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1996 | 63 | 3 | 247-272
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Representation formulae for (C₀) m-parameter operator semigroups

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Some general representation formulae for (C₀) m-parameter operator semigroups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C₀) one- and m-parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.
Twórcy
autor
  • Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152, U.S.A.
  • Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152, U.S.A.
Bibliografia
  • [1] P. L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer, New York, 1967.
  • [2] P. L. Butzer and L. Hahn, A probabilistic approach to representation formulae for semigroups of operators with rates of convergence, Semigroup Forum 21 (1980), 257-272.
  • [3] W. Z. Chen and M. Zhou, Freud-Butzer-Hahn type quantitative theorem for probabilistic representations of (C₀) operator semigroups, Approx. Theory Appl. 9 (1993), 1-8.
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  • [5] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London, 1969.
  • [6] S. Eisenberg and B. Wood, Approximating unbounded functions with linear operators generated by moment sequences, Studia Math. 35 (1970), 299-304.
  • [7] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Wiley, New York, 1968.
  • [8] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Providence, R.I., 1957.
  • [9] L. C. Hsu, Approximation of non-bounded continuous functions by certain sequences of linear positive operators of polynomials, Studia Math. 21 (1961), 37-43.
  • [10] J. Kisyński, Semi-groups of operators and some of their applications to partial differential equations, in: Control Theory and Topics in Functional Analysis, Vol. III, Internat. Atomic Energy Agency, Vienna, 1976, 305-405.
  • [11] W. Köhnen, Einige Saturationssätze für n-Parametrige Halbgruppen von Operatoren, Anal. Numér. Théor. Approx. 9 (1980), 65-73.
  • [12] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.
  • [13] D. Pfeifer, Probabilistic representations of operator semigroups - a unifying approach, Semigroup Forum 30 (1984), 17-34.
  • [14] D. Pfeifer, Approximation-theoretic aspects of probabilistic representations for operator semigroups, J. Approx. Theory 43 (1985), 271-296.
  • [15] D. Pfeifer, Probabilistic concepts of approximation theory in connexion with operator semigroups, Approx. Theory Appl. 1 (1985), 93-118.
  • [16] D. Pfeifer, A probabilistic variant of Chernoff's product formula, Semigroup Forum 46 (1993), 279-285.
  • [17] S. Y. Shaw, Approximation of unbounded functions and applications to representations of semigroups, J. Approx. Theory 28 (1980), 238-259.
  • [18] S. Y. Shaw, Some exponential formulas for m-parameter semigroups, Bull. Inst. Math. Acad. Sinica 9 (1981), 221-228.
  • [19] R. H. Wang, The Approximation of Unbounded Functions, Sciences Press, Peking, 1983 (in Chinese).
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Bibliografia
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