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Linear selections for the metric projection. (English) Zbl 0522.41027


MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50 Best approximation, Chebyshev systems
41A52 Uniqueness of best approximation
Full Text: DOI

References:

[1] Ando, T., Contractive projections in \(L_p\) spaces, Pacific J. Math., 17, 391-405 (1966) · Zbl 0192.23304
[2] Aronszajn, N.; Smith, K. T., Invariant subspaces of completely continuous operators, Ann. of Math., 60, 345-350 (1954) · Zbl 0056.11302
[3] Blatt, H. P.; Nürnberger, G.; Sommer, M., A characterization of pointwise-Lipschitz-continuous selections for the metric projection (February 1981), preprint · Zbl 0488.41034
[4] Brown, A. L., Best \(n\)-dimensional approximation to sets of functions, (Proc. London Math. Soc., 14 (1964)), 577-594 · Zbl 0129.04702
[5] Brown, A. L., On continuous selections for metric projections in spaces of continuous functions, J. Funct. Anal., 8, 431-449 (1971) · Zbl 0224.41013
[7] Cheney, E. W.; Wulbert, D. E., The existence and unicity of best approximations, Math. Scand., 24, 113-140 (1969) · Zbl 0186.18701
[8] Cheney, E. W.; Price, K. H., Minimal projections, (Talbot, A., Approximation Theory (1970), Academic Press: Academic Press New York), 261-289 · Zbl 0217.16202
[9] Daugavet, I. K., A property of completely continuous operators in the space \(C\), Uspehi Mat. Nauk, 18, 157-158 (1963) · Zbl 0138.38603
[10] Deutsch, F. R.; Maserick, P. H., Applications of the Hahn-Banach theorem in approximation theory, SIAM Rev., 9, 516-530 (1967) · Zbl 0166.10501
[11] Deutsch, F.; Kenderov, P., When does the metric projection admit a continuous selection?, (Cheney, E. W., Approximation Theory III (1980), Academic Press: Academic Press New York), 327-333 · Zbl 0482.41029
[12] Deutsch, F.; Kenderov, P., Continuous selections and approximate selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal., 14 (1983) · Zbl 0518.41031
[13] Fakhoury, H., Sélections linéaires associées au théorème de Hahn-Banach, J. Funct. Anal., 11, 436-452 (1972) · Zbl 0252.46023
[14] Fakhoury, H., Existence d’une projections continue de meilleure approximation dans certains espaces de Banach, J. Math. Pures Appl., 53, 1-16 (1974) · Zbl 0286.46023
[15] Foias, C.; Singer, I., Points of diffusion of linear operators and almost diffuse operators in spaces of continuous functions, Math. Z., 87, 434-450 (1965) · Zbl 0132.09904
[16] Hirschfeld, R. A., On best approximations in normed vector spaces, II, Nieuw Arch. Wisk., 6, 99-107 (1958) · Zbl 0089.31404
[17] Holmes, R. B., A Course on Optimization and Best Approximation, (Lecture Notes in Mathematics No. 257 (1972), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0234.46016
[18] Holmes, R. B.; Kripke, B. R., Smoothness of approximation, Michigan Math. J., 15, 225-248 (1968) · Zbl 0177.16201
[19] James, R. C., Inner products in normed linear spaces, Bull. Amer. Math. Soc., 53, 559-566 (1947) · Zbl 0041.43701
[20] James, R. C., Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61, 265-292 (1947) · Zbl 0037.08001
[22] Kripke, B. R.; Rivlin, T. J., Approximation in the metric of \(L^1(X, μ)\), Trans. Amer. Math. Soc., 115, 101-122 (1965) · Zbl 0135.35101
[23] Krüger, H., A remark on the lower semi-continuity of the set-valued metric projection, J. Approx. Theory, 28, 83-86 (1980) · Zbl 0428.41025
[24] Lazar, A. J., Spaces of affine continuous functions on simplexes, Trans. Amer. Math. Soc., 134, 503-525 (1968) · Zbl 0174.17102
[25] Lazar, A. J.; Morris, P. D.; Wulbert, D. E., Continuous selections for metric projections, J. Funct. Anal., 3, 193-216 (1969) · Zbl 0174.17101
[26] Liapunov, A., Sur les fonctions-vecteurs complètement additives, Bull. Acad. Sci. USSR Ser. Math., 4, 465-478 (1940) · Zbl 0024.38504
[27] Lindenstrauss, J., Extension of Compact Operators, Mem. Amer. Math. Soc., 48 (1964) · Zbl 0141.12001
[28] Lindenstrauss, J.; Tzafriri, L., On the complemented subspaces problem, Israel J. Math., 9, 263-269 (1971) · Zbl 0211.16301
[29] Michael, E., Continuous selections, I, Ann. of Math., 63, 361-382 (1956) · Zbl 0071.15902
[30] Morris, P. D., Metric projections onto subspaces of finite codimension, Duke Math. J., 35, 799-808 (1968) · Zbl 0167.42301
[31] Morris, P. D., Chebyshev subspaces of \(L_1\) with linear metric projections, J. Approx. Theory, 29, 231-234 (1980) · Zbl 0466.41015
[32] Nürnberger, G., Schnitte für die metrische Projektion, J. Approx. Theory, 2, 196-219 (1977) · Zbl 0379.41016
[33] Nürnberger, G., Continuous selections for the metric projection and alternation, J. Approx. Theory, 28, 212-226 (1980) · Zbl 0426.41035
[34] Nürnberger, G.; Sommer, M., Weak Chebyshev subspaces and continuous selections for the metric projection, Trans. Amer. Math. Soc., 238, 129-138 (1978) · Zbl 0389.41017
[35] Nürnberger, G.; Sommer, M., Characterization of continuous selections of the metric projection for spline functions, J. Approx. Theory, 22, 320-330 (1978) · Zbl 0384.41008
[36] Rudin, W.; Smith, K. T., Linearity of best approximation: A characterization of ellipsoids, (Proc. Nederl. Akad. Wet. Ser. A, 64 (1961)), 97-103 · Zbl 0098.08002
[37] Singer, I., The Theory of Best Approximation and Functional Analysis, (CBMS 13 (1974), SIAM: SIAM Philadelphia) · Zbl 0291.41020
[39] Stoer, J., Über die Existenz linearer Approximationsoperatoren, (Collatz, L.; Meinardus, G.; Unger, H., Funktionalanalysis, Approximationstheorie, Numerische Mathematik (1967), Birkhäuser: Birkhäuser Basel) · Zbl 0153.16401
[40] Wulbert, D. E., Some complemented function spaces in \(C(X)\), Pacific J. Math., 24, 589-602 (1968) · Zbl 0157.44102
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