In [3], Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for functions on the sphere, which ensure that the corresponding system is a frame. We strengthen a similar result in [3] by providing a complete and detailed proof.

中文翻译：

---

球体上的连续小波框架：重新审视群论方法

在 [3] 中，Antoine 和 Vandergheynst 提出了一种用于球体上连续小波框架的群论方法。该框架是通过应用与洛伦兹群表示相关的酉算子从单个所谓的可容许函数构建的，洛伦兹群是 Iwasawa 分解的幂零因子的平方可积模。我们证明了球面上函数的充要条件，保证相应的系统是一个框架。我们通过提供完整和详细的证明来加强 [3] 中的类似结果。