球面意味着中心在偶数维超平面上
Spherical means with centers on a hyperplane in even dimensions
摘要：
给定R^n上的一个实值函数，研究了在以超平面为中心的球面上由其球面平均恢复函数的问题。Bukhgeim和Kardakov的一篇旧文章导出了奇n情形的一个反演公式，非常简单和经济。我们应用他们的方法推导出一个反演公式的偶数n的情况下。我们的反演公式的一个特点，为偶数n的情况下，是它不需要傅里叶变换的平均值或希尔伯特变换的使用，不像以前已知的反演公式为偶数n的情况下。一路上，我们扩展等距单位的Bukhgeim和Kardakov奇数n，波动方程的解决方案，甚至n的情况下。
英文摘要：
Given a real valued function on R^n we study the problem of recovering the function from its spherical means over spheres centered on a hyperplane. An old paper of Bukhgeim and Kardakov derived an inversion formula for the odd n case with great simplicity and economy. We apply their method to derive an inversion formula for the even n case. A feature of our inversion formula, for the even n case, is that it does not require the Fourier transform of the mean values or the use of the Hilbert transform, unlike the previously known inversion formulas for the even n case. Along the way, we extend the isometry identity of Bukhgeim and Kardakov for odd n, for solutions of the wave equation, to the even n case.
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