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Network approximation of input-output maps and functionals. (English) Zbl 0876.94055
Summary: We give results concerning the problem of approximating the input-output maps of nonlinear discrete-time approximately finite-memory systems. Here the focus is on the linear dynamical parts of the approximating structures, and we give examples showing that these linear parts can be derived from a single prespecified function that meets certain conditions. This is done in the context of an approximation theorem in which attention is focused on what we call “basic sets”. We also consider the related but very different problem of approximating nonlinear functionals using lattice operations or the usual linear ring operations. For this problem we give criteria, not just sufficient conditions, for approximation on compact subsets of reflexive Banach spaces (any Hilbert space is a reflexive Banach space).

MSC:
94C05 Analytic circuit theory
41A30 Approximation by other special function classes
93C05 Linear systems in control theory
46C99 Inner product spaces and their generalizations, Hilbert spaces
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[1] G. Bachman and L. Narici,Functional Analysis, New York: Academic Press, 1966. · Zbl 0141.11502
[2] T. Chen and H. Chen, Approximations of continuous functionals by neural networks with application to dynamical systems,IEEE Transactions on Neural Networks, vol. 4, no. 6, pp. 910-918, November 1993. · doi:10.1109/72.286886
[3] ?, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its applications to dynamic systems,IEEE Transactions on Neural Networks, vol. 6, no. 4, pp. 911-917, July 1995. · doi:10.1109/72.392253
[4] G. Cybenko, Approximation by superposition of a single function,Mathematics of Control, Signals and Systems, vol. 2, pp. 303-314, 1989. · Zbl 0679.94019 · doi:10.1007/BF02551274
[5] B. De Vries, J. C. Principe, and P. Guedes de Oliveira, Adaline with adaptive recursive memory,Proceedings of IEEE-SP Workshop on Neural Networks for Signal Processing, pp. 101-110, 1991.
[6] H. N. Mhaskar and C. A. Micchelli, Approximation by superposition of sigmoidal and radial basis functions,Advances in Applied Mathematics, vol. 3, pp. 350-373, 1992. · Zbl 0763.41015 · doi:10.1016/0196-8858(92)90016-P
[7] J. Park and I. W. Sandberg, Approximation and radial-basis function networks,Neural Computation, vol. 5, no. 2, pp. 305-316, March 1993. · doi:10.1162/neco.1993.5.2.305
[8] W. Rudin,Fourier Analysis on Groups, New York: Interscience, 1962. · Zbl 0107.09603
[9] I. W. Sandberg, Structure theorems for nonlinear systems,Multidimensional Systems and Signal Processing, vol. 2, no. 3, pp. 267-286, 1991. (See also the Errata in vol. 3, no. 1, p. 101, 1992). A conference version of the paper appears inIntegral Methods in Science and Engineering-90, Proceedings of the International Conference on Integral Methods in Science and Engineering, Arlington, Texas, May 15-18, 1990, ed. A. H. Haji-Sheikh, New York: Hemisphere Publishing, pp. 92-110, 1991. · Zbl 0754.93032 · doi:10.1007/BF01952236
[10] ?, Approximation theorems for discrete-time systems,IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, vol. 38, no. 5, pp. 564-566, May 1991.
[11] ?, Approximately-finite memory and the theory of representations,International Journal of Electronics and Communication, vol. 46, pp. 191-199, April 1992.
[12] ? Uniform approximation and the circle criterion,IEEE Transactions on Automatic Control, vol. 38, no. 10, pp. 1450-1458, October 1993. · Zbl 0790.93091 · doi:10.1109/9.241560
[13] ?, General structures for classification,IEEE Transactions on Circuits andSystems-I: Fundamental Theory and Applications, vol. 41, no. 5, pp. 372-376, May 1994. · Zbl 0808.94005 · doi:10.1109/81.296334
[14] ?, Notes on weighted norms and network approximation of functionals,IEEE Transactions on Circuits and Systems I, vol. 43, no. 7, pp. 600-601, July 1996. · doi:10.1109/81.508182
[15] M. H. Stone, A generalized Weierstrass approximation theorem, InStudies in Modern Analysis, ed. R. C. Buck, vol. 1 ofMAA Studies in Mathematics, pp. 30-87, Englewood Cliffs, N.J.: Prentice-Hall, March 1962.
[16] V. Volterra,Theory of Functionals and of Integral and Integro-Differential Equations, New York: Dover, 1959. · Zbl 0086.10402
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