The λ-cosine transforms with odd kernel and the hemispherical transform

We review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S ^n-1 in ℝ ⁿ . These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S ^n-1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical Funk- Radon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L ^p -functions.