A complete understanding of the origin of the prestellar core mass function (CMF) is crucial. Two major features of the prestellar CMF are 1) a broad peak below 1 M⊙, presumably corresponding to a mean gravitational fragmentation scale, and 2) a characteristic power-law slope, very similar to the Salpeter slope of the stellar initial mass function (IMF) at the high-mass end. While recent Herschel observations have shown that the peak of the prestellar CMF is close to the thermal Jeans mass in marginally supercritical filaments, the origin of the power-law tail of the CMF/IMF at the high-mass end is less clear. In 2001, Inutsuka proposed a theoretical scenario in which the origin of the power-law tail can be understood as resulting from the growth of an initial spectrum of density perturbations seeded along the long axis of star-forming filaments by interstellar turbulence. Here, we report the statistical properties of the line-mass fluctuations of filaments in the Pipe, Taurus, and IC 5146 molecular clouds observed with Herschel for a sample of subcritical or marginally supercritical filaments using a 1D power spectrum analysis. The observed filament power spectra were fitted by a power-law function (Ptrue(s) ∝ sα) after removing the effect of beam convolution at small scales. A Gaussian-like distribution of power-spectrum slopes was found, centered at α̅corr = -1.6 ± 0.3. The characteristic index of the observed power spectra is close to that of the 1D velocity power spectrum generated by subsonic Kolomogorov turbulence (-1.67). Given the errors, the measured power-spectrum slope is also marginally consistent with the power spectrum index of -2 expected for supersonic compressible turbulence. With such a power spectrum of initial line-mass fluctuations, Inutsuka's model would yield a mass function of collapsed objects along filaments approaching dN/dM ∝ M- 2.3 ± 0.1 at the high-mass end (very close to the Salpeter power law) after a few free-fall times. An empirical correlation, P0.5(s0) ∝ ⟨NH2⟩1.4 ± 0.1, was also found between the amplitude of each filament power spectrum P(s0) and the mean column density along the filament ⟨NH2⟩. Finally, the dispersion of line-mass fluctuations along each filament σMline was found to scale with the physical length L of the filament roughly as σMline ∝ L0.7. Overall, our results are consistent with the suggestion that the bulk of the CMF/IMF results from the gravitational fragmentation of filaments. Appendices are available in electronic form at http://www.aanda.org