Relative Version of the Titchmarsh Convolution Theorem

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.