Frequency Response Function (FRF)-based control synthesis methods for Linear Time-Invariant (LTI) systems have been widely used in control theory and industry. Recently, there has been renewed interest in these methods, employing numerical optimization tools to enhance their performance. In this letter, we analyze the convergence properties of two data-driven frequency based H(2 )and H-infinity synthesis methods (Karimi and Kammer, 2017), (Schuchert et al., 2024) as the controller order increases. We demonstrate that, in the limit, an optimal controller can be designed using only the FRF, providing an estimate of the best achievable performance or enabling the design of an optimal high-order controller.