In the study of spectral synthesis S-sets and C-sets (see Rudin [3]; Reiter [2] uses the terminology Wiener sets and Wiener-Ditkin sets respectively) have been discussed extensively. A new concept of Ditkin sets was introduced and studied by Stegeman in [4] so that, in Reiter’s terminology, Wiener-Ditkin sets are precisely sets which are both Wiener sets and Ditkin sets. The importance of such sets in spectral synthesis and their connection to the C-set-S-set problem (see Rudin [3]) are mentioned there. In this paper we study local properties, unions and intersections of Ditkin sets. (Warning: Usually in the literature “Ditkin set” means “C-set”, but we follow the terminology of Stegeman.) Our results include: (i) if each point of a closed set E has a closed relative Ditkin neighbourhood, then E is a Ditkin set; (ii) any closed countable union of Ditkin sets is a Ditkin set; (iii) if is a Ditkin set, then is a Ditkin set if and only if and are Ditkin sets; and (iv) if are Ditkin sets with disjoint boundaries then is a Ditkin set.
@article{bwmeta1.element.bwnjournal-article-cmv69i2p271bwm,
author = {T. Muraleedharan and K. Parthasarathy},
title = {On Ditkin sets},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {271-274},
zbl = {0841.43011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv69i2p271bwm}
}
Muraleedharan, T.; Parthasarathy, K. On Ditkin sets. Colloquium Mathematicae, Tome 70 (1996) pp. 271-274. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv69i2p271bwm/
[000] [1] T. K. Muraleedharan and K. Parthasarathy, On unions and intersections of sets of synthesis, Proc. Amer. Math. Soc., to appear. | Zbl 0839.43007
[001] [2] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968. | Zbl 0165.15601
[002] [3] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.
[003] [4] J. D. Stegeman, Some problems on spectral synthesis, in: Proc. Harmonic Analysis (Iraklion, 1978), Lecture Notes in Math. 781, Springer, Berlin, 1980, 194-203.