  Let $\phi$ be a nontrivial function of $L^1(\RR)$. For each $s\geq 0$ we put
\begin{eqnarray*} p(s)=-\log \int_{|t|\geq s}|\phi (t)|dt. \end{eqnarray*} If
$\phi$ satisfies \begin{equation} \lim_{s\to \infty}\frac{p(s)}{s}=\infty
,\label{170506.1} \end{equation} we obtain asymptotic estimates of the size of
small-valued sets $B_{\epsilon}=\{x\in\RR : |\hat{\phi}(x)|\leq \epsilon,
|x|\leq R_{\epsilon}\}$ of Fourier transform \begin{eqnarray*}
\hat{\phi}(x)=\int_{-\infty}^{\infty}e^{-ixt}\phi (t)dt, x\in \RR,
\end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function
\begin{eqnarray*} p^{*}(t)=\sup_{s\geq 0}[st-p(s)], t\geq 0. \end{eqnarray*}
Applying these results, we give an explicit estimate for the error of
Tikhonov's regularization for the solution $f$ of the integral convolution
equation \begin{eqnarray*} \int_{-\infty}^{\infty}f(t-s)\phi (s)ds =g(t),
\end{eqnarray*} where $f,g \in L^2(\RR)$ and $\phi$ is a nontrivial function of
$L^1(\RR)$ satisfying condition (\ref{170506.1}), and $g,\phi$ are known
non-exactly. Also, our results extend some results of \cite{tld} and
\cite{tqd}.
