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1999 | 134 | 1 | 1-33
Tytuł artykułu

Complexifications of real Banach spaces, polynomials and multilinear maps

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We give a unified treatment of procedures for complexifying real Banach spaces. These include several approaches used in the past. We obtain best possible results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. These estimates provide generalizations and show sharpness of previously obtained inequalities.
Słowa kluczowe
Czasopismo
Rocznik
Tom
134
Numer
1
Strony
1-33
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-01-15
poprawiono
1998-07-06
Twórcy
  • Departamento de Análisis, Facultad de Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain, gustavo@sunam1.mat.ucm.es
  • Mathematics Department, National Technical University, Zografou Campus 157 80, Athens, Greece, ysarant@math.ntua.gr
autor
  • Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242 U.S.A., tonge@mcs.kent.edu
Bibliografia
  • [1] A. Alexiewicz and W. Orlicz, Analytic operations in real Banach spaces, Studia Math. 14 (1953), 57-78.
  • [2] R. Aron, B. Beauzamy and P. Enflo, Polynomials in many variables: Real vs complex norms, J. Approx. Theory 74 (1993), 181-198.
  • [3] R. Aron and J. Globevnik, Analytic functions on $c_0$, Rev. Mat. Univ. Complut. Madrid 2 (1989), 27-33.
  • [4] R. Aron, M. Lacruz, R. Ryan and A. Tonge, The generalized Rademacher functions, Note Mat. 12 (1992), 15-25.
  • [5] C. Benítez and Y. Sarantopoulos, Characterization of real inner product spaces by means of symmetric bilinear forms, J. Math. Anal. Appl. 180 (1993), 207-220.
  • [6] C. Benítez, Y. Sarantopoulos and A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (1998), 395-408.
  • [7] S. N. Bernstein, Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, Mémoires publiés par la Classe des Sciences de l'Académie de Belgique 4 (1912).
  • [8] J. Bochnak, Analytic functions in Banach spaces, Studia Math. 35 (1970), 273-292.
  • [9] J. Bochnak and J. Siciak, Polynomials and multilinear mappings in topological vector spaces, ibid. 39 (1971), 59-76.
  • [10] M. M. Day, Normed Linear Spaces, 3rd ed., Ergeb. Math. Grenzgeb. 21, Springer, 1973.
  • [11] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995.
  • [12] R. Duffin and A. C. Shaeffer, Some properties of functions of exponential type, Bull. Amer. Math. Soc. 44 (1938), 236-240.
  • [13] P. Erdős, Some remarks on polynomials, ibid. 53 (1947), 1169-1176.
  • [14] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953/56), 1-79.
  • [15] L. A. Harris, Bounds on the derivatives of holomorphic functions of vectors, in: Colloque d'Analyse (Rio de Janeiro, 1972), L. Nachbin (ed.), Actualités Sci. Indust. 1367, Hermann, Paris, 1975, 145-163.
  • [16] D. H. Hyers, Polynomial operators, in: Topics in Mathematical Analysis, Th. M. Rassias (ed.), World Sci., 1989, 410-444.
  • [17] M. Lacruz, Four aspects of modern analysis, Ph.D. thesis, Kent State Univ., 1991.
  • [18] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, 1977.
  • [19] G. G. Lorentz, Approximation of Functions, Chelsea, New York, N.Y., 1986.
  • [20] R. D. Mauldin (ed.), The Scottish Book. Mathematics from the Scottish Café, Birkhäuser, 1981.
  • [21] A. D. Michal and M. Wyman, Characterization of complex couple spaces, Ann. of Math. 42 (1941), 247-250.
  • [22] I. P. Natanson, Constructive Function Theory, vol. I, Ungar, New York, 1964.
  • [23] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., 1978; North-Holland, 1980.
  • [24] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
  • [25] H.-J. Rack, A generalization of an inequality of V. Markov to multivariate polynomials, J. Approx. Theory 35 (1982), 94-97.
  • [26] H.-J. Rack, A generalization of an inequality of V. Markov to multivariate polynomials, II, ibid. 40 (1984), 129-133.
  • [27] M. Reimer, On multivariate polynomials of least deviation from zero on the unit cube, ibid. 23 (1978), 65-69.
  • [28] Y. Sarantopoulos, Bounds on the derivatives of polynomials on normed spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991), 307-312.
  • [29] R. Schatten, A Theory of Cross-Spaces, Princeton Univ. Press, 1950.
  • [30] J. Siciak, Wiener's type sufficient conditions in $ℂ^N$, Univ. Iagell. Acta Math. 35 (1997), 47-74.
  • [31] A. E. Taylor, Additions to the theory of polynomials in normed linear spaces, Tôhoku Math. J. 44 (1938), 302-318.
  • [32] A. E. Taylor, Analysis in complex Banach spaces, Bull. Amer. Math. Soc. 49 (1943), 652-659.
  • [33] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite Dimensional Operator Ideals, Pitman Monographs Surveys Pure Appl. Math. 38, Longman Sci. Tech., 1989.
  • [34] C. Visser, A generalization of Chebyshev's inequality to polynomials in more than one variable, Indag. Math. 8 (1946), 310-311.
  • [35] J. Wenzel, Real and complex operator ideals, Quaestiones Math. 18 (1995), 271-285.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv134i1p1bwm
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