Modern control synthesis methods rely on accurate models to derive a performant controller. Obtaining a good model is often a costly step, and has led to a renewed interest in data-driven synthesis methods. Frequency-response-based synthesis methods have been proposed, as they rely on easy-to-obtain frequency-response data and offer flexible tuning approaches. Such methods formulate the objective as a minimization over the whole spectrum, which is problematic as only a finite number of frequency points can be considered when solving the problem using numerical solvers. Most methods require sampling the frequency response to obtain a trackable formulation, but this sampling process loses all stability and performance guarantees. By studying the inter-frequency behavior of such methods, bounds on the spectrum errors can be derived. Using these bounds, a novel algorithm is proposed to design a SISO controller using only a finite number of frequency response data, which guarantees closed-loop stability and robust performance.