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Best approximation in normed linear spaces

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Constructive Aspects of Functional Analysis

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 57))

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Abstract

Here we want to present briefly some results, problems and directions of research in the modern theory of best approximation, i.e. in which the methods of functional analysis are applied in a consequent manner. In this theory the functions to be approximated and the approximating functions are regarded as elements of certain normed linear (or, more generally, of certain metric) spaces of functions and best approximation amounts to finding “nearest points”. The advantages and a brief history of this modern point of view have been described in the Introduction to the monograph [82] and we shall not repeat them here; the material which will be presented in the sequel will be convincing enough, we hope, to prove again that the theory of best approximation in normed linear spaces constitutes both a rigorous theoretical foundation for the existing classical and more recent results in various concrete spaces and a powerfull tool for obtaining new results, solving the new problems which appear.

Since June 1966, when the Romanian version of the monograph [82] has gone to print, the theory of best approximation in normed linear spaces has developed rapidly and the number of papers in this field is growing continuously. However, except the expository paper up to 1967, by A. L. Garkavi [31], which appeared in 1969, and the bibliography [23] compiled by F. Deutsch and J. Lambert in 1970, we know of no other survey material on these new developments. One of the aims of our course is to fill this gap to a certain extent, by presenting much new material which appeared after the monograph [82]. In this respect the present course, though self-contained, may be regarded as an up to date complement to the monograph [82] however, the bibliography does not aim at begin complete, but wants merely to give useful orientation to the reader. Naturally, since another aim of the course is to introduce the non-specialits to this field, some overlapping with the material of the monograph [82] is unavoidable; however, even this part is presented here in a slightly improved way.

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References

  1. N. I. Ahiezer, Lectures on the theory of approximation. Second edition, Moscow (1965) [Russian].

    Google Scholar 

  2. E. Asplund, Čebyšev sets in Hilbert space. Trans. Amer. Math. Soc. 144 (1969), 236–240.

    MathSciNet  Google Scholar 

  3. D. A. Ault, F. R. Deutsch, PD. Morris and J. E. Olson, Interpolating subspaces in approximation theory. J. Approx. Theory 3 (1970), 164–182.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. H. Biggs, F. R. Deutsch, R. E. Huff, P. D. Morris and J. E. Olson, Interpolating subspaces in l 1 -spaces (to appear).

    Google Scholar 

  5. J. Blatter, Zur Stetigkeit von mengenwertigen metrischen Projektionen. Schriften desRheinisch-Westf. Inst. fur Instrum. Math. Univ. Bonn, Ser.A,.No. 16(1967), 17–38.

    MathSciNet  Google Scholar 

  6. J Blatter, Approximation und Selection. Habilitationsschrift, Bonn (1969).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. O. Brandt, Geometrische Approximations theorie in normier-ten Vektorräumen. Schriften des Rheinisch-Westf. Inst, für Instrum. Math. Univ. Bonn. ser. A, No. 18 (1968), 1–36.

    Google Scholar 

  9. B. Brosowski, Nicht-lineare Tschebyscheff-Approximation. Bibliogr. Inst. Hochschulskripten Bd. 808/808a, Mannheim (1968).

    Google Scholar 

  10. B. Brosowski, Einige Bemerkungen zum verallgemeinerten Kolmo-goroffschen Kriterium. Funktionalanalytische Methoden der numerischen Mathematik. ISNM 12, Birkhäuser Verlag (1969), 25.–34.

    Google Scholar 

  11. B. Brosowski, Nichtlineare Approximation in normierten Vek-torräumen. Abstract spaces and approximation. ISNM 10,Birkhäuser-Verlag (1969), 140–159.

    MathSciNet  Google Scholar 

  12. B. Brosowski, Fixpunktsätze in der Approximations theorie. Mathematica II(34) (1969), 195–220.

    MathSciNet  Google Scholar 

  13. B. Brosowski, K.-H. Hoffmann, E. Schäfer und H. Weber, Stetigkeitssätze für Metrische Projektionen. Iterationsverfahren. Numerische Mathematik. Approximations-theorie. ISNM 15, Birkhäuser-Verlag (1970), 11–17.

    Google Scholar 

  14. B. Brosowski, K.-H. Hoffmann, E. Schäfer und H. Weber, Metrische Projektionen auf lineare Teilräume von Co [Q, h]. Iterationsverfahren. Numerische Mathematik. Approximationstheorie. ISNM 15, Birkäuser-Ver -lag (1970), 19–27.

    Google Scholar 

  15. B. Brosowski und R. Wegmann, Charakterisierung bester Approximationen in normierten Vektorräumen. J. Approx. Theory 3 (1970), 369–397.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Browder, Multivalued monotone nonlinear mappings and duality mappings in Banach spaces. Trans. Amer. Math. Soc. 118(1965), 338–351.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. A. L. Brown, On continuous selections for metric projections in spaces of continuous functions (to appear).

    Google Scholar 

  19. E. W. Cheney, Introduction to approximation theory. Mc Graw Hill, New York (1966).

    MATH  Google Scholar 

  20. E.W. Cheney and D. E. Wulbert, Existence and unicity of best approximations. Math. Scand. 24(1969), 113–140.

    MATH  MathSciNet  Google Scholar 

  21. M.M. Day, Normed linear spaces. Springer-Verlag, Berlin-Göttingen-Heidelberg (1962).

    MATH  Google Scholar 

  22. F. R. Deutsch and P. H. Maserick. Applications of the Hann-Banach theorem in approximation theory. SLAM Rev. 9 (1967), 516–530.

    Article  MATH  MathSciNet  Google Scholar 

  23. F. R. Deutsch and J. Lambert, A bibliography on metric projections (mimeographed).

    Google Scholar 

  24. A. I. Dubovitskii and A. A. Miliutin, Extremum problems in the presence of restrictions. Ž. Vycisl. Mat. i Mat. Fiz. 5(1965), 395–453 [Russian].

    MathSciNet  Google Scholar 

  25. N. Dunford and J. Schwartz, Linear operators. Part. I :General theory. Interscience Publ., New York (1958).

    Google Scholar 

  26. M. Edelstein, Nearest points of sets in uniformly convex Banach spaces. J. London Math. Soc. 43(1968), 375–377.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Edelstein, A note on nearest points. Quarterly J. Math. 21 (1970), 403–406.

    Article  MATH  MathSciNet  Google Scholar 

  28. N. V. Efimov and S. B. Stečkin, Some properties of Čebyšev sets. Doklady Akad. Nauk SSSR 18 (1958), 17–19[Russian].

    Google Scholar 

  29. G. Ewald, D. G. Larman and C.A.Rogers, The directions of the line segments and of the n-dimensional balls on the boundary of a convex body in Euclidean space (to appear).

    Google Scholar 

  30. Ky Fan and I. Glicksberg, Some geometric properties of the spheres in a normed linear space. Duke Math. J. 52(1958), 553–568.

    Article  MathSciNet  Google Scholar 

  31. A. L. Garkavi, The theory of best approximation in normed linear spaces. Mathematical analysis 1967, Moscow (1969), 75–132 [Russian].

    Google Scholar 

  32. A.L. Garkavi, The problem of Helly and best approximation in the space of continuous functions. Izevestija Akad. Nauk SSSR 31 (1967), 641–656 [Russian].

    MATH  MathSciNet  Google Scholar 

  33. A.L. Garkavi, Compact admitting Čebyšev systems of measures. Matem. Sbornik 74 (116) (1967), 209–217 [Russian].

    MathSciNet  Google Scholar 

  34. A. L. Garkavi. Characterization of Čebyšev subspaces of finite codimension L Matem. Zametki 7(1970), 155–163 (1970) [Russian].

    MathSciNet  Google Scholar 

  35. G. Godini, Best approximation in normed linear spaces by elements of convex cones. Studii si cercet. mat. 21(1969), 931–936 (1969) [Romanian].

    MATH  MathSciNet  Google Scholar 

  36. A. Haar, Die Minkowskische Geometrie und die Annäherung an stetige Funktionen. Math. Ann. 78(1918). 294–311.

    Article  MathSciNet  Google Scholar 

  37. S. Ia. Havinson, On approximation by elements of convex sets. Doklady Akad. Nauk SSSR 172(1967), 294–297 [Russian].

    MathSciNet  Google Scholar 

  38. R. B. Holmes, On the continuity of best approximation operators. Symp. on infinite dimensional topology. Princeton University Press (to appear).

    Google Scholar 

  39. R. B. Holmes and B. R. Kripke, Smoothness of approximation. Michigan Math. J. 15(1968), 225–248.

    Article  MATH  MathSciNet  Google Scholar 

  40. Y. Ikebe, A characterization of Haar Subspaces in C [a, b]. Proc. Japan Acad. 44(1968), 219–220.

    Article  MATH  MathSciNet  Google Scholar 

  41. Y. Ikebe., A characterization of best Tchebycheff approximations in function spaces. Proc. Japan Acad. 44(1968) 485–488.

    Article  MATH  MathSciNet  Google Scholar 

  42. A. D. Ioffe and V. M. Tihomirov, Duality of convex functions and extremal problems. Uspehi Mat. Nauk 23,6(144) (1968), 51–116 [Russian].

    MathSciNet  Google Scholar 

  43. R. C. James, Characterizations of reflexivity. Studia Math. 23(1964), 205–216.

    MATH  MathSciNet  Google Scholar 

  44. M. I. Kadec, Topological equivalence of uniformly convex spaces. Uspehi Mat. Nauk. 10, 4(66) (1955), 137– 141 [Russian].

    MathSciNet  Google Scholar 

  45. V. Klee, The support property of a convex set in a linear normed space. Duke Math. J. 15 (1948), 767–772.

    Article  MATH  MathSciNet  Google Scholar 

  46. V. Klee, Research problem no. 5, Bull. Amer. Math. Soc. 63 (1957), 419.

    Article  MathSciNet  Google Scholar 

  47. V. Klee, Convexity of Chebyshev sets, Math. Ann. 142 (1961), 292–304.

    Article  MATH  MathSciNet  Google Scholar 

  48. V. Klee, Remarks on nearest points in normed linear spaces. Proc. Coll. on convexity (Copenhagen, 1965), Univ. of Copenhagen (1966), 168–176.

    Google Scholar 

  49. A. N. Kolmogorov, A remark on the polynomials or P. L. Čebyšev deviating the least from a given function. Uspehi Mat. Nauk 3,1(23) (1948), 216–221 [Russian].

    MathSciNet  Google Scholar 

  50. C.A. Kottman and Bor-Luh Lin, On the weak continuity of metric projections. Michigan Math. J. 17(1970), 401–404.

    Article  MATH  MathSciNet  Google Scholar 

  51. B. R. Kripke and T. J. Rivlin, Approximation in the metric of L1(X,μ) Trans. Amer. Math. Soc. 115 (1965), 101–122.

    MathSciNet  Google Scholar 

  52. p.j. Laurent, Théorèms de caractérisation en approximation convexe. Mathematica 10 (33) (1968), 95–111.

    MathSciNet  Google Scholar 

  53. P.J. Laurent, Conditions nécessaires pour une meilleure appromation non linéaire dans un espace normé. Compte rendus Acad. Sci. (Paris) Sér. A-B 269(1969), A 245–A 248.

    MathSciNet  Google Scholar 

  54. P.J. Laurent and Pham-Dinh-Tuan, Global approximation of a compact set by elements of a convex set in a normed space. Numer. Math. 15 (1970), 137–150.

    Article  MATH  MathSciNet  Google Scholar 

  55. A.J. Lazar, Spaces of affine continuous functions on simplexes. Trans, amer. Math. Soc. 134(1968,), 503–525.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  57. J. Lindenstrauss, Extension of compact operators. Memoirs. Amer. Math. Soc. 48 (1964).

    Google Scholar 

  58. J. Lindenstrauss, On nonlinear projections in Banach spaces. Michigan Math. J. 11(1964), 263–287.

    Article  MATH  MathSciNet  Google Scholar 

  59. J. Lindenstrauss, On nonseparable reflexive Banach spaces. Bull. Amer. Math. Soc. 72(1966), 967–970.

    Article  MATH  MathSciNet  Google Scholar 

  60. J. Lindenstrauss and L. Tzafriri, On the complemented subspace problem (to appear).

    Google Scholar 

  61. G. G. Lorentz, Approximation of functions. Holt, Rmehart and Winston, New York (1966).

    MATH  Google Scholar 

  62. G. Meinardus, Approximation of functions, theory and numerical methods. Springer-Verlag, Berlin-Heidelberg New Jork (1967).

    MATH  Google Scholar 

  63. E. A. Michael, Continuous selections. I. Ann. Math. 63(1956), 361–382.

    Article  MathSciNet  Google Scholar 

  64. J.J. Moreau, Fonctionnelles convexes. Séminaire sur les équaions aux dérivées partielles. Collège de France, Paris (1966–67).

    Google Scholar 

  65. J. J. Moreau, Distance à un convexe d'un espace normé et caracterisation des points proximaux. Séminaire d'analyse unilatérale. Fac. Sci. Montpellier, 2(1969), expose no 6.

    Google Scholar 

  66. P.D.Morris, Metric projections onto subspaces of finite codimension. Duke Math. J. 35 (1968), 799–808.

    Article  MATH  MathSciNet  Google Scholar 

  67. J. E. V. Ošman, Continuity of metric projections and some geometric properties of the unit sphere in a Banach space, Doklady Akad. Nauk SSSR (1969), 34–36. [Russian].

    Google Scholar 

  68. R. R. Phelps, Convex sets and nearest points. Proc. Amer. Math. Soc. 8(1957), 790–797.

    Article  MATH  MathSciNet  Google Scholar 

  69. W. Pollul, Reflexivität und Existenz-Teilräume in der linearen Approximationstheorie. Schriften der Gas. für Math, und Datenverarbeitung, Bonn (to appear)

    Google Scholar 

  70. J. R. Rice, The approximation of functions. Vol. I;Linear theory. Vol. II:Non-linear and multivariate theory. Addison-Wesley, Reading, Mass. -London-Don Mills,Ont. (1964 and 1969).

    Google Scholar 

  71. J. R. Rice, Nonlinear approximation. Approximation of functions (Ed. by H. L. Garabedian). Elsevier. Amsterdam-London-New York (1965), 111–133.

    Google Scholar 

  72. J. R. Rice, Non-linear approximation. II. Curvature in Minkowski geometry and local uniquenness. Trans. Amer.Math. Soc. 128(1967), 437–459.

    MATH  MathSciNet  Google Scholar 

  73. W. W. Rogosinski, Continuous linear functiohals on subspaces of Lpand C. Proc. London Math. Soc. 6,22(1956), 175–190.

    Article  MathSciNet  Google Scholar 

  74. Ch. Roumieu, Sur quelques problèmes d' approximation. Sémki. math. fac. sci. Montpellier (1966).

    Google Scholar 

  75. G.Š. Rubinstein, On an extremal problem in a linear normed space. Sibirak. Mat. Ž. 6(1965), 711–714 [Russian]

    Google Scholar 

  76. I. Singer, Properties of the surface of the unit cell and applications to the solution of the problem of uniqueness of the polynomial of best approximation in arbitrary Banach spaces. Studii si cercet. mat. 7(1956), 95–145 [Romanian]

    Google Scholar 

  77. I. Singer, Caractérisation des éléments de meilleure approximation dans un espace de Banach quelconque. Acta Sci. Math. 17(1956), 181–189

    MATH  Google Scholar 

  78. I. Singer, On best approximation of continuous functions. Math. Ann. 140 (1960), 165–168.

    Article  MATH  MathSciNet  Google Scholar 

  79. I: Singer, On best approximation of continuous functions.II. Rev. math, pures et Appl. 6(1961), 507–511.

    MATH  Google Scholar 

  80. I. Singer, Some remarks on approximative compactness.Rev. roum. math, pures et appl. 9(1964), 167–177.

    MATH  Google Scholar 

  81. I. Singer, On the extension of continuous linear functionals and best approximation in normed linear spaces. Math. Ann. 159(1965), 344–355.

    Article  MATH  MathSciNet  Google Scholar 

  82. I. Singer, Best approximation in normed linear spaces by elements of linear subspaces. Publ. House Acad.Soc. Rep. Romania, Bucharest (1967) [Romanian]. English translation : Publ. House Acad. Soc. Rep. Romania, Bucharest and Springer-Verlag, Berlin-Heidelberg-New York (1970).

    Google Scholar 

  83. I. Singer, Some open problems on best approximation in normed linear spaces. Séminaire Choquet, 6 annéUniversité de Paris (1966/67), exposé no. 12.

    Google Scholar 

  84. I. Singer, On metric projections onto linear subspaces of normed linear spaces. Proc. Confer, on “Projections and related topics” held in Clemson. Aug. 1967 Preliminary Edition (January 1968).

    Google Scholar 

  85. I. Singer, Remark on a paper of Y. Ikebe. Proc. Amer. Math. Soc. 21 (1969),24–26.

    Article  MATH  MathSciNet  Google Scholar 

  86. I. Singer, On normed linear spaces which are proximinal in every superspace. J. Approx. Theory (to appear).

    Google Scholar 

  87. K. Tatarkiewicz, Une théorie généralisée de la meilleure approximation. Ann. Univ. Mariae Curie-Sklodowska 6 (1952), 31–46.

    MathSciNet  Google Scholar 

  88. L. P. Vlasov, On Čebyšev sets. Doklady Akad. Nauk SSSR 173 (1967), 491–494 [Russian].

    MathSciNet  Google Scholar 

  89. L.P. Vlasov, Approximatively convex sets in uniformly smooth spaces. Mat. Zametki 1(1967), 443–449 [Russian].

    MathSciNet  Google Scholar 

  90. L. P. Vlasov, On Čebyšev and approximatively convex sets. Mat. Zametki 2(1967), 191–200 [Russian].

    MathSciNet  Google Scholar 

  91. L. P. Vlasov, Čebyšev sets and some generalizations of them Mat. Zametki 3(1968), 59–69 [Russian].

    MATH  MathSciNet  Google Scholar 

  92. L. P. Vlasov, Approximative properties of sets in Banach spaces. Mat. Zametki 7 (1970), 593–604 [Russian].

    MATH  MathSciNet  Google Scholar 

  93. L. P. Vlasov, Almost convex and Čebyšev sets. Mat. Zametki 8 (1970) 545–550 [Russian].

    MATH  MathSciNet  Google Scholar 

  94. D. E. Wulbert, Continuity of metric projections. Trans. Amer. Math. Soc. 134 (1968), 335–343.

    Article  MATH  MathSciNet  Google Scholar 

  95. D. E. Wulbert, Convergence of operators and Korovkin's theorem. J. Approx. Theory 1(1968), 8–18.

    Article  MathSciNet  Google Scholar 

  96. D. E. Wulbert, Differential theory for non-linear approximation. (preprint).

    Google Scholar 

  97. D. E. Wulbert, Uniqueness and differential characterization of approximations from manifolds of functions. Amer. J. Math, (to appear).

    Google Scholar 

  98. D. E. Wulbert, Nonlinear approximation with tangential characterization (to appear).

    Google Scholar 

  99. S. I. Zuhovitskii, On minimal extensions of linear functionals in the space of continuous functions. Izvestija Akad. Nauk SSSR 21(1957), 409–422 [Russian].

    Google Scholar 

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    Ivan Singer

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Singer, I. (2011). Best approximation in normed linear spaces. In: Geymonat, G. (eds) Constructive Aspects of Functional Analysis. C.I.M.E. Summer Schools, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10984-3_6

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