We present a systematic method of removing endpoint-discontinuity artifacts in the Fourier analysis of spectral segments, enabling the more accurate extraction of information. This principal-component-removal approach differs from a previous version by using extrapolated (or extended) data outside rather than inside the spectral range. This not only allows coefficients to be accessed to the while-noise limit with no distortion of the segment, but also generates interpolated coefficients for improved analytic insight. Examples are provided.