Plane waves on a sphere and some applications

Fourier analysis on a sphere of dimension $d\geqslant 2$ is developed in a form that recalls Fourier analysts in flat space to the maximal extent possible. Plane waves – the kernels of the corresponding integral transforms – are generalized functions, for which a regularization is defined; completeness relations, finite-difference equations, and a composition theorem are found. In the case $d=4$, these functions are used for the transition to the Euclidean expressions in the commutator functions of field theory with a curved momentum space; for $d=3$, they are used to describe a class of states in a Coulomb field.