The Hilbert transform (HT) is an important method for signal demodulation and instantaneous frequency (IF) estimation. The modulus of the analytic signal constructed by the HT is regarded as the amplitude, and the derivative of the instantaneous phase of the extracted pure frequency modulation signal is the IF. When the spectrums of the amplitude function and the oscillation term overlap so as not to satisfy the Bedrosian condition, the instantaneous amplitude (IA) and frequency calculated by the Hilbert transform will contain errors. The recursive Hilbert transform (RHT) is an effective method to overcome this problem. The RHT regards the pure frequency modulation signal obtained by the previous HT as a new signal and recursively computes its Hilbert transform until convergence. The final pure frequency modulation signal of the recursive procedure has the same zero-crossing points as the original signal, and its corresponding quadrature error signal vanishes. We emphasize the convergence analysis of the algorithm and study the convergent tendency of the quadrature error signal in each recursive step. The key to the proof is that the discrete Fourier transform value of the quadrature error signal is regarded as a vector, and the length/norm of the vector decreases with the recursion process. Finally, three examples are used to demonstrate the effective application of this method in signal demodulation, IF identification and damped vibrating signal analysis, which indicate the application potential of the RHT method in mono-component signal processing.