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On the universal convergence sets

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Summary

In this paper we study various universal convergence sets, in connection with some problem arising from the Korovkin approximation theory. This analysis is made in the context of topological vector lattices, of commutative Banach algebras, of Banach algebras with an involution and of C *-algebras. Some applications and examples are given in specific function spaces and function algebras.

References

  1. [1]

    F. Altomare -C. Boccaccio,On Korovkin-type theorems in spaces of continuous complex-valued functions, Boll. Un. Mat. Ital., (6)1-B (1982), pp. 75–86.

  2. [2]

    F.Altomare,Quelques remarques sur les ensembles de Korovkin dans les espaces des fonctions continues complexes, Séminaire Initiation à l'Analyse: G.Choquet - M.Rogalski - J.Saint Raymond, 20e Année, 1980–81, Exp. no. 4, 12 pp., Publ. Math. Univ. Pierre et Marie Curie.

  3. [3]

    F. Altomare,On the Korovkin approximation theory in commutative Banach algebras, Rendiconti di Matematica, (4), Serie VII, Vol.2 (1982), pp. 755–767.

  4. [4]

    F.Altomare,Frontières abstraites et convergence de familles filtrées de formes linéaires sur les algèbres de Banach commutatives, Séminaire Initiation à l'Analyse, G.Choquet - M.Rogalski - J.Saint Raymond, 21 Année, 1981–82, Publ. Math. Pierre et Marie Curie.

  5. [5]

    G. Aquaro,Un criterio di compattezza per insiemi di applicazioni non continue, Rend. Accad. Sc. Fisiche e Mat. della Soc. Naz. di Sc., Lettere ed Arti in Napoli, Serie 4, Vol. XXVI (1959), pp. 1–8.

  6. [6]

    R. Arens -I. M. Singer,Generalized analytic functions, Trans. Amer. Math. Soc.,81, (1956), pp. 379–393.

  7. [7]

    H. Bauer -K. Donner,Korovkin approximation in C 0 (X), Math. Ann.,226 (1978) pp. 225–237.

  8. [8]

    H.Berens - G. G.Lorentz,Theorems of Korovkin type for positive linear operators on Banach lattices, Approx. Theory, Proc. Intern. Symp. Univ. Texas, Austin, Texas, (1973), pp. 1–30.

  9. [9]

    F. F.Bonsall - J.Duncan,Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer Verlag, 1973.

  10. [10]

    N. Bourbaki,Théories spectrales, Ch. I et II, Hermann Paris, 1967.

  11. [11]

    J. Dixmier,Les C *-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964.

  12. [12]

    H. O. Flösser,A Korovkin-type theorem in locally convex M-spaces, Proc. of the Amer. Math. Soc.,72, n. 3 (1978), pp. 456–460.

  13. [13]

    H. O. Flösser,Korovkin closures of finite sets, Archiv der Mathematik, Vol.32 (1979), pp. 600–608.

  14. [14]

    H. O. Flösser -R. Irmisch -W. Roth,Infimum-stable convex cone and approximation, Proc. London Math. Soc., (3)42 (1981), pp. 104–120.

  15. [15]

    H. O. Flösser,Sequences of positive contractions on AM-spaces, J. Approx. Theory,31 (1981), n. 2, pp. 118–137.

  16. [16]

    B. V. Limaye -M. N. N. Namboodiri,Korovkin-type approximation on C *-algebras, J. Approx. Theory,34 (1982), pp. 237–246.

  17. [17]

    G. Maltese,Integral representation theorems via Banach algebras, L'enseignement mathématique, T. XXV, fasc. 3–4 (1979), pp. 273–284.

  18. [18]

    C. E. Rickart,General theory of Banach algebras, D. Van Nostrand Company Inc., N. Y., 1960.

  19. [19]

    A. G. Robertson,A Korovkin theorem for Schwarz maps on C *-algebras, Math. Z.,156 (1977), pp. 205–207.

  20. [20]

    E. Scheffold,Über konvergenz linearer operatoren, Mathematica (Cluj),20 (43) (1977), n. 2, pp. 193–198.

  21. [21]

    Sin-Ei Takahasi,Korovkin's theorems for C *-algebras, J. Approx. Theory,27 (1979), pp. 197–202.

  22. [22]

    M.Wolff,On the theory of approximation by positive operators in vector lattices, Funct. Anal. Surveys and recent results, Proc. Conf. Paderbon, 1976, North-Holland Publ. Comp., 1977.

  23. [23]

    M. Wolff,On the universal Korovkin closure of subsets in vector lattices, J. Approx. Theory,22 (1978), pp. 243–253.

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Work performed out under the auspices of the G.N.A.F.A. (C.N.R.) and of Ministero della Pubblica Istruzione (60%) for the year 1982.

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Altomare, F. On the universal convergence sets. Annali di Matematica pura ed applicata 138, 223–243 (1984). https://doi.org/10.1007/BF01762545

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Keywords

  • Function Space
  • Specific Function
  • Approximation Theory
  • Vector Lattice
  • Banach Algebra