关于连续函数和光滑函数的乘法映射
On Multiplicative Maps of Continuous and Smooth Functions
摘要：
本文研究了定义在流形上的实值或复值函数族上的乘法双射的一般形式。在真实的情况下，我们证明了它本质上是由一个复合定义的基础流形的一个同构（在家庭的连续函数多一点的自由）。我们的结果在实际情况下大多是已知定理的单扩张。然后，我们表明，在复杂的情况下，唯一允许的额外的自由是复共轭。最后，我们应用这些结果来刻画某些函数空间之间的傅里叶变换。
英文摘要：
In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a diffeomorphism of the underlying manifold (with a bit more freedom in families of continuous functions). Our results in the real case are mostly simple extensions of known theorems. We then show that in the complex case, the only additional freedom allowed is complex conjugation. Finally, we apply those results to characterize the Fourier transform between certain function spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
