A Helson set of uniqueness but not of synthesis

T. Körner

Colloquium Mathematicae (1991)

  • Volume: 62, Issue: 1, page 67-71
  • ISSN: 0010-1354

Abstract

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In [3] I showed that there are Helson sets on the circle 𝕋 which are not of synthesis, by constructing a Helson set which was not of uniqueness and so automatically not of synthesis. In [2] Kaufman gave a substantially simpler construction of such a set; his construction is now standard. It is natural to ask whether there exist Helson sets which are of uniqueness but not of synthesis; this has circulated as an open question. The answer is "yes" and was also given in [3, pp. 87-92] but seems to have got lost in the depths of that rather long paper. Furthermore, the proof depends on the methods of [3], which few people would now wish to master. The object of this note is to give a proof using the methods of [2].

How to cite

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Körner, T.. "A Helson set of uniqueness but not of synthesis." Colloquium Mathematicae 62.1 (1991): 67-71. <http://eudml.org/doc/210100>.

@article{Körner1991,
abstract = {In [3] I showed that there are Helson sets on the circle 𝕋 which are not of synthesis, by constructing a Helson set which was not of uniqueness and so automatically not of synthesis. In [2] Kaufman gave a substantially simpler construction of such a set; his construction is now standard. It is natural to ask whether there exist Helson sets which are of uniqueness but not of synthesis; this has circulated as an open question. The answer is "yes" and was also given in [3, pp. 87-92] but seems to have got lost in the depths of that rather long paper. Furthermore, the proof depends on the methods of [3], which few people would now wish to master. The object of this note is to give a proof using the methods of [2].},
author = {Körner, T.},
journal = {Colloquium Mathematicae},
keywords = {Helson sets on the circle; uniqueness; synthesis; set of multiplicity; Dirichlet set; pseudomeasure},
language = {eng},
number = {1},
pages = {67-71},
title = {A Helson set of uniqueness but not of synthesis},
url = {http://eudml.org/doc/210100},
volume = {62},
year = {1991},
}

TY - JOUR
AU - Körner, T.
TI - A Helson set of uniqueness but not of synthesis
JO - Colloquium Mathematicae
PY - 1991
VL - 62
IS - 1
SP - 67
EP - 71
AB - In [3] I showed that there are Helson sets on the circle 𝕋 which are not of synthesis, by constructing a Helson set which was not of uniqueness and so automatically not of synthesis. In [2] Kaufman gave a substantially simpler construction of such a set; his construction is now standard. It is natural to ask whether there exist Helson sets which are of uniqueness but not of synthesis; this has circulated as an open question. The answer is "yes" and was also given in [3, pp. 87-92] but seems to have got lost in the depths of that rather long paper. Furthermore, the proof depends on the methods of [3], which few people would now wish to master. The object of this note is to give a proof using the methods of [2].
LA - eng
KW - Helson sets on the circle; uniqueness; synthesis; set of multiplicity; Dirichlet set; pseudomeasure
UR - http://eudml.org/doc/210100
ER -

References

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  1. [1] J.-P. Kahane, Séries de Fourier Absolument Convergentes, Springer, 1970. Zbl0195.07602
  2. [2] R. Kaufman, M-sets and distributions, Astérisque 5 (1973), 225-230. Zbl0281.43006
  3. [3] T. W. Körner, A pseudofunction on a Helson set. I, ibid., 3-224. Zbl0281.43004

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