论文标题
$ c^\ infty $ - 同构属性，用于一类奇异加权的X射线变换
The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms
论文作者
论文摘要
我们研究了封闭的Euclidean磁盘$ {\ Mathbb d} $的X射线转换的一个独立偶相常规操作员家族，该家族是通过考虑特定的特定重量加权的$ l^2 $拓扑来获得的。我们首先根据正交磁盘（或广义Zernike）多项式恢复了众所周知的奇异值分解，然后证明每个实现都是$ c^\ infty（{\ mathbb d}）$的同构函数。作为推论：我们给出一些范围的特征；我们展示了如何将这种正常运算符的选择表示为两个杰出差异操作员的功能。我们还表明，同构特性还具有一类恒定的圆形，圆形对称的简单表面。这些结果允许设计功能上下文，其中在Fréchet和Hilbert Spaces编码特定边界行为的情况下，在X射线变换中构建的正常运算符是可逆的。
We study a one-parameter family of self-adjoint normal operators for the X-ray transform on the closed Euclidean disk ${\mathbb D}$, obtained by considering specific singularly weighted $L^2$ topologies. We first recover the well-known Singular Value Decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of $C^\infty({\mathbb D})$. As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the X-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior.
