {\bm \eta} \centerdot
{\mathcal S}_{\bm \eta} {\bf y}^{(2)}_{N\ell}=
    \frac{-1}{P_{N}(0)}{Y}_{N \ell}({\bm \eta}), \ \
      {\bm \eta} \times
 {\mathcal S}_{\bm \eta} {\bf y}^{(2)}_{N\ell}=
    \frac{-1}{P_{N}(0)} \frac{ {\bf y}^{(3)}_{N \ell}({\bm \eta})}  {N(N+1)},
    \\
    \text{for} \  N=2j+1&:
   {\bm \eta} \centerdot {\mathcal S}_{\bm \eta}
   {\bf y}^{(3)}_{N\ell}=0, \
   {\bm \eta} \times {\mathcal S}_{\bm \eta} {\bf y}^{(3)}_{N\ell}
   = \frac{-1}{P_{N-1}(0)} \frac{{\bf y}^{(2)}_{N\ell}({\bm \eta})}{N} .
 \end{align*}

Based on the above, we get
 \begin{align*}
  {\mathcal S }_{\bm \theta} {\bm \eta} \centerdot
  {\mathcal F}_{\bm \eta}  {\bf y}^{(2)}_{N\ell}
  -
  {\mathcal F}_{\bm \theta} {\bm \eta} \centerdot {\mathcal S}_{\bm \eta}
      {\bf y}^{(2)}_{N\ell}
 & = \left \{
 \begin{array}{ll}
  {Y}_{N \ell}({\bm \theta}), & N=2j   \\
  {Y}_{N \ell}({\bm \theta}) , & N=2j+1 ,  \\
 \end{array}
 \right .
 \\
 {\mathcal S }_{\bm \theta} {\bm \eta} \centerdot {\mathcal F}_{\bm \eta} {\bf y}^{(3)}_{N\ell}
 -
 {\mathcal F}_{\bm \theta} {\bm \eta} \centerdot {\mathcal S}_{\bm \eta}{\bf y}^{(3)}_{N\ell}
 &= \left \{
 \begin{array}{ll}
  0, & N=2j \\
  0, & N=2j+1 , \\
 \end{array}
 \right .
 \\
{\bm \theta} \centerdot
   {\mathcal S }_{\bm \theta} {\bm \eta} \times {\mathcal F}_{\bm \eta} {\bf y}^{(2)}_{N\ell}
 -{\bm \theta} \centerdot
 {\mathcal F}_{\bm \theta} {\bm \eta} \times {\mathcal S}_{\bm \eta} {\bf y}^{(2)}_{N\ell}
  &= \left \{
 \begin{array}{ll}
  0, & N=2j \\
  0 & N=2j+1 , \\
 \end{array}
 \right .
 \\
 {\bm \theta} \centerdot
  {\mathcal S }_{\bm \theta} {\bm \eta} \times {\mathcal F}_{\bm \eta} {\bf y}^{(3)}_{N\ell}
 -{\bm \theta} \centerdot
 {\mathcal F}_{\bm \theta} {\bm \eta} \times {\mathcal S}_{\bm \eta}{\bf y}^{(3)}_{N\ell}
  &= \left \{
 \begin{array}{ll}
  {Y}_{N \ell}({\bm \theta}), & N=2j \\
  {Y}_{N \ell}({\bm \theta}), & N=2j+1 . \\
 \end{array}
 \right .
  \end{align*}
\section{Conclusion}
 This paper is devoted to the study of Funk--Minkowski
 transform ${\mathcal F}$
  and  Hilbert type spherical convolution ${\mathcal S}$.  We provide inversion
 formulas for  two F--M transforms ${\mathcal F }f$ and ${\mathcal F} \nabla f$. In this case both  even and
 odd parts of the function $f$ are determined. Also, the formulas
 for decomposition of a tangent vector field on the sphere into
 divergence-free and curl-free parts
 with the participation of operators ${\mathcal F}$ and
 ${\mathcal S}$
 are  derived. In the process
 of obtaining and proving all formulas, the spherical multipliers
 approach  is used.

\bigskip

config: arxiv


TITLE: Common name of $\sum_n 1/(x^p + n^p)$?
QUESTION [2 upvotes]: Famously, one has that the cotangent can be written as the Fourier sum
$$\pi z \cot \pi z = 1 - 2z^2 \sum_{n=1}^\infty \frac{1}{n^2 -z^2}$$
demonstrating evenly spaced poles alone the $x$-axis. Similarly, the hyperbolic cotangent has poles along the imaginary axis:
$$\pi z \coth \pi z = 1 + 2z^2 \sum_{n=1}^\infty \frac{1}{n^2 + z^2}$$
What about poles along rays on arbitrary roots of unity? Consider
$$ f_p(z) = \sum_{n=1}^\infty \frac{1}{n^p + z^p}$$
for integer $p$. This function clearly has evenly-spaced poles along rays: so, for $p=3$ the poles are on a tri-axial pattern, along the negative x-axis, and along the two rays at 60 degrees.
The question is: does this function have a common name, or related to some "obvious" function with a common name, or generally studied in some particular setting? It clearly participates in a rich set of identities involving Dirichlet characters, cyclotomic polynomials, Ramanujan-style identities, and so on. But I don't recall spotting it in any books on combinatorics.
(Conversely, how about functions with zeros along rays?)

REPLY [1 votes]: As pointed out in a comment, one can "easily" show that
$$
\sum_{n=1}^\infty \frac{1}{n^p+z^p} =
\lim_{s\to 1}  \sum_{k=0}^{p-1}
e^{i2\pi k/p} \zeta(s, ze^{i2\pi k/p})
$$
where $\zeta(s,a)$ is the Hurwitz zeta function. I suppose that this answer is enough to send me on my merry way, since both the Hurwitz zeta and the roots of unity can be taken to be "well understood" ingredients.


TITLE: Is gauge invariance necessary to have Ward identity hold for off-shell amplitudes?
QUESTION [3 upvotes]: In this other SE post: Is it really proper to say Ward identity is a consequence of gauge invariance? it is shown that the on-shell Ward identity is a consequence of global $U(1)$ symmetry for QED. This is very logical because the Ward identity is a consequence of the global symmetry and the contact terms are zero once you use the LSZ formula to extract the S-matrix matrix element. But it is also important to know how the the Ward identity behaves off-shell. For example when we consider the self-energy of a photon we know that
$$q_{\mu} \Pi^{\mu \nu} (q) = 0 \tag{1}$$
even if $q$ is off-shell. This is what protects the mass of the photon after renormalization.
Let us therefore write the Ward identity in the most general case:
$$\begin{equation}
\label{eq1}
\partial_{\mu} \langle T j^{\mu} (x) F_{1} (x_1)...F_{N} (x_n)\rangle = -i \sum_{j=1}^{n} \delta^4 (x-x_i) \langle TF_{1} (x_1)...\delta F_{j}(x_j)...F_{N} (x_n)\rangle,\tag{2}
\end{equation}$$
where $F$ can be either a fermion field $\psi$ or a photon field $A^{\mu}$. Let us consider the case that for every fermion field there is an antifermion field. This is what happens in all diagrams since fermion fields inside Green's function are appear in couples inside currents. By using global $U(1)$ we get that the variations of photon fields are automatically zero, every fermion brings a $+i\theta$, and every antifermion brings a $-i\theta$, therefore the right hand side of Ward's identity is zero. This implies that even off-shell $q_{\mu} \Pi^{\mu \nu} (q)$. Therefore it is not gauge symmetry that protects the mass of the photon but global symmetry. If this is true it would imply that QED need not be a gauge theory to be renormalizable and we can add a mass term for the photon: $ \mathcal{L}_{\gamma mass}=\frac{1}{2} m_{\gamma}^2 A^{\mu}A_{\mu} $ that breaks gauge symmetry but preserves global symmetry. Also, because of the Ward identity the photon propagator after renormalization would be:
$$\begin{equation}
D^{\mu \nu} = \frac{-i g^{\mu \nu}}{(q^2-m_{\gamma}^2)\left(1+ \Pi(q^2)\right)},\tag{3}
\end{equation}$$
where $\Pi^{\mu \nu} (q) \equiv (g^{\mu \nu} q^2-q^{\mu} q^{\nu}) \Pi(q^2)$. This would imply that $U(1)$ symmetry preserves the mass of the photon even if $m_{\gamma} \neq 0$. All of this looks extremely suspicious to me and I am pretty sure I must have made a mistake somewhere in the derivation of the off-shell Ward identity but I can't tell where, and why assuming gauge symmetry would fix such mistake.

REPLY [0 votes]: The Ward identities (2) for connected$^1$ correlators (with appropriate contact terms, derived via the Schwinger-Dysons equations for a global symmetry) hold off-shell, cf. Ref. 1.

However, the transversality (1) of the photon 1-particle irreducible (1PI) vacuum polarization/self-energy is a consequence of local gauge symmetry (and appropriate class of gauge-fixing conditions), as explained in e.g. this and this Phys.SE posts.

References:

M.E. Peskin & D.V. Schroeder, An Intro to QFT; Section 9.6.

--
$^1$ The Ward identity (2) is originally for not-necessarily-connected correlators, but one can identify connected components, cf. the linked cluster theorem, to make it a statement about connected correlators.


TITLE: Little O Computation: $4n\:\in \:o\left(n^2\right)$
QUESTION [1 upvotes]: For this, I know you can use calculus techniques and set $\lim _{x\to \infty }\left(\frac{4n}{n^2}\right)$ and get 0. Though, I'm wanting to know how you can get a specific constant and k values.

REPLY [2 votes]: Since you don't want to use the limit, we can in this case use the definition (see here for details):
$$
f(n)=o(g(n)) \stackrel{\text{def}}\equiv \forall k>0\,\exists N\ge 0\, \forall n\ge N\,[\:f(n)\le k\cdot g(n)\:]
$$
In other words, the definition is exactly the same as for big-O, except that the inequality eventually holds for all positive constant multiples $k$.
With this in mind, choose any positive $k$. Then to have
$$
4n\le k\:n^2
$$
we need the equivalent
$$
\frac{4}{k}\le n
$$
so for any positive $k$ we'll have $4n\le n^2$ for all $n\ge N = 4/k$.


TITLE: Equal to the same are equal to each other - Is it geometry?
QUESTION [2 upvotes]: Is this geometry or algebra  "things equal to the same thing are equal to each other"?
thank you in advance....
Joseph Whelan  CPA
Bedford, NY

REPLY [0 votes]: This is an axiom of equality.
As Umberto P. suggests, it is closely related to the definition of equality wherein we say that equality is a predicate which satisfies reflexivity, symmetry and transitivity.
If you say algebra is study of functions and operations and geometry is study of relations, this might be close to, but not in, geometry.
It is far more fundamental than geometry.


TITLE: Show that the digit $d$($0<d<10$) occurs as a first digit in power of 2 with frequency $log_{10} \frac{d+1}{d}$.
QUESTION [3 upvotes]: Show that the digit $d$($0<d<10$) occurs as a first digit in power of $2 $ with frequency $log_{10} \frac{d+1}{d}$.
I think that this can be proved using Birkhoff Ergodic theorem. Any help?

REPLY [1 votes]: Although one answer has been posted, I wanted to put some light on the way you have asked the question. The flavour would be the same as Mr alex.jordan.
Now observe if $d$ be the first digit of a number $n$ then $\exists r \in \Bbb N$ s.t $d10^r \leq 2^n < (d+1)10^{r}$.
Taking $\log_{10}$ both side we get $\log_{10}d + r \leq \log_{10}n< \log_{10}(d+1)+r $ Now when we take $(\mod 1)$ we'll get

$d10^r \leq n\log_{10}2 < (d+1)10^{r}$ iff $n\log_{10}2(\mod 1) \in [\log_{10}d, \log_{10}(d+1)]$

Now consider the map $T:[0,1) \to [0,1)$ s.t $x \mapsto x+ \alpha$ where $\alpha=\log_{10}2$ is irrational.
Then you check that the map $T$ is ergodic.
Consider $I=[\log_{10}d, \log_{10}(d+1)] \subseteq [0,1)$ and $f=\chi_I$
Now, Birkhoff ergodic theorem along with the equivalent conditions of ergodicity states that $\lim_{n \to \infty} \frac 1n |\{0 \leq j \leq n-1: T^j(x) \in I\}|=\lim_{n \to \infty} \frac 1n \sum_{j=0}^{n-1}f(T^j(x))=\lim_{n \to \infty} \frac 1n \sum_{j=0}^{n-1}\chi_I(T^j(x))\to \mu(I)$ for almost every $x \in [0,1)$.
Now the notable thing is that here for irrational rotation the fact happens for all $x \in [0,1)$ in particular putting $x=0$ we get

$\lim_{n \to \infty} \frac 1n |\{0 \leq j \leq n-1: T^j(x) \in I\}|=\lim_{n \to \infty} \frac 1n |\{0 \leq j \leq n-1:$ the first digit of $2^i$ is $d\}|=\mu(I)=\log_{10}\frac{d+1}d$

Hence proved.


TITLE: If monochrome laser strikes a mirror perpendicularly, how would I know whether there is destructive interference?
QUESTION [0 upvotes]: Suppose I have a setup where there are 2 mirrors parallel to each other and I let in a laser beam of a single wavelength of red color through a tiny little hole in one of the mirror so that the beam would strike the other mirror at right angle. Suppose the laser bounces back and forth and the mirrors remain perfectly still, how do I check if there is destructive interference or not?

REPLY [0 votes]: A schematic diagram of an arrangement which will enable you to view the interference fringes might look like this.

Please note that to see the fringes as described below the mirrors have to be optically flat, in other words flat to within fractions of the wavelength of the light emitted by the laser.
You will note that the inclined mirror reflects light from the laser down onto the parallel mirror and then allows light which is reflected from those parallel mirrors to pass through and hit the screen where the interference fringes can be viewed.
In the arrangement as shown you will see circular fringes whose radius will vary as the separation between the mirrors varies.
As the separation of the mirrors is decreased fringes seem to be "created" at the centre and appear to move outwards from the centre.
When the parallel mirrors are very close (a few wavelengths) to one another then the radius of the fringes becomes larger but you are unlikely to see a uniform illumination (or lack of it) across the whole screen.
The width/sharpness of the fringes will depend on the number of significant reflection there are between the mirrors and as the reflectivity $R$ of the mirrors goes up the fringes become sharper.

This increase in sharpness as there are more significant reflections between the parallel mirrors is equivalent to the increase in sharpness in the fringes produced from multiple slit interference as one goes from two slits to many thousands of slits which is called a diffraction grating.
A simple experimental set up to observe such fringes using two microscope slides is described here with the separation of the two reflecting surface (the sides of a microscope slide) changed by heating the microscope slide with a hair dryer.
The reflectivity of microscope slides is low so to achieve the very sharp slides you need to use a Fabry-Perot etalon which is described here.