# Representation formulae for (C₀) m-parameter operator semigroups

Mi Zhou; George A. Anastassiou

Annales Polonici Mathematici (1996)

- Volume: 63, Issue: 3, page 247-272
- ISSN: 0066-2216

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topMi Zhou, and George A. Anastassiou. "Representation formulae for (C₀) m-parameter operator semigroups." Annales Polonici Mathematici 63.3 (1996): 247-272. <http://eudml.org/doc/262824>.

@article{MiZhou1996,

abstract = {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.},

author = {Mi Zhou, George A. Anastassiou},

journal = {Annales Polonici Mathematici},

keywords = {multi-parameter operator semigroups; representation formulae; multivariate approximation; sum of random vectors; Banach space; multiplier enlargement method; rate of convergence; second modulus of continuity; inequalities; -parameter operator semigroups; probabilistic approach},

language = {eng},

number = {3},

pages = {247-272},

title = {Representation formulae for (C₀) m-parameter operator semigroups},

url = {http://eudml.org/doc/262824},

volume = {63},

year = {1996},

}

TY - JOUR

AU - Mi Zhou

AU - George A. Anastassiou

TI - Representation formulae for (C₀) m-parameter operator semigroups

JO - Annales Polonici Mathematici

PY - 1996

VL - 63

IS - 3

SP - 247

EP - 272

AB - 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.

LA - eng

KW - multi-parameter operator semigroups; representation formulae; multivariate approximation; sum of random vectors; Banach space; multiplier enlargement method; rate of convergence; second modulus of continuity; inequalities; -parameter operator semigroups; probabilistic approach

UR - http://eudml.org/doc/262824

ER -

## References

top- [1] P. L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer, New York, 1967. Zbl0164.43702
- [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. Zbl0452.60009
- [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. Zbl0784.41015
- [4] K. L. Chung, On the exponential formulas of semi-group theory, Math. Scand. 10 (1962), 153-162. Zbl0106.31201
- [5] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London, 1969. Zbl0176.00502
- [6] S. Eisenberg and B. Wood, Approximating unbounded functions with linear operators generated by moment sequences, Studia Math. 35 (1970), 299-304. Zbl0199.11601
- [7] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Wiley, New York, 1968. Zbl0155.23101
- [8] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Providence, R.I., 1957. Zbl0078.10004
- [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. Zbl0102.05003
- [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. Zbl0449.41008
- [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. Zbl0532.60007
- [14] D. Pfeifer, Approximation-theoretic aspects of probabilistic representations for operator semigroups, J. Approx. Theory 43 (1985), 271-296. Zbl0572.47028
- [15] D. Pfeifer, Probabilistic concepts of approximation theory in connexion with operator semigroups, Approx. Theory Appl. 1 (1985), 93-118. Zbl0605.47042
- [16] D. Pfeifer, A probabilistic variant of Chernoff's product formula, Semigroup Forum 46 (1993), 279-285. Zbl0808.47028
- [17] S. Y. Shaw, Approximation of unbounded functions and applications to representations of semigroups, J. Approx. Theory 28 (1980), 238-259. Zbl0452.41020
- [18] S. Y. Shaw, Some exponential formulas for m-parameter semigroups, Bull. Inst. Math. Acad. Sinica 9 (1981), 221-228. Zbl0467.47036
- [19] R. H. Wang, The Approximation of Unbounded Functions, Sciences Press, Peking, 1983 (in Chinese).

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