Geodesic
Relative Version of the Titchmarsh Convolution Theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 1, pp. 31-38.

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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. 
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E. A. Gorin; D. V. Treschev. Relative Version of the Titchmarsh Convolution Theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 1, pp. 31-38. http://geodesic.mathdoc.fr/item/FAA_2012_46_1_a2/

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