The optimum approximation of vector-signals and estimation of the velocity of an object causing Doppler shift

In many applications of signal processing such as target-detection or remote-sensing, it is necessary to estimate an unknown object-signal given by a linear combination of data-signals obtained by given observation-systems. In this paper, we present the optimum approximation of signals f(t) expressed by linear combinations of extended inverse transforms, including many known mathematical inverse transforms, of the components of generalized spectrum-vectors F(ω) in a given Hilbert space. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations. Through the process of the proof, it is shown that the above approximation gives, at the same time, the optimum approximation of vector signals f(t) determined by F(ω). We indicate that this approximation gives a favorable method in Doppler-radar that is useful to estimate the velocity of a target object having Doppler shift.