Fourier Analysis and Order Structure

1. Fourier分析的研究（a）将F.和M.Riesz定理推广到超群和非Abel紧群的情形。在此之前，这些考虑仅在阿贝尔情况下进行。(b)Bochner定理说两个Riesz集的乘积也是Riesz集。在忠诚紧Abel群的情形下，H. Yamaguchi的Bochner定理成立.这一次，我们可以证明Bochner定理在非Abel紧群中成立。(c)圆盘超群中的傅立叶分析可以被许多人发展。我们可以将这些理论推广到离散超群的情形.有序结构研究(a)具有Fatou性质的赋范线性空间是完备的（Riesz-Fisher定理）在一般的序拓扑线性空间中是不成立的。我们找到了Riesz-Fisher定理在空间中成立的一个充分必要条件。(b)定义了序有界子集的广义上确界（或广义下确界）.研究了广义上确界（或广义下确界）的性质.我们构造了一个广义上确界和广义下确界的理论。我们发现分配律成立当且仅当空间是Riesz空间。(c)优化理论通常只考虑真实的值函数。对取值于序拓扑线性空间的函数构造了一个优化理论。从实际应用的角度来看，这一理论是非常重要的。

1. Study on Fourier Analysis(a) We can extend the F.and M.Riesz theorem in the case of hypergroup and non-Abelian compact group. Before, these considerations are made in only Abelian cases.(b) Bochner's theorem says that product of two Riesz sets is also a Riesz set. In the case of loyally compact Abelian group, Bochner's thearem is true by H.Yamaguchi. This time, we can prove that Bochner's theorem is valid in non-Abelian compact group.(c) Fourier Analysis in Disk hypergroup can be developed by many peoples. We can extend these theory in the case of discrete hyper group.2. Study on Order Structure.(a) Normed linear space with Fatou property is complete (Riesz-Fisher's theorem) is not true in general ordered topological inear space. We find a necessary and sufficient condition tWat the Riesz-Fisher's theorem is valid in the space.(b) We defined generalized suprernum ( or generalized infimum) for a order bounded sub set. We investigate properties of generalized suprem ( or generalized infimum). We construct a theory of generalized supremum and generalized infimum. We find that distributive law is valid if and only if the space is a Riesz space.(c) Optimization theory usually considers only real valued functions. We construct an optimization theory for function whose value is taken in an ordered topological linear space. For application, this theory is very important.