Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog; Peer Chr. Kunstmann

Commentationes Mathematicae Universitatis Carolinae (2015)

  • Volume: 56, Issue: 2, page 159-168
  • ISSN: 0010-2628

Abstract

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We prove the existence of functions , the Fourier series of which being universally divergent on countable subsets of . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on .

How to cite

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Herzog, Gerd, and Kunstmann, Peer Chr.. "Universally divergent Fourier series via Landau's extremal functions." Commentationes Mathematicae Universitatis Carolinae 56.2 (2015): 159-168. <http://eudml.org/doc/270134>.

@article{Herzog2015,
abstract = {We prove the existence of functions $f\in A(\mathbb \{D\})$, the Fourier series of which being universally divergent on countable subsets of $\mathbb \{T\} = \partial \mathbb \{D\}$. The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $\mathbb \{T\}\setminus \lbrace 1\rbrace $.},
author = {Herzog, Gerd, Kunstmann, Peer Chr.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Fourier series; universal functions; Landau's extremal functions; Fourier series; universal functions; Landau's extremal functions},
language = {eng},
number = {2},
pages = {159-168},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Universally divergent Fourier series via Landau's extremal functions},
url = {http://eudml.org/doc/270134},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Herzog, Gerd
AU - Kunstmann, Peer Chr.
TI - Universally divergent Fourier series via Landau's extremal functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 2
SP - 159
EP - 168
AB - We prove the existence of functions $f\in A(\mathbb {D})$, the Fourier series of which being universally divergent on countable subsets of $\mathbb {T} = \partial \mathbb {D}$. The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $\mathbb {T}\setminus \lbrace 1\rbrace $.
LA - eng
KW - Fourier series; universal functions; Landau's extremal functions; Fourier series; universal functions; Landau's extremal functions
UR - http://eudml.org/doc/270134
ER -

References

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  1. Anderson J.M., Clunie J., Pommerenke Ch., On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. Zbl0292.30030MR0361090
  2. Beise H.-P., Müller J., On the boundary behaviour of Taylor series in Hardy and Bergman spaces, preprint, 2014. 
  3. Bernal-González L., 10.1007/s00020-012-1984-6, Integral Equations Operator Theory 74 (2012), 271–279. Zbl1282.47008MR2983066DOI10.1007/s00020-012-1984-6
  4. Fejér L., 10.1007/BF02543853, Acta Math. 49 (1926), 183–190. MR1555240DOI10.1007/BF02543853
  5. Grosse-Erdmann K.-G., 10.1090/S0273-0979-99-00788-0, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345–381. Zbl0933.47003MR1685272DOI10.1090/S0273-0979-99-00788-0
  6. Grosse-Erdmann K.-G., Peris Manguillot A., Linear Chaos, Universitext, Springer, London, 2011. Zbl1246.47004MR2919812
  7. Kahane J.-P., 10.1007/BF02791536, J. Anal. Math. 80 (2000), 143–182. Zbl0961.42001MR1771526DOI10.1007/BF02791536
  8. Katsoprinakis E., Nestoridis V., Papachristodoulos C., Universality and Cesàro summability, Comput. Methods Funct. Theory 12 (2012), 419–448. Zbl1262.30071MR3058515
  9. Landau E., Gaier D., Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie, Springer, Berlin, 1986. Zbl0601.30001MR0869998
  10. Müller J., 10.1016/j.crma.2010.10.026, C.R. Math. Acad. Sci. Paris 348 (2010), 1155–1158. Zbl1204.42008MR2738918DOI10.1016/j.crma.2010.10.026
  11. Pogosyan N.B., Universal Fourier series, Uspekhi Mat. Nauk 38 (1983), 185–186. Zbl0525.42003MR0693740
  12. Rudin W., Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987. Zbl1038.00002MR0924157
  13. Wintner A., The theorem of Eneström and the extremal functions of Landau-Schur, Math. Scand. 5 (1957), 236–240. Zbl0081.07103MR0098192
  14. Zygmund A., Trigonometric Series. Vol. I, II, Cambridge University Press, Cambridge, 2002. MR1963498

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