# On Ditkin sets

T. Muraleedharan; K. Parthasarathy

Colloquium Mathematicae (1996)

- Volume: 69, Issue: 2, page 271-274
- ISSN: 0010-1354

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topMuraleedharan, T., and Parthasarathy, K.. "On Ditkin sets." Colloquium Mathematicae 69.2 (1996): 271-274. <http://eudml.org/doc/210340>.

@article{Muraleedharan1996,

abstract = {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 $E_1 ∩ E_2$ is a Ditkin set, then $E_1 ∩ E_2$ is a Ditkin set if and only if $E_1$ and $E_2$ are Ditkin sets; and (iv) if $E_1, E_2$ are Ditkin sets with disjoint boundaries then $E_1 ∩ E_2$ is a Ditkin set.},

author = {Muraleedharan, T., Parthasarathy, K.},

journal = {Colloquium Mathematicae},

keywords = {locally compact abelian group; Fourier algebra; Fourier-Gelfand transform; Ditkin set},

language = {eng},

number = {2},

pages = {271-274},

title = {On Ditkin sets},

url = {http://eudml.org/doc/210340},

volume = {69},

year = {1996},

}

TY - JOUR

AU - Muraleedharan, T.

AU - Parthasarathy, K.

TI - On Ditkin sets

JO - Colloquium Mathematicae

PY - 1996

VL - 69

IS - 2

SP - 271

EP - 274

AB - 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 $E_1 ∩ E_2$ is a Ditkin set, then $E_1 ∩ E_2$ is a Ditkin set if and only if $E_1$ and $E_2$ are Ditkin sets; and (iv) if $E_1, E_2$ are Ditkin sets with disjoint boundaries then $E_1 ∩ E_2$ is a Ditkin set.

LA - eng

KW - locally compact abelian group; Fourier algebra; Fourier-Gelfand transform; Ditkin set

UR - http://eudml.org/doc/210340

ER -

## References

top- [1] T. K. Muraleedharan and K. Parthasarathy, On unions and intersections of sets of synthesis, Proc. Amer. Math. Soc., to appear. Zbl0839.43007
- [2] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968. Zbl0165.15601
- [3] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.
- [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.

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