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

Abstract

top
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.

How to cite

top

Mi 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. [1] P. L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer, New York, 1967. Zbl0164.43702
  2. [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. [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. [4] K. L. Chung, On the exponential formulas of semi-group theory, Math. Scand. 10 (1962), 153-162. Zbl0106.31201
  5. [5] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London, 1969. Zbl0176.00502
  6. [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. [7] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Wiley, New York, 1968. Zbl0155.23101
  8. [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. [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. [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. [11] W. Köhnen, Einige Saturationssätze für n-Parametrige Halbgruppen von Operatoren, Anal. Numér. Théor. Approx. 9 (1980), 65-73. Zbl0449.41008
  12. [12] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953. 
  13. [13] D. Pfeifer, Probabilistic representations of operator semigroups - a unifying approach, Semigroup Forum 30 (1984), 17-34. Zbl0532.60007
  14. [14] D. Pfeifer, Approximation-theoretic aspects of probabilistic representations for operator semigroups, J. Approx. Theory 43 (1985), 271-296. Zbl0572.47028
  15. [15] D. Pfeifer, Probabilistic concepts of approximation theory in connexion with operator semigroups, Approx. Theory Appl. 1 (1985), 93-118. Zbl0605.47042
  16. [16] D. Pfeifer, A probabilistic variant of Chernoff's product formula, Semigroup Forum 46 (1993), 279-285. Zbl0808.47028
  17. [17] S. Y. Shaw, Approximation of unbounded functions and applications to representations of semigroups, J. Approx. Theory 28 (1980), 238-259. Zbl0452.41020
  18. [18] S. Y. Shaw, Some exponential formulas for m-parameter semigroups, Bull. Inst. Math. Acad. Sinica 9 (1981), 221-228. Zbl0467.47036
  19. [19] R. H. Wang, The Approximation of Unbounded Functions, Sciences Press, Peking, 1983 (in Chinese). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.