A function approximation based on an orthonormal wave function expansion in a complex space is derived. Although a probability density function (PDF) cannot always be expanded in an orthogonal series in a real space because a PDF is a positive real function, the function approximation can approximate an arbitrary PDF with high accuracy. It is applied to an actor-critic method of reinforcement learning to derive an optimal policy expressed by an arbitrary PDF in a continuous-action continuous-state environment. A chaos control problem and a PDF approximation problem are solved using the actor-critic method with the function approximation, and it is shown that the function approximation can approximate a PDF well and that the actor-critic method with the function approximation exhibits high performance.

This paper introduces a new recursive factorization of the polynomial, 1-zN, over the real numbers when N is an even composite integer. The recursive factorization is applied for efficient computation of the discrete Fourier transform (DFT) and the cyclic convolution of real sequences with highly composite even length.

THe decimation-in-time (DIT) and the decimation-in-frequency (DIF) algorithms are the most well-known fast algorithms for computing the discrete Fourier transform(DFT). These algorithms constitute the basis of the fast Fourier transform (FFT) implementations, including the pipeline implementation and other parallel configurations. This paper derives an alternative generalization of the algorithms which applies for sequences whose lengths are not a power of two. The treatment is consistent with the radix-two DIF and DIT algorithms, and the generalization is useful for utilizing the accumulated technologies of the FFT algorithm for such sequences.

Relating to the problem of suppressing the immanent redundancy contained in an image with out vitiating the quality of the resultant approximation, the interpolation of multi-dimensional signal is widely discussed. The minimization of the approximation error is one of the important problems in this field. In this paper, we establish the optimum interpolatory approximation of multi-dimensional orthogonal expansions. The proposed approximation is superior, in some sense, to all the linear and the nonlinear approximations using a wide class of measures of error and the same generalized moments of these signals. Further, in the fields of information processing, we sometimes consider the orthonormal development of an image each coefficient of which represents the principal featurr of the image. The selection of the orthonormal bases becomes important in this problem. The Fisher's criterion is a powerful tool for this class of problems called declustering. In this paper, we will make some remarks to the problem of optimizing the Fisher's criterion under the condition that the quality of the approximation is maintained.

Extended interpolatory approximations are discussed for some classes of n-dimensional stochastic signals expressed as the orthogonal expansions with respect to a given set of orthonormal functions. We assume that the norm of the weighted mutual correlation function of the signal is smaller than a given positive number. The presented approximation has the minimum measure of approximation error among all the linear and nonlinear statistical approximations using the similar measure of error and the same generalized moments of these signals.