Let \(\Pi_{\pm} = \{z \in \mathbb C : z = t +\, \text{is},\, t \in [- \pi, \pi],\, \pm s > 0\}\) be upper and lower half-strips in the complex plane \(\mathbb C\). According to the author, the periodic Riemann problem is the following: find a pair \(\Phi_{+,-}(z)\) of analytic functions in \(\Pi_{\pm}\) whose boundary values as \(s \to 0_{\pm}\) satisfy the linear relation \(\Phi +(t) = G(t)\Phi^{ -}(t) + g(t)\), \(t\in [- \pi, \pi]\), on the interval \([- \pi, \pi]\), where \(G(t)\) and \(g(t)\) are functions defined on \([- \pi, \pi]\) such that \(G(-\pi) = G(\pi)\) and \(g(-\pi ) = g(\pi)\).
Solvability conditions depending on the index of the problem are given.
It is shown that this problem can be used to study the solvability of discrete convolution equations.