Toeplitz operators of finite interval type and the table method

We solve a Riemann–Hilbert problem with almost periodic coefficient G  , associated to a Toeplitz operator TGTG in a class which is closely connected to finite interval convolution equations, based on a generalization of the so-called table method. The explicit determination of solutions to that problem allows one to establish necessary and sufficient conditions for the invertibility of the corresponding Toeplitz operator, and to determine an appropriate factorization of G  , providing explicit formulas for the inverse of TGTG. Some unexpected properties of the Fourier spectrum of the solutions are revealed which are not apparent through other approaches to the same problem.