  We lay the foundations for the construction of analytic expressions for
Fourier-domain gravitational waveforms produced by eccentric, inspiraling
compact binaries in a post-circular or small-eccentricity approximation. The
time-dependent, "plus" and "cross" polarizations are expanded in Bessel
functions, which are then self-consistently re-expanded in a power series about
zero initial eccentricity to eighth order. The stationary phase approximation
is then employed to obtain explicit analytic expressions for the Fourier
transform of the post-circular expanded, time-domain signal. We exemplify this
framework by considering Newtonian-accurate waveforms, which in the
post-circular scheme give rise to higher harmonics of the orbital phase and
amplitude corrections both to the amplitude and the phase of the Fourier domain
waveform. Such higher harmonics lead to an effective increase in the inspiral
mass reach of a detector as a function of the binary's eccentricity e_0 at the
time when the binary enters the detector sensitivity band. Using the largest
initial eccentricity allowed by our approximations (e_0 < 0.4), the mass reach
is found to be enhanced up to factors of approximately 5 relative to that of
circular binaries for Advanced LIGO, LISA, and the proposed Einstein Telescope
at a signal-to-noise ratio of ten. A post-Newtonian generalization of the post
circular scheme is also discussed, which holds the promise to provide
"ready-to-use" Fourier-domain waveforms for data analysis of eccentric
inspirals.
