Uniform convergence of double trigonometric series

Péter Kórus

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 3, page 225-243
  • ISSN: 0862-7959

Abstract

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It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone or general monotone double sequences. In this paper we give new sufficient conditions for the uniformity of the regular convergence of sine-cosine and double cosine series, which are necessary as well in case the coefficients are non-negative. The new results also bring necessary and sufficient conditions for the uniform regular convergence of double trigonometric series in complex form.

How to cite

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Kórus, Péter. "Uniform convergence of double trigonometric series." Mathematica Bohemica 138.3 (2013): 225-243. <http://eudml.org/doc/260640>.

@article{Kórus2013,
abstract = {It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone or general monotone double sequences. In this paper we give new sufficient conditions for the uniformity of the regular convergence of sine-cosine and double cosine series, which are necessary as well in case the coefficients are non-negative. The new results also bring necessary and sufficient conditions for the uniform regular convergence of double trigonometric series in complex form.},
author = {Kórus, Péter},
journal = {Mathematica Bohemica},
keywords = {sine series; cosine series; double sine series; sine-cosine series; double cosine series; uniform convergence; regular convergence; general monotone sequence; general monotone double sequence; supremum bounded variation; double sine series; double cosine series; uniform convergence; regular convergence; general monotone double sequence; supremum bounded variation},
language = {eng},
number = {3},
pages = {225-243},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Uniform convergence of double trigonometric series},
url = {http://eudml.org/doc/260640},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Kórus, Péter
TI - Uniform convergence of double trigonometric series
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 3
SP - 225
EP - 243
AB - It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone or general monotone double sequences. In this paper we give new sufficient conditions for the uniformity of the regular convergence of sine-cosine and double cosine series, which are necessary as well in case the coefficients are non-negative. The new results also bring necessary and sufficient conditions for the uniform regular convergence of double trigonometric series in complex form.
LA - eng
KW - sine series; cosine series; double sine series; sine-cosine series; double cosine series; uniform convergence; regular convergence; general monotone sequence; general monotone double sequence; supremum bounded variation; double sine series; double cosine series; uniform convergence; regular convergence; general monotone double sequence; supremum bounded variation
UR - http://eudml.org/doc/260640
ER -

References

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  1. Chaundy, T. W., Jolliffe, A. E., The uniform convergence of a certain class of trigonometrical series, Proc. London Math. Soc. 15 (1916), 214-216. (1916) MR1576557
  2. Dyachenko, M., Tikhonov, S., General monotone sequences and convergence of trigonometric series, Topics in Classical Analysis and Applications in Honor of Daniel Waterman. World Scientific Hackensack, NJ L. De Carli et al. (2008), 88-101. (2008) Zbl1167.42303MR2569380
  3. Dyachenko, M., Tikhonov, S., 10.4064/sm193-3-5, Stud. Math. 193 (2009), 285-306. (2009) Zbl1169.42001MR2515585DOI10.4064/sm193-3-5
  4. Kórus, P., 10.1007/s10474-010-9217-4, Acta Math. Hung. 128 (2010), 369-380. (2010) Zbl1240.42015MR2670995DOI10.1007/s10474-010-9217-4
  5. Kórus, P., 10.1007/s10998-011-8205-y, Period. Math. Hung. 63 (2011), 205-214. (2011) Zbl1265.42007MR2853213DOI10.1007/s10998-011-8205-y
  6. Kórus, P., Móricz, F., 10.4064/sm193-1-4, Stud. Math. 193 (2009), 79-97. (2009) Zbl1167.42002MR2506415DOI10.4064/sm193-1-4
  7. Móricz, F., 10.1007/BF01994074, Acta Math. Hung. 41 (1983), 161-168. (1983) MR0704536DOI10.1007/BF01994074
  8. Tikhonov, S., 10.1016/j.jmaa.2006.02.053, J. Math. Anal. Appl. 326 (2007), 721-735. (2007) Zbl1141.42004MR2277815DOI10.1016/j.jmaa.2006.02.053
  9. Tikhonov, S., 10.1016/j.jat.2007.05.006, J. Approx. Theory 153 (2008), 19-39. (2008) Zbl1215.42002MR2432551DOI10.1016/j.jat.2007.05.006
  10. Zhak, I. E., Shneider, A. A., Conditions for uniform convergence of double sine series, Russian Izv. Vyssh. Uchebn. Zaved., Mat. 53 (1966), 44-52. (1966) 
  11. Zhou, S. P., Zhou, P., Yu, D. S., 10.1007/s11425-010-3138-0, Sci. China, Math. 53 (2010), 1853-1862. (2010) Zbl1211.42006MR2665519DOI10.1007/s11425-010-3138-0

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