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Approximation by bernstein systems

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Abstract

LetK denote an arbitrary compact subset of realL 2. Letf be any causal continuous function onL 2. Then there is a linear differential system:ż(t)=A(t)z(t)+B(t)u(t), and a memoryless polynomic state to output mapw(t)=φ(z(t),t) such that the system,\(\hat f\), thereby computed satisfies

$$\mathop {\sup }\limits_{u \in K} ||f(u) - \hat f(u)||< \varepsilon$$

whereε >0 is arbitrary. This and other results are developed.

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References

  1. V. Volterra, Theory of Functionals, Dover Publications Inc., New York, 1959.

    Google Scholar 

  2. E. Bedrosian andS. O. Rice, The output properties of Volterra systems driven by harmonic and Gaussian inputsProc. IEEE,59 (1971), 1688–1707.

    Google Scholar 

  3. R. B. Parente, Nonlinear differential equations and analytic systems theory,SIAM J. Appl. Math. 18 (1970), 41–66.

    Google Scholar 

  4. H. L. Van Trees, Functional techniques for the analysis of the nonlinear behavior of phaselocked loops,Proc. IEEE 52, 894–911.

  5. R. W. Brockett, Volterra series and gemoetric control theory,Automatica 12 (1976).

  6. C. Bruni, G. Dipillo, andG. Koch, On the Mathematical Models of Bilinear Systems,Ricerche di Automatica 2 (1976), 11–26.

    Google Scholar 

  7. A. H. Krener, Linearization and bilinearization of control systems,Proc. 1974 Allerton Conf. on Cir. and Sys. Th. (1974), 834–843.

  8. E. G. Gilbert,Volterra Series and the Response of Nonlinear Differential Systems, Trans. Conf. on Inf. Sci. and Systems, Johns Hopkins University, March 1976.

  9. R. S. Cobbold, Theory and Applications of Field-Effect Transistors, Wiley-Interscience, Section 7.1.3, New York, 1970.

  10. K. Weierstrass, Uber die analystische darshell-bankeit sogenannter willkürlicher funktionen reeler argumente,Math. Werke III Bd. (1903).

  11. M. Frechet, Sur les fonctionelles continues,Ann. de l'Ecole Normale Sup. Third Series27 (1910).

  12. S. Bernstein, DéMonstration du Théoréme de Weierstrass, fondeé sur le calcul des probabilitiés,Com. Soc. Math. (Kharkow)13 (1912–13).

  13. M. H. Stone The Generalized Weierstrass Approximation Theorem,Math. Mag. 21 (1948), 167–183.

    Google Scholar 

  14. P. M. Prenter, A. Weierstrass Theorem for real separable Hilbert spaces,J. Approximation Theory 3, (1970), 341–351.

    Google Scholar 

  15. P. M. Prenter, A Weierstrass Theorem for real normed linear space,Bull. American Math. Soc. 75 (1969), 860–862.

    Google Scholar 

  16. N. W. Ahmed, An approximation Theorem for Continuous Functions onL p spaces, TR-73–18, Univ. of Ottawa, Ottawa, Can., Nov. 1975.

  17. W. A. Porter, T. M. Clark, andR. M. DeSantis, Causality structure and the Weierstrass Theorem,J. Math. Anal. Appli. 52 (1975).

  18. W. A. Porter, The common causality structure of multilinear maps and their multipower forms,J. Math. Anal. Appl. 57 (1977).

  19. W. A. Porter, Data interpolation, causality structure, and system identification,Information and control 29 (1975).

  20. W. A. Porter, Causal Realization from Input-Output pairs,SIAM J. Control, to appear 1976.

  21. R. Saeks,Resolution Space, Operations and Systems Lecture Notes, Springer-Verlag, New York, 1973.

    Google Scholar 

  22. I. Z. Gohberg and M. G. Krein, Theory of Volterra operators in Hilbert space and applications,Amer. Math. Soc. 24 (1970).

  23. G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, 1953.

  24. R. W. Brockett, Nonlinear systems and differential geometry,Proceeding IEEE 64 (1976), 61–72.

    Google Scholar 

  25. D. Jackson, The theory of approximation,Amer. Math. Soc. Coll. Publ. 11 (1930), New York.

  26. Ch. J. DeLa Vallée-Poussin, Lecons sur l'approximation, Paris, 1919.

  27. T. Popoviciu, Sur l'approximation des fonctions convexes d'ordre supérieur,Mathematica (Cluj) 10 (1935), 49–54.

    Google Scholar 

  28. M. Schetzen, Measurement of the kernals of a nonlinear system of finite order,Int. J. Control 1 (1965), 257–263.

    Google Scholar 

  29. W. A. Porter, Causal parameter identification, to appearInt. J. Sys. Sci. (1978).

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Authors and Affiliations

  1. Department of Electrical Engineering, Louisiana State University, 70803, Baton Rouge, LA, USA

    William A. Porter

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  1. William A. Porter
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Supported in part by the United States Air Force Office of Scientific Research, Grant No. 78-3500.

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Porter, W.A. Approximation by bernstein systems. Math. Systems Theory 11, 259–274 (1977). https://doi.org/10.1007/BF01768480

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  • DOI: https://doi.org/10.1007/BF01768480

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