Generalized Minkowski-Funk transforms and some arithmetical
properties of the associated Legendre functions
The Cauchy problem for the Euler-Poisson-Darboux equation on the sphere and various transforms of integral geometry (including those of Minkowski and Funk) give rise to a family of fractional integrals associated with a spherical cap of a fixed radius $\theta$. These fractional integrals are called the generalized Minkowski-Funk transforms.
An investigation of injectivity, invertibility and boundedness of these transforms in Sobolev spaces leads to small denominators for spherical harmonics expansions and to some delicate problems related to arithmetical properties of zeros of the associated Legendre functions.
The problems of such a type can be stated also for Jacobi (or Gegenbauer) polynomials. For example, how does the number of common zeros of such polynomials, say, $P_j (t), t=\cos \theta$, depend on $\theta$? In other words, how many $j$'s satisfy the equation $P_j (t)=0$ for $t$ fixed? The results are different depending on whether $\theta$ is a rational or irrational multiple of $\pi$. In some simplest cases the usual techniques of diophantine approximations are applicable.
The new number-theoretical meaning is given to the classical Schneider-Berenstein-Zalcman
results on injectivity of the Pompeiu transform on the sphere. Curious
examples are exhibited.