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Constrained best approximation in Hilbert space

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Abstract

In this paper we study the characterization of the solution to the extremal problem inf{‖x‖x ∈C ∩M}, wherex is in a Hilbert spaceH, C is a convex cone, andM is a translate of a subspace ofH determined by interpolation conditions. We introduce a simple geometric property called the “conical hull intersection property” that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation. Our approach naturally lends itself to considering the data cone as opposed to the constraint cone. A nice characterization of the solution occurs, for example, if the data vector associated withM is an interior point of the data cone.

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References

  1. V. E. Anderson, P. A. Ivert (1987):Constrained interpolants with minimal W k,p-norm. J. Approx. Theory,49:283–288.

    Google Scholar 

  2. A. Ben-Israel (1969):Linear equations and inequalities on finite-dimensional, real or complex, vector spaces: a unified theory. J. Math. Anal. Appl.,27:367–389.

    Google Scholar 

  3. C. de Boor (1973):The quasi-interpolant as a tool in elementary polynomial spline theory. In: Approximation Theory (G. G. Lorentz, ed.). New York: Academic Press, 269–276.

    Google Scholar 

  4. J. M.Borwein, H.Wolkowicz (to appear):A simple constraint qualification in infinite-dimensional programming. Math Programming.

  5. J. P. Butler, J. A. Reeds, S. V. Dawson (1981):Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing. SIAM. J. Numer. Anal.,10:381–397.

    Google Scholar 

  6. C. K. Chui (1988): Multivariate Splines: Theory and Applications. CBMS-NSF Lecture Series. Philadelphia: SIAM.

    Google Scholar 

  7. J. Diestel (1984): Sequences and Series in Banach Spaces. New York: Springer-Verlag.

    Google Scholar 

  8. A. L.Dontchev, G.Illiev, M. M.Konstantinov (1985):Constrained interpolation via optimal control. In: Proceedings of the Fourteenth Spring Conference of the Union of Bulgarian mathematicians, Sunny Beach, April 6–9, 1985, pp. 385–392.

  9. P. R. Halmos (1967): A Hilbert Space Problem Book. New York: Van-Nostrand.

    Google Scholar 

  10. R. B. Holmes (1975): Geometric Functional Analysis and Its Applications. New York: Springer-Verlag.

    Google Scholar 

  11. G. Illiev, W. Pollul (1984):Convex interpolation with minimal L ∞-norm of the second derivative. Math Z.,186:49–56.

    Google Scholar 

  12. G.Illiev, W.Pollul (1984):Convex interpolation by functions with minimal L p -norm (1<p<∞)of the kth derivative. In: Proceedings of the Thirteenth Spring Conference of the Union of Bulgarian Mathematicians, Sunny Beach, April 6–9, 1984.

  13. L.Irvine (1985): Constrained Interpolation. Masters Thesis, Old Dominion University.

  14. L. D. Irvine, S. P. Marin, P. W. Smith (1986):Constrained interpolation and smoothing. Constr. Approx.,2:129–151.

    Google Scholar 

  15. D. G. Luenberoer (1968): Optimization by Vector Space Methods. New York: Wiley.

    Google Scholar 

  16. C. A. Micchelli, P. W. Smith, J. Swetits, J. D. Ward (1985):Constrained L p Approximation. Constr. Approx.,1:93–102.

    Google Scholar 

  17. C. A. Micchelli, F. Utreras (1988):Smoothing and interpolation in a convex subset of a Hilbert space. SIAM J. Sci. Statist. Comput.,9:728–746.

    Google Scholar 

  18. D. Moskovitz, L. L. Dines (1939):Convexity in a linear space with an inner product. Duke Math. J.,5:520–534.

    Google Scholar 

  19. G.Opfer, H. J.Oberle (to appear):The derivation of cubic splines with obstacles by methods of optimization and optimal control. Numer. math.

  20. R. T. Rockafellar (1970): Convex Analysis. Princeton, NJ: Princeton University Press.

    Google Scholar 

  21. P. W. Smith, J. D. Ward (1988):Distinguished solutions to an L ∞ minimization problem. Approx. Theory Appl.,4:29–40.

    Google Scholar 

  22. P. W. Smith, H. Wolkowicz (1986):A nonlinear equation for linear programming. Math Programming,34:235–238.

    Google Scholar 

  23. J. Ward (1986):Some constrained approximation problems. in: Approximation Theory V (C. K. Chui, L. L. Schumaker, J. D. Ward eds.) New York: Academic Press, pp. 211–229.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 77843-3368, College Station Texas, USA

    Charles K. Chui

  2. Department of Mathematics, The Pennsylvania State University, 16802, University Park, Pennsylvania, USA

    Frank Deutsch

  3. Department of Mathematics, Texas A&M University, 77843-3368, College Station Texas, USA

    Joseph D. Ward

Authors
  1. Charles K. Chui
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  2. Frank Deutsch
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  3. Joseph D. Ward
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Communicated by Charles A. Micchelli.

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Chui, C.K., Deutsch, F. & Ward, J.D. Constrained best approximation in Hilbert space. Constr. Approx 6, 35–64 (1990). https://doi.org/10.1007/BF01891408

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  • DOI: https://doi.org/10.1007/BF01891408

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