For any channel, the existence of a unique Augustin mean is established for any positive order and probability mass function on the input set. The Augustin mean is shown to be the unique fixed point of an operator defined in terms of the order and the input distribution. The Augustin information is shown to be continuously differentiable in the order. For any channel and convex constraint set with finite Augustin capacity, the existence of a unique Augustin center and the associated van Erven-Harremoes bound are established. The Augustin-Legendre (A-L) information, capacity, center, and radius are introduced, and the latter three are proved to be equal to the corresponding Rényi-Gallager quantities. The equality of the A-L capacity to the A-L radius for arbitrary channels and the existence of a unique A-L center for channels with finite A-L capacity are established. For all interior points of the feasible set of cost constraints, the cost constrained Augustin capacity and center are expressed in terms of the A-L capacity and center. Certain shift-invariant families of probabilities and certain Gaussian channels are analyzed as examples.

中文翻译：

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奥古斯丁的能力和中心

对于任何通道，对于输入集上的任何正序和概率质量函数，都存在唯一的Augustin均值。显示奥古斯丁均值是根据顺序和输入分布定义的运算符的唯一不动点。奥古斯丁信息显示顺序是连续可区分的。对于具有有限奥古斯丁容量的任何通道和凸约束集，都建立了唯一奥古斯丁中心的存在以及相关的范·欧文-哈雷莫斯边界。介绍了Augustin-Legendre（AL）的信息，容量，中心和半径，并且后三个被证明等于相应的Rényi-Gallager量。对于任意信道，AL能力与AL半径相等，并且对于AL能力有限的信道，存在唯一的AL中心。对于可行的成本约束集合的所有内部点，以AL容量和中心表示受成本约束的Augustin容量和中心。作为示例分析了某些平移不变的概率族和某些高斯通道。