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Some frequency-domain controller design problems are solved using a finite number of frequency samples. Consequently, the performance and stability conditions are not guaranteed for the frequencies between the frequency samples. In this paper, all possible interpolants between the frequency samples of the open-loop system are bounded using convex constraints on a linearly parameterized controller. These constraints are integrated in a method which solves an H∞ control problem based on spectral models by convex optimization. The method is applied to a simulation example. It is shown how the added conservatism is reduced while the number of frequency samples is increased.