A one-radius theorem on a sphere with pricked point

We considers local properties of mean periodicity on the two-dimensional sphere $\mathbb{S}^2$. According to the classical properties of periodic functions, each function continuous on the unit circle $\mathbb{S}^1$ and possessing zero integrals over any interval of a fixed length $2r$ on $\mathbb{S}^1$ is identically zero if and only if the number $r/\pi$ is irrational. In addition, there is no non-zero continuous function on $\mathbb{R}$ possessing zero integrals over all segments of fixed length and their boundaries. The aim of this paper is to study similar phenomena on a sphere in $\mathbb{R}^3$ with a pricked point. We study smooth functions on $\mathbb{S}^2\setminus(0,0,-1)$ with zero integrals over all admissible spherical caps and circles of a fixed radius. For such functions, we establish a one-radius theorem, which implies the injectivity of the corresponding integral transform. We also improve the well-known Ungar theorem on spherical means, which gives necessary and sufficient conditions for the spherical cap belong to the class of Pompeiu sets on $\mathbb{S}^2$. The proof of the main results is based on the description of solutions $f\in C^{\infty}(\mathbb{S}^2\setminus(0,0,-1))$ of the convolution equation $(f\ast \sigma_r)(\xi)=0$, $\xi\in B_{\pi-r}$, where $B_{\pi-r}$ is the open geodesic ball of radius $\pi-r$ centered at the point $(0,0,1)$ on $\mathbb{S}^2$ and $\sigma_r$ is the delta-function supported on $\partial B_r$. The key tool used for describing $f$ is the Fourier series in spherical harmonics on $\mathbb{S}^1$. We show that the Fourier coefficients $f_k(\theta)$ of the function $f$ are representable by series in Legendre functions related with the zeroes of the function $P_\nu(\cos r)$. Our main results are consequence of the above representation of the function $f$ and the corresponding properties of the Legendre functions. The results obtained in the work can be used in solving problems associated with ball and spherical means.