Inconsistent treatment of DC component

[keatonb]

I am attempting to compute a spectral window by taking the power spectrum of ones sampled as the time series. I expect a value of 1 at frequency=0, but usually I get zero power at zero frequency... but not always! I set fit_offset=False and center_data=False to try to retain the the DC value. Here's an example:

import numpy as np
import matplotlib.pyplot as plt
import gatspy.periodic as gp

time = np.linspace(0,9,1e5)
nyq=1./(2.*np.median(np.diff(time)))
deltaf=(1./time[-1])/4.5
freq,sw=gp.lomb_scargle_fast.lomb_scargle_fast(time,np.ones(time.size)/time.size,
                                               f0=0.,df=deltaf,Nf=nyq/deltaf,fit_offset=False,
                                               center_data=False)

plt.plot(freq,sw,label='oversample by 4.5')

deltaf=(1./time[-1])/5.
freq,sw=gp.lomb_scargle_fast.lomb_scargle_fast(time,np.ones(time.size)/time.size,
                                               f0=0.,df=deltaf,Nf=nyq/deltaf,fit_offset=False,
                                               center_data=False)
plt.plot(freq,sw,label='oversample by 5')
plt.title('why this different behavior at zero frequency?')
plt.legend()
plt.xlim(0,.5)
plt.xlabel('frequency')
plt.ylabel('spectral window')
plt.show()

This seems to only affect the power at exactly frequency = 0, and the result appears to be sensitive to some precise relationship between the frequency sampling and the time sampling. Am I correct to expect power=1 for frequency=0? Is the "spectral window" a well defined quantity for the Lomb-Scargle periodogram (though I expect it to reproduce the Fourier transform result for the evenly-sampled example above)?

[keatonb]

Perhaps this is a result of the hardcoded

if np.isnan(power[0]) or np.isinf(power[0]):
    power[0] = 0

where that condition only happens sometimes. Should the forced behavior be different when center_data=False?

[keatonb]

I believe the desired power at f = 0 is
power[0] = np.mean(y ** 2 - (y - np.mean(y)) ** 2) / np.mean(y ** 2)

I'll open a pull request.