A note on formal power series

Xiao-Xiong Gan; Dariusz Bugajewski

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 4, page 595-604
  • ISSN: 0010-2628

Abstract

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In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series g is convex and balanced which implies that the subset 𝕏 ¯ g consisting of formal power series which can be composed by a formal power series g possesses such properties. We also provide a necessary and sufficient condition for the superposition operator T g to map 𝕏 ¯ g into itself or to map 𝕏 g into itself, respectively.

How to cite

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Gan, Xiao-Xiong, and Bugajewski, Dariusz. "A note on formal power series." Commentationes Mathematicae Universitatis Carolinae 51.4 (2010): 595-604. <http://eudml.org/doc/247132>.

@article{Gan2010,
abstract = {In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset $\overline\{\mathbb \{X\}\}_g $ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator $T_g$ to map $\overline\{\mathbb \{X\}\}_g$ into itself or to map $\{\mathbb \{X\}\}_g$ into itself, respectively.},
author = {Gan, Xiao-Xiong, Bugajewski, Dariusz},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {composition; end behavior of convergence of power series; convex and balanced set; formal power series; composition; end behavior; convergence of power series; convex set; balanced set; formal power series},
language = {eng},
number = {4},
pages = {595-604},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on formal power series},
url = {http://eudml.org/doc/247132},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Gan, Xiao-Xiong
AU - Bugajewski, Dariusz
TI - A note on formal power series
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 4
SP - 595
EP - 604
AB - In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset $\overline{\mathbb {X}}_g $ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator $T_g$ to map $\overline{\mathbb {X}}_g$ into itself or to map ${\mathbb {X}}_g$ into itself, respectively.
LA - eng
KW - composition; end behavior of convergence of power series; convex and balanced set; formal power series; composition; end behavior; convergence of power series; convex set; balanced set; formal power series
UR - http://eudml.org/doc/247132
ER -

References

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  1. Henrici P., Applied and Computational Complex Analysis, John Wiley and Sons, New York, 1988. Zbl1107.30300MR0372162
  2. Remmert R., Theory of Complex Functions, Fouth corrected printing, Springer, New York, Berlin, Heidelberg, 1998. Zbl0780.30001MR1084167
  3. Raney G., 10.1090/S0002-9947-1960-0114765-9, Trans. Amer. Math. Soc. 94 (1960), no. 3, 441–451. Zbl0131.01402MR0114765DOI10.1090/S0002-9947-1960-0114765-9
  4. Cheng C.C., McKay J., Towber J., Wang S.S., Wrigh D., 10.1090/S0002-9947-97-01781-9, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1769–1782. MR1390972DOI10.1090/S0002-9947-97-01781-9
  5. Constantine G.M., Savits T.H., 10.1090/S0002-9947-96-01501-2, Trans. Amer. Math. Soc. 348 (1996), no. 2, 503–520. Zbl0846.05003MR1325915DOI10.1090/S0002-9947-96-01501-2
  6. Li H., 10.1090/S0002-9947-97-01514-6, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1437–1446. Zbl0990.11073MR1327259DOI10.1090/S0002-9947-97-01514-6
  7. Chaumat J., Chollet A.M., 10.1090/S0002-9947-01-02733-7, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1691–1703. Zbl0965.13015MR1806723DOI10.1090/S0002-9947-01-02733-7
  8. Eakin P.M., Harris G.A., 10.1007/BF01391465, Math. Ann. 229 (1977), 201–210. MR0444651DOI10.1007/BF01391465
  9. Gan X., Knox N., 10.1155/S0161171202107150, Int. J. Math. and Math. Sci. 30 (2002), no. 12, 761–770. Zbl0998.13010MR1917671DOI10.1155/S0161171202107150
  10. Neelon T.S., 10.1090/S0002-9939-97-03894-X, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2531–2535. Zbl0890.32004MR1396991DOI10.1090/S0002-9939-97-03894-X
  11. Neelon T.S., On solutions to formal equations, Bull. Belg. Math. Soc. 7 (2000), no. 3, 419–427. Zbl0982.13011MR1788146
  12. Droste M., Zhang G., 10.1016/S0890-5401(03)00066-X, Inform. and Comp. 184 (2003), no. 2, 369–383. MR1987985DOI10.1016/S0890-5401(03)00066-X
  13. Pravica D., Spurr M., Unique summing of formal power series solutions to advanced and delayed differential equations, Discrete Contin. Dyn. Syst. 2005, suppl., 730–737. Zbl1155.34371MR2192733
  14. Sibuya Y., 10.1016/S0022-0396(02)00083-9, J. Differential Equations 190 (2003), no. 2, 559–578. Zbl1029.34079MR1970042DOI10.1016/S0022-0396(02)00083-9
  15. Gan X., A generalized chain rule for formal power series, Commun. Math. Anal. 2 (2007), no. 1, 37–44. Zbl1166.26304MR2332968
  16. Lang S., 10.1007/978-1-4757-1871-3, Second edition, Graduate Texts in Mathematics, 103, Springer, New York, 1985. Zbl0933.30001MR0788885DOI10.1007/978-1-4757-1871-3
  17. Stromberg K.R., Introduction to Classical Real Analysis, Wadsworth International, Belmont, Calif., 1981. Zbl0454.26001MR0604364

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