Applied Functional Analysis related to Mathematical Information Theory

数学信息论最初是由C.E.香农于1948年创立的。从那时起，它发展了数学基础，使通信系统确定。在这项研究中，我们有四个结果的有限块长度容量的离散时间高斯信道的反馈.本文证明了反馈容量C_（n，FB，Z）的稳定性。即C_<n，FB，Z>（αP_1+βP_2）≥ αC_<n，FB，Z>（P_1）+βC_<n，FB，Z>（P_2）。当功率约束P相对较大时，我们给出了分块白色反馈容量C_<n，FB，Z>（P）的有用上限。即对任意P>P_0={mr_m-（r_1+r_2+.+r_m）}/n，C_n，FB，Z>（P）[小于或等于]（C_n，Z>（P））/（P_0）P，C_n，FB，Z>（P）[小于或等于]C_n，Z>（P_0）+1/2logP/（P_0）。给出了log（1+t^）的算子凸性<-1>.作为其应用，我们证明了无反馈容量C_n，.（P）。即C_（n，Z）（P）≤ αC_（n，Z）（P）+βC_（n，Z）（P）。证明了反馈容量C_&lt;n，FB，&gt;的类凸性。（P）。即存在P_1，P_2[大于或等于]0（P=αP_1+βP_2）使得C_&lt;n，FB，Z&gt;（P）[小于或等于]αC_&lt;n，FB，Z_1&gt;（P_1）+βC_&lt;n，FB，Z_2&gt;（P_2）.

The mathematical information theory was originally founded by C.E.Shannon in 1948. Since then, it has developed the mathematical foundations to make the communication systems certain. In this research we have four results on the finite block length capacity of discrete time Gaussian channels with feedback.1. We proved the concavity of feedback capacity C_<n, FB,Z>( ). That is,C_<n, FB,Z>(αP_1+βP_2)【greater than or equal】αC_<n, FB,Z>(P_1)+βC_<n, FB,Z>(P_2).2. We gave the useful upper bound to blockwise white feedback capacity C_<n, FB,Z>(P) when the power constraint P is relatively large. That is, for any P>P_0={mr_m-(r_1+r_2+…+r_m)}/n,C_<n, FB,Z>(P)【less than or equal】(C_<n, Z>(P))/(P_0)P,C_<n, FB,Z>(P)【less than or equal】C_<n, Z>(P_0)+1/2logP/(P_0).3. We gave the operator convexity of log(1+t^<-1>). As its application we proved the convexity of nonfeedback capacity C_n, .(P). That is,C_<n, Z>(P)【less than or equal】αC_<n, Z_1>(P)+βC_<n, Z_2>(P).4. We proved the convex-likeness of feedback capacity C_<n, FB,>.(P). That is, there exist P_1, P_2【greater than or equal】0(P=αP_1+βP_2) such thatC_<n, FB,Z>(P)【less than or equal】αC_<n, FB,Z_1>(P_1)+βC_<n, FB,Z_2>(P_2).In future we will try to prove the convexity of feedback capacity.