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Continuous Selections in Chebyshev Approximation

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Parametric Optimization and Approximation

Abstract

A survey on continuous selections for the set-valued metric projection onto finite-dimensional subspaces of C[a,b] is given. Some applications are discussed.

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References

  1. Berdyshev, V.I. (1975) Metric projection onto finite-dimensional subspaces of C and L, Hath. Zametki 18, 473–488 (Russian).

    Google Scholar 

  2. Blatt, H.-P., Nürnberger, G. and M. Sommer (1981–1982) A characterization of pointwise-Lipschitz-continuous selections for the metric projection. Numer. Funct. Anal, and Optimiz. 4 (2), 101–121.

    Article  Google Scholar 

  3. Blatt, H.-P., Nürnberger, G. and M. Sommer (1980) Pointwise-Lipschitz-continuous selections for the metric projection, in “Approximation Theory III” (E.W. Cheney, ed.), Academic Press, New York, 223–228.

    Google Scholar 

  4. Blatter, J. (1967) Zur Stetigkeit von mengenwertigen metrischen Projektionen. Schr. Rheinisch-Westfälischen Inst. Instrum. Math. Univ. Bonn, Ser. A, Nr. 16.

    Google Scholar 

  5. Blatter, J., Morris, P.D. and D.E. Wulbert (1968) Continuity of the set-valued metric projection. Math. Annalen 178, 12–24.

    Article  Google Scholar 

  6. Blatter, J. and L.L. Schumaker (1983) Continuous selections and maximal alternators for spline approximation. J. Approx. Theory 38, 71–80.

    Article  Google Scholar 

  7. Blatter, J. and L.L. Schumaker (1982) The set of continuous selections of a metric projection in C(X). J. Approx. Theory 36, 141–155.

    Article  Google Scholar 

  8. Brosowski, B. and R. Wegmann (1973) On the lower semicontinuity of the set-valued metric projection. J. Approx. Theory 8, 84–100.

    Article  Google Scholar 

  9. Brosowski, B., Deutsch, F. and G. Nürnberger (1980) Parametric approximation. J. Approx. Theory 29, 261–277.

    Article  Google Scholar 

  10. Brown, A.L. (1964) Best n-dimensional approximation to sets of functions. Proc. London Math. Soc. 14, 577–594.

    Article  Google Scholar 

  11. Brown, A.L. (1971) On continuous selections for metric projections in spaces of continuous functions. J. Functional Anal. 8 431–449.

    Article  Google Scholar 

  12. Brown, A.L. (1982) An extension to Mairhuber’s theorem. On metric projections and discontinuity of multivariate best uniform approximation. J. Approx. Theory 36, 156–172.

    Article  Google Scholar 

  13. Descloux, J. (1963) Approximations in LP and Chebyshev approximations. J. Soc. Indust. Appl. Math. 11, 1017–1026.

    Article  Google Scholar 

  14. Deutsch, F. and J. Lambert (1980) On continuity of metric projections. J. Approx. Theory 29, 116–131.

    Article  Google Scholar 

  15. Deutsch, F. and P. Kenderov (1980) When does the metric projection admit a continuous selection?, in “Approximation Theory III” (E.W. Cheney, ed.), Academic Press, New York, 327–333.

    Google Scholar 

  16. Deutsch, F. and P. Kenderov (1983) Continuous selections and approximate selections for set-valued mappings and applications to metric projections. SIAM J. Math. Anal, and Appl. 14.

    Google Scholar 

  17. Deutsch, F. (1983) A survey of metric selections, to appear in “Contemporary Mathematics” Ced. by R. Sine), Amer. Math. Soc. Providence.

    Google Scholar 

  18. Franchetti, C. and E.W. Cheney (1980) Best approximation problems for multivariate functions. Preprint.

    Google Scholar 

  19. Freud, G. (1958) Eine Ungleichung für Tschebyscheffsche Approximationspolynome. Acta. Sci. Math. (Szeged) 19, 162–164.

    Google Scholar 

  20. Garkavi, A.L. (1965) Almost Cebyshev systems of continuous functions. Izv. Vyss. Ucebn. Zaved. Matematika 45, 36–44 (Russian) [English transi, in: Amer. Math. Soc. Transl. 96 (1970), 177-187].

    Google Scholar 

  21. Krüger, H. (1980) A remark on the lower semicontinuity of the set-valued metric projection. J. Approx. Theory 28, 83–86.

    Article  Google Scholar 

  22. Lazar, A.J., Morris, P.D. and D.E. Wulbert (1969) Continuous selections for metric projections. J. Functional Anal. 3, 193–216.

    Article  Google Scholar 

  23. Mairhuber, J.C. (1956) On Haar’s theorem concerning Chebyshev approximation problems having unique solutions. Proc. Amer. Math. Soc. 7, 609–615.

    Google Scholar 

  24. Meinardus, G. (1967) Approximation of Functions: Theory and Numerical Methods, Springer, New York.

    Book  Google Scholar 

  25. Michael, E. (1956) Selected selection theorems. Amer. Math. Monthly 63, 233–237.

    Article  Google Scholar 

  26. Nürnberger, G. (1975) Dualität von Schnitten für die metrische Projektion und von Fortsetzungen kompakter Operatoren. Dissertation, Erlangen.

    Google Scholar 

  27. Nürnberger, G. (1977) Schnitte für die metrische Projektion. J. Approx. Theory 20 (1977), 196–219.

    Article  Google Scholar 

  28. Nürnberger, G. (1980) Nonexistence of continuous selections of the metric projection and weak Chebyshev systems. SIAM J. Math. Anal. 11, 460–467.

    Article  Google Scholar 

  29. Nürnberger, G. (1980) Continuous selections for the metric projection and alternation. J. Approx. Theory 28, 212–226.

    Article  Google Scholar 

  30. Nürnberger, G. and M. Sommer (1978) Weak Chebyshev subspaces and continuous selections for the metric projection. Trans. Amer. Math. Soc. 238, 129–138.

    Google Scholar 

  31. Nürnberger, G. and M. Sommer (1978) Characterization of continuous selections of the metric projection for spline functions. J. Approx. Theory 22, 320–330.

    Article  Google Scholar 

  32. Nürnberger, G. and M. Sommer (1983) A Remez type algorithm for spline functions. Numer. Math. 41, 117–146.

    Article  Google Scholar 

  33. Nürnberger, G., Schumaker, L.L., Sommer, M. and H. Strauß, Approximation by generalized splines. To appear.

    Google Scholar 

  34. Respess, J.R. and E.W. Cheney (1982) On Lipschitzian proximity maps, in “Nonlinear Analysis and Applications” (ed. by J.H. Burry and S.P. Singh), Vol. 80, Lecture Notes in Pure and Applied Hath., Dekker, New York.

    Google Scholar 

  35. Rice, J.R. (1969) The Approximation of Functions, Vol. II, Addision-Wesly, Reading, Massachusetts.

    Google Scholar 

  36. Singer, I. (1974) The Theory of Best Approximation and Functional Analysis, CBMS 13, SIAM, Philadelphia.

    Book  Google Scholar 

  37. Sommer, M. (1979) Continuous selections of the metric projection for 1-Chebyshev spaces. J. Approx. Theory 26, 46–53.

    Article  Google Scholar 

  38. Sommer, M. (1980) Characterization of continuous selections for the metric projection for generalized splines. SIAM J. Math. Anal. 11, 23–40.

    Article  Google Scholar 

  39. Sommer, M. (1982) Existence of pointwise-Lipschitz-continuous selections of the metric projection for a class of Z-spaces. J. Approx. Theory 34, 115–130.

    Article  Google Scholar 

  40. Sommer, M. (1980) Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces. Trans. Amer. Math. Soc. 260, 403–409.

    Article  Google Scholar 

  41. Sommer, M. (1980) Continuous selections for metric projections, in “Quantitative Approximation” (ed. by R. DeVore and K. Scherer), Academic Press, New York, 301–317.

    Google Scholar 

  42. Sommer, M. (1982) Characterization of continuous selections of the metric projection for a class of weak Chebyshev spaces. SIAM J. Math. Anal. 13, 280–294.

    Article  Google Scholar 

  43. Sommer, M. (1983) Lp-approximations and Chebyshev approximations in subspaces of spline functions, in “Approximation and Optimization in Mathematical Physics” (ed. by B. Brosowski and E. Martensen), Peter Lang, Frankfurt, Bern, 105–139.

    Google Scholar 

  44. Sommer, M. Continuous selections and convergence of best Lp-approximations in subspaces of spline functions. To appear in Numer. Funct. Anal, and Optimiz.

    Google Scholar 

  45. Sommer, M. and H. Strauß (1977) Eigenschaften von schwach tschebyscheffsehen Räumen. J. Approx. Theory 21, 257–268.

    Article  Google Scholar 

  46. Stover, V. (1981) The strict approximation and continuous selections for the metric projection. Dissertation, San Diego.

    Google Scholar 

  47. Strauß, H. Characterization of strict approximations in subspaces of spline functions. To appear in Numer. Funct. Anal, and Optimiz.

    Google Scholar 

  48. Strauß, H. An Algorithm for the computation of strict approximations in subspaces of spline functions. To appear in J. Approx. Theory.

    Google Scholar 

  49. Wegmann, R. (1973) Some properties of the peak-set-mapping. J. Approx. Theory 8, 262–284.

    Article  Google Scholar 

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Nürnberger, G., Sommer, M. (1984). Continuous Selections in Chebyshev Approximation. In: Brosowski, B., Deutsch, F. (eds) Parametric Optimization and Approximation. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 72. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6253-0_16

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  • DOI: https://doi.org/10.1007/978-3-0348-6253-0_16

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-6255-4

  • Online ISBN: 978-3-0348-6253-0

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