7 Reducing the amplitude of peaks in $G_{N}(\nu)$

The spectral window also helps to decide when new measurements should be made to reduce the annoying spikes in the spectral window. We make use of the simulated data in Sect. 4 as an example. A practical procedure could be to determine the polar coordinates $\rho,\theta$ (point (1) in Fig. 5, where we used as a concrete example the simulated data of Sect. 4) of the complex number $(1/N)\sum_{k=1}^{N} \exp(i 2\pi \nu_{\rm h} t_{\rm k})$for the annoying frequency $\nu_{\rm h}$. Point (1) is the center of gravity of the observing points. The value of $G_{N}(\nu_{\rm h})$ is the distance of point (1) to the origin. Then, the new measurement should be taken at

$$t_{N+1}= \left(k + \frac{\pi + \theta}{2 \pi}\right) \frac{1}{\nu_{\rm h}}$$ (4)

for some integer value k compatible with the other constraints. This produces the largest reduction of the peak.

  

Figure 5: The points $\exp(i 2\pi \nu_{\rm h} t_{\rm k})$ are displayed in the complex plane (small open squares) for the frequency $\nu_{\rm h}=3.99$ corresponding to the highest peak of $G_{N}(\nu)$ ($1 < \nu < 50$, in Fig. 1). The black square (1) represents the center of gravity of these points, while the open square (2) opposed to (1) on the circle, is the best position for tN+1 (modulo $2\pi \nu_{\rm h}$) to reduce the amplitude of GN at $\nu_{\rm h}$

Acknowledgements

We would like to thank S. Paltani, G. Burki, F. Kienzle, C. Fluetsch, D. Kurtz and the reviewer A. Milsztajn for their interesting discussions and comments. Furthermore, we thank very warmly D. Minniti for his helpful and efficient collaboration.