Title: A simple range characterization for spherical mean transform in even dimensions Show Chinese title

Title: 球面平均变换在偶数维中的简单范围特征

Abstract: The paper presents a new and simple range characterization for the spherical mean transform of functions supported in the unit ball in even dimensions. It complements the previous work of the same authors, where they solved an analogous problem in odd dimensions. The range description in even dimensions consists of symmetry relations, using a special kind of elliptic integrals involving the coefficients of the spherical harmonics expansion of the function in the range of the transform. The article also introduces a pair of original identities involving normalized Bessel functions of the first and the second kind. The first result is an integral cross-product identity for Bessel functions of integer order, complementing a similar relation for Bessel functions of half-integer order obtained in the aforementioned work of the same authors. The second result is a new Nicholson-type identity. Both of these relations can be considered as important standalone results in the theory of special functions. Finally, as part of the proof of one of the theorems, the authors derive an interesting equality involving elliptic integrals, which may be of independent interest. Show Chinese abstract

Abstract: 本文提出了一种新的且简单的球均值变换在偶数维单位球内支集函数的范围特征化方法。 这补充了同一作者之前的工作，在该工作中他们解决了奇数维类似问题。 偶数维中的范围描述包括对称性关系，使用涉及变换范围内函数球谐波展开系数的特殊椭圆积分。 文章还引入了一对关于第一类和第二类归一化贝塞尔函数的新恒等式。 第一个结果是关于整数阶贝塞尔函数的积分交叉乘积恒等式，补充了同一作者在上述工作中获得的关于半整数阶贝塞尔函数的类似关系。 第二个结果是一种新的Nicholson型恒等式。 这两条恒等式可以被视为特殊函数理论中的重要独立结果。 最后，作为其中一个定理证明的一部分，作者推导出一个有趣的涉及椭圆积分的等式，这可能具有独立的研究价值。