A stability estimate for convolution equation (Q1855779)

The author gives an estimate for the solution of a convolution equation \[ \int_R k(x-y)f(y)dy=F(x), x\in R \tag{1} \] where \(f(x), F(x) \in L_2(R)\). Theorem. Assume that \(\int_R |\widehat{f}(\xi)|^2 (1+|\xi|^2) d\xi \leq M\). Then for the solution of the equation (1), the following estimate is valid \(\|f\|_2\leq \frac{1}{k_c}\|F\|_2+\frac{1}{c}M\) where \(k_c=\inf_{|\xi|\leq c} |\widehat{k}(\xi)|>0\). Here \(\widehat{f}(\xi)\) is the Fourier transform of \(f(y).\)