In this paper, it is shown a base of the new trend of algorithm mathematically that treats a historical reason of continuous discrimination in the world as well as its solution by introducing new concepts of parallel world that includes an invisible set of errors as its companion. With respect to a matrix operator-filter bank that the ma-trix operator-analysis-filter bank H and the matrix operator-sampling-filter bank S are given, firstly, we introduce the detail algorithm to derive the optimum matrix operator-synthesis-filter bank Z that minimizes all the worst-case measures of the matrix operator-error-signals$\mathbf{E}(\omega)=\mathbf{F}(\omega)-\mathbf{Y}(\omega)$between the matrix operator-input-signals$\mathbf{F}(\omega)$and the matrix operator-output-signals$\mathbf{Y}(\omega)$of the matrix operator-filter bank at the same time. Further, feedback is introduced to the above approximation theory and it is indicated that introducing conversations with feedback do not superior automatically to the accumulation of existing knowledge of signal prediction. Secondly, the concept of category in the field of mathematics is applied to the above optimum signal approximation and is indicated that the category-based approximation theory is applied to the set-theoretic con-sideration of the recognition of human. Based on this discussion, it is shown naturally, why the narrow perception that tends to create isolation shows an apparent advantage in the short term and, often, why such narrow thinking becomes intimate with discriminatory action in a human group. Throughout these considerations, it is presented that, in order to abolish easy and intimate discriminatory behavior, it is important create a parallel world of conception where we share the set of invisible error signals including the words and the consciousness of both worlds.

Telemedicine has become an important issue in connection with the recent epidemic of infectious diseases. We present the optimum operator filter bank that minimizes all the worst-case measures for the error between the input-operator and the output-operator of the filter bank at the same time. In the remote surgery, multiple cameras are often scaled and rotated at various lengths and angles. This shows that these images have undergone some kind of affine transformations and are transmitted to the remote displays for doctors. We present the optimum operator filter bank giving affine isomorphic transmission of matrix images. Further, it is shown that the presented optimal approximation theory can be applied to the two-mode analysis of the buckling damage in patients' bone fractures and injuries due to an accident or error of the remote arms. Based on a one-to-one correspondence between input-operators, output-operators and error-operators in this approximation, we show, in almost all digital remote processing, it is appropriate to use the optimum approximation in this paper.

This paper introduces a new parallel imaging method based on a combination of the optimum interpolation approximation (OIA) and compressed sensing (CS) method. The proposed method is a kind of improved SENSE method because all sensitivity maps of receiver coils are assumed to be known. Firstly, we introduce a theory of the OIA that minimizes the supremum value of any measure of approximation error. Secondly, a practical method of computing the optimum interpolation functions of the OIA is presented. Because the proof of the optimality is based on the set theory, the OIA is optimized for a group of MR images, not for particular ones. Thirdly, we present a new combinational method of the OIA and the CS method. In this method, the sufficiently accurate reconstruction of a particular target MR image by the CS method is used as the weighting function of the OIA. Because the accuracy of the OIA tends to improve as the weighting function approaches to the target MR image, the proposed combinational method becomes optimum for reconstructing the particular target MR image. Finally, we compare the reconstruction accuracy of the proposed combinational method with that of the CS-only method.

With respect to a set of input matrix-signals F(ω) satisfying that a weighted absolute squared sum of the aliasing components in the frequency domain is smaller than a given positive constant and a set of output matrix-signals Y(ω) of a matrix-filterbank Y(ω)=α[F(ω)] that has a given analysis-matrix H(ω) with column array of sub-band matrix-filters H_1(ω), H_2(ω),...,H_M-1(ω) in it, we present the optimum approximation that minimizes all the worst-case measures of matrix-error-signals E(ω)=F(ω)-Y(ω) in the frequency domain Ω={ω} at the same time. We use the sample values of the output of the analysis-matrix H(ω) at the uniform sample points in the time-domain. The optimum approximation presented in this paper succeeds to the simultaneous minimization of all the upper-limit measures sup_F(ω)εΘβ{E(ω)}, where Θand β are the given set of input matrix-signals and an arbitrary operator of the error-matrix E(ω) in the frequency domain. It is assumed the weight approximates the finite amplitude of the low-pass characteristic of the on-line or the wireless communication environment and, by this restriction, we can escape unwanted disturbance-noise in the frequency domain having a high peak with a finite energy.

In this paper, with respect to band-limited signals f(t) contained in a given set of signals Ξ, we present very general optimum continuous running approximations that minimize various continuous worst-case measures of running approximation error simultaneously. Firstly, as a means of proof, we consider a multi-legged-type signal m(t) that is a combined-signal of a given infinite number of one-dimensional band-limited signals hn(t) (n=...,-2,-1,0,1,2,...) in Ξ. A series of given finite segments σn(t) of hn(t) are arranged sequentially and make backbone of m(t). Sets of two other signals of hn(t) make feet of m(t). With respect to these hn(t), we consider a series of the Kida's optimum approximations, gn(t), each of which uses a given finite number of generalized sample values of hn(t). Also, we define a similar multi-legged-type approximation y(t) of m(t) using these gn(t). Secondly, under a slight modification of Ξ, when the backbone of m(t) itself is a band-limited signal f(t) in Ξ, we prove that backbone of y(t) becomes the corresponding optimum continuous running approximation of f(t). Although this paper treats pure theoretical topics, we believe that it is important for multi-paths communication and signal processing systems, such as MIMO systems, to present a fundamental method of constructing the optimum running approximation including various extended multi-paths transmission systems without using difficult high-level mathematical concept.

For a set of signals that each signal is defined by means of a certain spectrum-vector composed of a finite number of extended Fourier transforms of component waves, one of the authors presents an extended optimum approximation but a running approximation is not treated. In this paper, we show the outline of the result given in as a premise of the arguments, firstly. Then, under the conditions that the required time-interval in the approximation is wide but limited and the measures of error are continuous, we present the optimum running approximation for this set of signals by using a certain one-to-one correspondence between the error in the wide time-interval and the error in its small segment. It is shown that the presented running approximation minimizes various worst-case measures of approximation error simultaneously and the corresponding interpolation functions are obtained by solving sets of linear equations having constant coefficient-matrices. Finally, we present an example for a multi-input one-output system having separate pass band to eliminate the prescribed noise band.

We derive a new numerical solution of linear in-homogeneous partial differential equations (PDEs) from the optimum interpolation approximation theory on generalized multidimensional filter banks, based on the similarity between the linear inhomogeneous PDEs and the generalized multidimensional filter banks. We will prove that the proposed numerical solution satisfies a given linear inhomogeneous PDE and given initial/boundary conditions at all given sample points, based on the discrete orthogonality of the approximation theory. Because the numerical solution becomes the optimum approximation of the unknown exact solution of the given linear inhomogeneous PDE in the meaning of the optimum interpolation approximation theory, we can consider that the numerical solution is with high degree of accuracy.

The optimum space-limited interpolation functions are presented which minimize simultaneously the wide variety of measures of error defined independently in each separate block in the space variable domain. For this approximation, certain reciprocal relation holds. Although the quantization of the decimated sample values is contained in this discussion, the proposed approximation has the favorable property stated above. An interesting equivalent transformation for filter banks having a tree structure is considered. As applications, we present several examples of cosine modulated filter bank and linear phase filter banks designed by simple iterative linear approximation based on the reciprocal relation.

Extended interpolatory approximation method is presented for the n-dimensional (n-D) waves whose spectrums have the weighted Lpnorms smaller than a given positive number. The sample points are selected from a certain subset of the whole vertices of n-D parallelepipeds arranged periodically in Rn. It is assumed that the sample values contain small statistical error. The proposed method minimizes the measure of error which corresponds to the envelope of the standard deviations of the approximation errors. Some important conditions for the convergence of the proposed approximation are presented. Application to the design of n-D FIR filters is also given.