Solvability criterion for convolution equations on a half-strip
The author obtains a criterion for the solvability of the following homogeneous convolution equation defined on a halfstrip: \[ \int\limits_{\partial D_{\sigma}}f(\omega+\tau)g(\omega)d\omega=0, \; \tau\le 0, \; g\in L^2(-\infty,0). \] Here \(D_{\sigma}=\left\{z: \; \mathfrak{Re} z<0, \; |\mathfrak{Im} z|<\sigma \right\}, \; \sigma>0\). This is a generalization of the classical equation \[ \int\limits_{-\infty}^{0}f(u+\tau)g(u)du=0, \; \tau\le 0, \; g\in L^2(-\infty,0). \] The considered equation has direct applications in signal theory, in the study of cyclic functions and in information theory. The proof is based on a new decomposition property of weighted Hardy spaces.
scientific article; zbMATH DE number 7382591
scientific article