Inversion and characterization of the hemispherical transform
Inversion and characterization of the hemispherical transform
复制标题
半球变换的反演和表征
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发表时间：
1999
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通讯作者：
B. Rubin
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作者：
B. Rubin
得到了半球变换（FM）（x）= M {y ∈ Sn：x）的显式反演公式。y ≥ 0}，x <$Sn，其中Sn是n+1中的n维单位球面，n ≥ 2，M是Sn上的有限Borel测度。如果M关于Sn上的勒贝格测度是绝对连续的，即，d M（y）=f（y）dy，我们写（F f）（x）= f x.y> 0 f（y）dy，并考虑以下情况：（a）f <$C∞（Sn）;（B）f <$Lp（Sn），1 ≤ p < ∞;（c）f <$C（Sn）。在情形（a）中，我们的反演公式涉及Laplace-Beltrami算子的某个多项式。在其余的情况下，采用相关的小波变换。刻画了F的值域，研究了F在Sobolev空间Lp γ（Sn）尺度上的作用.对于zonalf λ L1（S2），半球变换F被P. Funk（1916）明确地反演;我们在更高的维度中重现他的论证。
Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃Sn :x. y ≥ 0},x ∃Sn, whereSn is thendimensional unit sphere in ℝn+1,n ≥ 2, and Μ is a finite Borel measure onSn. If Μ is absolutely continuous with respect to Lebesgue measuredy onSn, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫x.y> 0f(y)dy and consider the following cases: (a)f ∃C∞(Sn); (b)f ∃ Lp(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 ofF is characterized and the action in the scale of Sobolev spacesLpγ (Sn) is studied. For zonalf ∃ L1(S2), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions.