Journal d’Analyse Mathématique

, Volume 77, Issue 1, pp 105–128 | Cite as

Inversion and characterization of the hemispherical transform

  • Boris Rubin


Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S n :x. y ≥ 0},xS n, whereS n is thendimensional unit sphere in ℝn+1,n ≥ 2, and Μ is a finite Borel measure onS n. If Μ is absolutely continuous with respect to Lebesgue measuredy onS n, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫ x.y> 0 f(y)dy and consider the following cases: (a)fC (Sn); (b)f ∃ Lp(S n), 1 ≤ p < ∞; and (c)fC(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 ofF is characterized and the action in the scale of Sobolev spacesL p γ (Sn) is studied. For zonalf ∃ L1(S 2), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions.


Sobolev Space Singular Integral Operator Inversion Formula Fractional Integral Zonal Function 
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Copyright information

© Hebrew University of Jerusalem 1999

Authors and Affiliations

  • Boris Rubin
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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