The dependence of the generalized Radon transform on defining measures
The dependence of the generalized Radon transform on defining measures
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广义 Radon 变换对定义测度的依赖
DOI：
10.1090/s0002-9947-1980-0552261-8
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发表时间：
1980
影响因子：
1.3
通讯作者：
E. T. Quinto
中科院分区：
文献类型：
--
作者：
E. T. Quinto
ABsmAcr。Guillemin证明了广义Radon变换R及其对偶R‘是傅里叶积分算子，R’是椭圆伪微分算子。本文研究了Radon变换对定义测度的依赖关系。在一般情况下，我们用测度来计算rr作为伪微分算子的符号，并给出了rr被微分算子可逆的定义测度的必要条件。然后用一般测度检验RX中点和超平面上的Radon变换，并根据定义测度计算rr的符号。最后，如果R'R是RI上的平移不变算子，那么我们证明R'R是可逆的，并且我们的条件等价于（R‘R）’是微分算子。
ABsmAcr. Guillemin proved that the generalized Radon transform R and its dual R' are Fourier integral operators and that R'R is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of R'R as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for R'R to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in RX with general measures and we calculate the symbol of R'R in terms of the defining measures. Finally, if R'R is a translation invariant operator on RI then we prove that R'R is invertible and that our condition is equivalent to (R'R)' being a differential operator.