The problem of analytic continuation of a function f from the unit circle to a circle with radius \(r>1\) is ill-posed. It can be regularized by prescribing a bound \(| f(z)| \leq \beta\) on a circle \(| z| =R\) for some \(R>r\). The paper describes and analyzes an algorithm which from N data \(g_ j\) for the values of f at the Nth roots of unity \(\omega^ j\) calculates estimates for the N values \(f(r\omega^ j)\). This transformation is a multiplication in Fourier space and can be effected by two fast Fourier transforms. Given a bound \(\epsilon\) for the error in the data \(g_ j\), a bound \(\tau\) for the truncation error of replacing f by an N-term Laurent polynomial and a bound \(\beta\) for f on the circle \(| z| =R\) the algorithm also yields a bound for the error of the estimated \(f(r\omega^ j)\). This bound is of the order \(\epsilon^{1-\theta}\) with \(\theta =\log r/\log R\) and is based on Hadamard’s three circles theorem.