Frequency response function measurements take a central place in the instrumentation and measurement field because many measurement problems boil down to the characterisation of a linear dynamic behaviour. The major problems to be faced are leakage- and noise errors. The local polynomial method (LPM) was recently presented as a superior method to reduce the leakage errors with several orders of magnitude while the noise sensitivity remained the same as that of the classical windowing methods. At the resonance frequencies, where often most information about the system is to be retrieved, the dominating error is the interpolation error. In this paper it is shown that the interpolation error for sufficiently low damping is bounded by (BLPM/B3dB)^(R+2) with BLPM the local bandwidth of the LPM, R the degree of the local polynomial that is selected to be even (user choices), and B3dB the 3dB bandwidth of the resonance, which is a system property.