发表机构
Tata Institute of Fundamental Research（塔塔基础研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对奇数维球平均变换，提出一种不依赖已有结果的直接构造性证明，验证了基于对称条件的值域刻画充分性，并能显式构造原像函数。
AI 中文摘要
本文关注奇数维欧几里得空间中球平均变换（SMT）的值域刻画。在最近的一项工作中，得到了一个更简洁的、刻画奇数维SMT值域的描述，该描述涉及作用于值域中函数的球谐展开系数上的某些线性常微分算子所满足的对称关系。该刻画的充分性部分是利用Agranovsky-Kuchment-Quinto已有的值域刻画结果证明的。在当前工作中，我们提供了一个直接且构造性的证明，以说明上述对称条件的充分性，且无需借助任何已有的值域刻画结果。作为额外优势，给定一个满足值域条件的函数$g$，我们的方法能够显式构造一个函数$f$，使得$f$的SMT等于$g$。
英文摘要
This work focuses on range characterization of spherical mean transform (SMT) in odd-dimension Euclidean space. In a recent work, a simpler description characterizing the range of SMT in odd dimensions was derived. This involved symmetry relations involving certain linear ordinary differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range. The sufficiency part of this characterization was shown using an existing range characterization result due to Agranovsky-Kuchment-Quinto. In the current work, we provide a direct and constructive proof showing the sufficiency of the aforementioned symmetry conditions and we obtain this without invoking any of the existing range characterization results. As an added advantage, given a function $g$ satisfying the range conditions, our approach provides an explicit construction of a function $f$ such that the SMT of $f$ is $g$.
