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Indicator Functions with Uniformly Bounded Fourier Sums and Large Gaps in the Spectrum

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Abstract

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term “Fourier sums” should be understood as “partial Fourier integrals”. A certain weighted version of the result is also provided. This version leads to a new Men\('\)shov-type correction theorem.

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Notes

  1. Formally, the claim is also true for discrete groups, but this is not interesting: the compact set K mentioned in the description of the result is not controlled, all this is about the behavior of Fourier transforms at infinity.

  2. By definition, this means that the operators of convolution with these functions converge pointwise to the identity on \(L^p(G)\), \(1\le p<\infty \).

  3. There is a slight inaccuracy in that proof in the Russian version of the paper, which was corrected in the English translation

  4. Not quite: the lowest upper bound in the lattice of measurable subsets of \(\Gamma \).

  5. Surely, the fact that now the weight \(w_1\) is no longer bounded by 1 from above does not present an obstruction.

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  1. St. Petersburg Department of the V. A. Steklov Math. Institute, 27 Fontanka, St. Petersburg, Russia, 191023

    S. V. Kislyakov & P. S. Perstneva

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  1. S. V. Kislyakov
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  2. P. S. Perstneva
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Correspondence to S. V. Kislyakov.

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Communicated by Sergij Tikhonov.

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Kislyakov, S.V., Perstneva, P.S. Indicator Functions with Uniformly Bounded Fourier Sums and Large Gaps in the Spectrum. J Fourier Anal Appl 27, 33 (2021). https://doi.org/10.1007/s00041-021-09840-3

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  • DOI: https://doi.org/10.1007/s00041-021-09840-3

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