关于一维庞培问题

Vivina Barutello, Camillo Costantini
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引用次数: 0

摘要

在不假定连通性的情况下,我们研究了实线子集的庞培性质。特别是,我们将研究重点放在有界(不相交)区间的有限联合上,并强调了与以下情况相对应的不同结果:假设相关函数在所有等距图像上或仅仅在域的所有平移图像上具有恒积分。虽然前一种集合都不具有后一种意义上的庞培性质,但我们提供了一个必要且充分的条件,以使两个区间的结合具有前一种意义上的庞培性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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On the one-dimensional Pompeiu problem

We investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding to the cases where the function in question is supposed to have constant integral on all isometric images, or just on all the translation-images of the domain. While no set of the previous kind enjoys the Pompeiu property in the latter sense, we provide a necessary and sufficient condition in order a union of two intervals to have the Pompeiu property in the former sense, and we produce some examples to give an insight of the complexity of the problem for three-interval sets.

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