Error of Tikhonov's regularization for integral convolution equations

Abstract

Let $ϕ$ be a nontrivial function of $L^1(\RR)$. For each $s\geq 0$ we put \begin{eqnarray*} p(s)=-\log \int_{|t|\geq s}|ϕ(t)|dt. \end{eqnarray*} If $ϕ$ 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 $B_ε=\{x\in\RR : |\hatϕ(x)|\leq ε, |x|\leq R_ε\}$ of Fourier transform \begin{eqnarray*} \hatϕ(x)=\int_{-\infty}^{\infty}e^{-ixt}ϕ(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)ϕ(s)ds =g(t), \end{eqnarray*} where $f,g \in L^2(\RR)$ and $ϕ$ is a nontrivial function of $L^1(\RR)$ satisfying condition (\ref{170506.1}), and $g,ϕ$ are known non-exactly. Also, our results extend some results of \cite{tld} and \cite{tqd}.