Fourier Analysis and Order Structure

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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.