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 Funktsional. Anal. i Prilozhen., 2012, Volume 46, Issue 1, Pages 31–38 (Mi faa3052)

Relative Version of the Titchmarsh Convolution Theorem

E. A. Gorina, D. V. Treschevb

a Moscow State (V. I. Lenin) Pedagogical Institute
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider the algebra $C_u=C_u(\mathbb{R})$ of uniformly continuous bounded complex functions on the real line $\mathbb{R}$ with pointwise operations and $\sup$-norm. Let $I$ be a closed ideal in $C_u$ invariant with respect to translations, and let $\operatorname{ah}_I(f)$ denote the minimal real number (if it exists) satisfying the following condition. If $\lambda>\operatorname{ah}_I(f)$, then $(\hat f - \hat g)|_V=0$ for some $g\in I$, where $V$ is a neighborhood of the point $\lambda$. The classical Titchmarsh convolution theorem is equivalent to the equality $\operatorname{ah}_I(f_1\cdot f_2)=\operatorname{ah}_I(f_1)+\operatorname{ah}_I(f_2)$, where $I = \{0\}$. We show that, for ideals $I$ of general form, this equality does not generally hold, but $\operatorname{ah}_I(f^n)=n\cdot\operatorname{ah}_I(f)$ holds for any $I$. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true.

Keywords: Titchmarsh's convolution theorem, estimation of entire functions, Banach algebra

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations

DOI: https://doi.org/10.4213/faa3052

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English version:
Functional Analysis and Its Applications, 2012, 46:1, 26–32

Bibliographic databases:

Document Type: Article
UDC: 517.987+517.51+517.53

Citation: E. A. Gorin, D. V. Treschev, “Relative Version of the Titchmarsh Convolution Theorem”, Funktsional. Anal. i Prilozhen., 46:1 (2012), 31–38; Funct. Anal. Appl., 46:1 (2012), 26–32

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3052
• https://doi.org/10.4213/faa3052
• http://mi.mathnet.ru/eng/faa/v46/i1/p31

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This publication is cited in the following articles:
1. A. V. Dymov, “Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom”, Izv. Math., 76:6 (2012), 1116–1149
2. S. M. Saulin, “Dissipation effects in infinite-dimensional Hamiltonian systems.”, Theoret. and Math. Phys., 191:1 (2017), 537–557
3. Dymov A.V., “Asymptotic Behavior of a Network of Oscillators Coupled to Thermostats of Finite Energy”, Russ. J. Math. Phys., 25:2 (2018), 183–199
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