AbstractThe general problem of unfolding is considered from the point of view of linear vector space theory as applied to the more specific problem of spectral unfolding. It is shown that the basis for many of the methods currently in use is an expansion of the unknown spectrumφ(E) in terms of some set of functionswn(E). The coefficients in the expansion are determined by the measured outputs of the detectors. The relationships between the various solutions obtained by using different sets ofwn(E) functions are explored. It is shown that the particular solution obtained by using the response functions of the detectors as thewn(E) is effectively an orthogonal decomposition ofφ(E) whereas all other expansions are nonorthogonal decompositions. As a result of these properties, the response function expansion, for example, has a bounded square deviation fromφ(E) and is less sensitive to errors in the measured detector outputs, whereas other expansions can lead to solutions that may differ violently fromφ(E). Conditions under which the latter situation can occur are of a fundamental nature and do not owe their origin to calculational inaccuracies.The square-wave solution is given particular attention and the theoretical basis is investigated of the standard practice of requiring an all positive solution with theoretical outputs that differ least from those measured. It is shown that the correct square-wave representation forφ(E) results in theoretical detector outputs that necessarily differ from those produced byφ(E) itself—possibly by a large amount. Thus, except for cases where this difference is known,a priori, to be small, there is no theoretical basis for this standard practice.