Harmonic Analysis and its application to measurements

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Measurement theory is a mathematical theory, which is motivated by Born's probabilistic interpretation of quantum mechanics. Particularly, it is related to the resolution of the identity, or, in terms of Fourier analysis, a theory of wavelet. Out research results are as follows.(1) Consider an N-particle classical system (e.g., N【approximately equal】10ィイD123ィエD1) with a Hamiltonian H on the phase space RィイD16NィエD1. Moreover, we consider the class of measurements such as we can get the information concerning NィイD2oィエD2 particles in this system (e.g., NィイD2oィエD2【less than or equal】10ィイD118ィエD1<<N) . We showed that this class of measurements also induces another kind of equilibrium statistical mechanics. In this equilibrium statistical mechanics, we can assert that the "equilibrium state" exists almost everywhere as a point of the energy surface Ω({(q,p)RィイD16NィエD1:H(q,p)=E}).(2) Quantum logic is, of course, motivated bu quantum mechanics. Thus, it is natural to consider that several logi … More c may be formulated in terms of measurements. In fact we showed that "fuzzy logic" is clarified and understood in the limit of measurement theory.(3) The theory of "measurement errors" is an important aspect of measurement theory. In fact, we showed that the "measurement error" in Heisenberg's uncertainty relation is similar to that of classical measurement.(4) The concept of "trajectories" is fundamental to classical mechanics. However, we showed that this concept is derived from that of simultaneous measurements.(5) Information theory is reconstructed in terms of measurements. This is natural since "information" should be obtained by measurements.(6) We consider that statistics is to analyze "measured data" for some purpose. Thus, it is viable to consider that statistics is described in terms of measurements. In faxt, we showed that factor analysis, regression analysis and Kalman filter are formulated in measurement theory.As mentioned above, measurement theory covers a great region, which may be called "information science". The technological aspect of measurement theory is, of course, important. However, in our research, we devoted ourselves to the scientific and mathematical aspect of measurement theory. Less