The problem of estimating the largest gain of an unknown linear and time-invariant filter is studied, known as the H-infinity -norm estimation problem. The approach presented in this paper is iterative and corresponds to the combination of two state-of-the-art methods: Power Iterations and Weighted Thompson Sampling. The combination is done by means of a well-known technique in communications known as a cyclic prefix, in which the last points of a signal are prepended to it. This allows to take considerably more exact measurements of the frequency response of the system at a set of equispaced frequencies. The discussion is complemented with a simulation study, showing that the proposed algorithm has an increased speed of convergence to the quantity of interest.