Manifold-valued generalized functions in full Colombeau spaces

Michael Kunzinger; Eduard Nigsch

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 4, page 519-534
  • ISSN: 0010-2628

Abstract

top
We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

How to cite

top

Kunzinger, Michael, and Nigsch, Eduard. "Manifold-valued generalized functions in full Colombeau spaces." Commentationes Mathematicae Universitatis Carolinae 52.4 (2011): 519-534. <http://eudml.org/doc/247052>.

@article{Kunzinger2011,
abstract = {We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.},
author = {Kunzinger, Michael, Nigsch, Eduard},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {algebras of generalized functions; manifold-valued generalized functions; full Colombeau algebras; generalized function; manifold-valued generalized function; full Colombeau algebra},
language = {eng},
number = {4},
pages = {519-534},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Manifold-valued generalized functions in full Colombeau spaces},
url = {http://eudml.org/doc/247052},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Kunzinger, Michael
AU - Nigsch, Eduard
TI - Manifold-valued generalized functions in full Colombeau spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 4
SP - 519
EP - 534
AB - We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.
LA - eng
KW - algebras of generalized functions; manifold-valued generalized functions; full Colombeau algebras; generalized function; manifold-valued generalized function; full Colombeau algebra
UR - http://eudml.org/doc/247052
ER -

References

top
  1. Abraham R., Marsden J.E., Ratiu T., 10.1007/978-1-4612-1029-0_8, Applied Mathematical Sciences, 75, Springer, New York, 1988. Zbl0875.58002MR0960687DOI10.1007/978-1-4612-1029-0_8
  2. Clarke C.J.S., Vickers J.A., Wilson J.P., 10.1088/0264-9381/13/9/013, Classical Quantum Gravity 13 (1996), no. 9, 2485–2498. Zbl0859.53074MR1410817DOI10.1088/0264-9381/13/9/013
  3. Colombeau J.-F., New Generalized Functions and Multiplication of Distributions, North-Holland Mathematics Studies, 84, North-Holland, Amsterdam, 1984. Zbl0761.46021MR0738781
  4. Colombeau J.-F., Elementary Introduction to New Generalized Functions, North-Holland Mathematics Studies, 113, North-Holland, Amsterdam, 1985. Zbl0584.46024MR0808961
  5. Colombeau J.-F., Meril A., 10.1006/jmaa.1994.1303, J. Math. Anal. Appl. 186 (1994), no. 2, 357–364. MR1292997DOI10.1006/jmaa.1994.1303
  6. Dapić N., Pilipović S., 10.1016/0019-3577(96)83722-0, Indag. Math. (N.S.) 7 (1996), no. 3, 293–309. MR1621385DOI10.1016/0019-3577(96)83722-0
  7. de Roever J.W., Damsma, M., 10.1016/0019-3577(91)90022-Y, Indag. Math. (N.S.) 2 (1991), no. 3, 341–358. MR1149687DOI10.1016/0019-3577(91)90022-Y
  8. Erlacher E., Grosser M., Inversion of a “discontinuous coordinate transformation” in general relativity, Applicable Analysis(to appear). MR2842608
  9. Grosser M., Farkas E., Kunzinger M., Steinbauer R., On the foundations of nonlinear generalized functions I and II, Mem. Amer. Math. Soc. 153 (2001), no. 729. MR1848157
  10. Grosser M., Kunzinger M., Oberguggenberger M., Steinbauer R., Geometric Theory of Generalized Functions with Applications to General Relativity, Kluwer, Dordrecht, 2001. Zbl0998.46015
  11. Grosser M., Kunzinger M., Steinbauer R., Vickers J.A., 10.1006/aima.2001.2018, Adv. Math. 166 (2002), no. 1, 50–72. Zbl0995.46054MR1882858DOI10.1006/aima.2001.2018
  12. Grosser M., Kunzinger M., Steinbauer R., Vickers J.A., A global theory of algebras of generalized functions II: tensor distributions, submitted, http://arxiv.org/abs/0902.1865. 
  13. Jelinek J., An intrinsic definition of the Colombeau generalized functions, Comment. Math. Univ. Carolin. 45 (1999), no. 1, 71–95. Zbl1060.46513MR1715203
  14. Jelinek J., Colombeau product of currents, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 437–462. Zbl1123.46025MR2174523
  15. Kriegl A., Michor P., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, 1997. Zbl0889.58001MR1471480
  16. Kunzinger M., 10.1007/s00605-002-0488-x, Monatsh. Math. 137 (2002), 31–49. Zbl1014.46054MR1930994DOI10.1007/s00605-002-0488-x
  17. Kunzinger M., Steinbauer R., 10.1023/A:1014554315909, Acta Appl. Math. 71 (2002), no. 2, 179–206. Zbl1012.46048MR1914741DOI10.1023/A:1014554315909
  18. Kunzinger, M., Steinbauer, R., Vickers, J.A., Intrinsic characterization of manifold-valued generalized functions, Proc. London Math. Soc. 87 (2003), no. 2, 451–470. Zbl1042.46050MR1990935
  19. Kunzinger M., Steinbauer R., Vickers J.A., 10.1090/S0002-9947-09-04621-2, Trans. Amer. Math. Soc. 361 (2009), 5177–5192. Zbl1184.46070MR2515808DOI10.1090/S0002-9947-09-04621-2
  20. Marsden J.E., Generalized Hamiltonian mechanics: A mathematical exposition of non-smooth dynamical systems and classical Hamiltonian mechanics, Arch. Rational Mech. Anal. 28 (1967/1968), 323–361. Zbl0155.51302MR0224935
  21. Nedeljkov M., Pilipović S., Scarpalezos D., The Linear Theory of Colombeau Generalized Functions, Pitman Research Notes in Mathematics, 385, Longman, Harlow, 1998. MR1638310
  22. Nigsch E., Approximation properties of local smoothing kernels, Integral Transform. Spec. Funct.(to appear). MR2801281
  23. Oberguggenberger M., Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, 259, Longman Scientific & Technical, Harlow, 1992. Zbl0818.46036MR1187755
  24. O'Neill B., Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, New York, 1983. Zbl0531.53051MR0719023
  25. Parker P.E., 10.1063/1.524224, J. Math. Phys. 20 (1979), no. 7, 1423–1426. Zbl0442.53064MR0538717DOI10.1063/1.524224
  26. Steinbauer R., Vickers J.A., 10.1088/0264-9381/23/10/R01, Classical Quantum Gravity 23 (2006), no. 10, R91–R114. Zbl1096.83001MR2226026DOI10.1088/0264-9381/23/10/R01
  27. Vernaeve H., 10.1007/s00605-009-0152-9, Monatsh. Math. 162 (2011), 225–237. MR2769888DOI10.1007/s00605-009-0152-9
  28. Vickers J., Wilson J., A nonlinear theory of tensor distributions, ESI-Preprint 566 (http://www.esi.ac.at/ESI-Preprints.html), 1998. 
  29. Vickers J.A., Wilson J.P., 10.1088/0264-9381/16/2/019, Classical Quantum Gravity 16 (1999), no. 2, 579–588. Zbl0933.83033MR1672476DOI10.1088/0264-9381/16/2/019

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.