ArticleOriginal scientific text

Title

On Ditkin sets

Authors , 1

Affiliations

  1. Ramanujan Institute, University of Madras, Madras 600 005, India

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 E1E2 is a Ditkin set, then E1E2 is a Ditkin set if and only if E1 and E2 are Ditkin sets; and (iv) if E1,E2 are Ditkin sets with disjoint boundaries then E1E2 is a Ditkin set.

Bibliography

  1. T. K. Muraleedharan and K. Parthasarathy, On unions and intersections of sets of synthesis, Proc. Amer. Math. Soc., to appear.
  2. H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968.
  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.
Pages:
271-274
Main language of publication
English
Received
1994-09-12
Accepted
1995-02-15
Published
1996
Exact and natural sciences