The Augustin Capacity and Center
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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 any convex constraint set with finite Augustin capacity, the existence of a unique Augustin center and associated van Erven-Harremoes bound are established.The Augustin-Legendre (A-L) information, capacity, center, and radius are introduced and the latter three is proved to be equal to the corresponding Renyi-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 the A-L capacity and center. Certain shift invariant families of probability measures and certain Gaussian channels are analyzed as examples.
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