  For over five decades the procedure termed maximum-entropy (M-E) has been
used to sharpen structure in spectra, optical and otherwise. However, this is a
contradiction: by modifying data, this approach violates the fundamental M-E
principle, which is to extend, in a model-independent way, trends established
by low-index Fourier coefficients into the white-noise region. The Burg
derivation, and indirectly the prediction-error equations on which sharpening
is based, both lead to the correct solution, although this has been
consistently overlooked. For a single Lorentzian line these equations can be
solved analytically. The resultant lineshape is an exact autoregressive model-1
(AR(1)) replica of the original, demonstrating how the M-E reconstruction
extends low-index Fourier coefficients to the digital limit and illustrating
why this approach works so well for lineshapes resulting from first-order decay
processes. By simultaneously retaining low-index coefficients exactly and
eliminating Gibbs oscillations, M-E noise filtering is quantitatively superior
to that achieved by any linear method, including the high-performance filters
recently proposed. Examples are provided.
