Large Toeplitz Systems with Quasi-polynomial Right-hand Sides

We consider the asymptotics of the solutions of large linear systems with Toeplitz matrices generated by a complex valued symbol which is infinitely differentiable, has no zeros on the unit circle, and whose winding number about the origin is zero. The emphasis is on quasi-polynomials as right-hand sides, in which case we show that the central fragment of the solution is asymptotically also a quasi-polynomial. Moreover, we establish asymptotic formulas that give specific components of the solution independently of the other components. We are greatly indebted to the referees for suggesting substantial simplifications in our original proofs and the constructive advice which helped to improve the exposition.