Inversion and characterization of the hemispherical transform

Explicit inversion formulas are obtained for the hemispherical transform (Fμ)(x) = μ{y ∈ Sⁿ: x · y ≥ 0}, x ∈ Sⁿ, where Sⁿ is the n-dimensional unit sphere in ℝⁿ⁺¹, n ≥ 2, and μ is a finite Borel measure on Sⁿ. If μ is absolutely continuous with respect to Lebesgue measure dy on Sⁿ, i.e., dμ(y) = f(y)dy, we write (F f)(x) = ∫_x·y>0 f(y)dy and consider the following cases: (a) f ∈ C^∞(Sⁿ); (b) f ∈ L^p(Sⁿ), 1 ≤ p < ∞; and (c) f ∈ C(Sⁿ). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining cases, the relevant wavelet transforms are employed. The range of F is characterized and the action in the scale of Sobolev spaces L_p^γ(Sⁿ) is studied. For zonal f ∈ L¹ (S²), the hemispherical transform F f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions.