Applied Functional Analysis related to Mathematical Information Theory

点击翻译按钮获取中文摘要

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.