Let $phi$ be a nontrivial function of $L^1(RR)$. For each $sgeq 0$ we put begin{eqnarray*} p(s)=-log int_{|t|geq s}|phi (t)|dt. end{eqnarray*} If $phi$ satisfies begin{equation} lim_{sto infty}frac{p(s)}{s}=infty ,label{170506.1} end{equation} we obtain asymptotic estimates of the size of small-valued sets $B_{epsilon}={xinRR : |hat{phi}(x)|leq epsilon, |x|leq R_{epsilon}}$ of Fourier transform begin{eqnarray*} hat{phi}(x)=int_{-infty}^{infty}e^{-ixt}phi (t)dt, xin RR, end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function begin{eqnarray*} p^{*}(t)=sup_{sgeq 0}[st-p(s)], tgeq 0. end{eqnarray*} Applying these results, we give an explicit estimate for the error of Tikhonovs 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}.
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