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}.
Categories: math.NA