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Functional Analysis and Its Applications

, Volume 46, Issue 1, pp 26–32 | Cite as

Relative version of the Titchmarsh convolution theorem

  • E. A. Gorin
  • D. V. Treschev
Article

Abstract

We consider the algebra C u = C u (ℝ) of uniformly continuous bounded complex functions on the real line ℝ with pointwise operations and sup-norm. Let I be a closed ideal in C u invariant with respect to translations, and let ah I (f) denote the minimal real number (if it exists) satisfying the following condition. If λ > ah I (f), then \(\left. {\left( {\hat f - \hat g} \right)} \right|_V = 0\) for some gI, where V is a neighborhood of the point λ. The classical Titchmarsh convolution theorem is equivalent to the equality ah I (f 1 · f 2) = ah I (f 1) + ah I (f 2), where I = {0}. We show that, for ideals I of general form, this equality does not generally hold, but ah I (f n ) = n · ah I (f) holds for any I. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true.

Key words

Titchmarsh’s convolution theorem estimation of entire functions Banach algebra 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Moscow State Pedagogical UniversityMoscowRussia
  2. 2.V. A. Steklov Mathematical InstituteMoscowRussia

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