Characterisation of threshold functions under the Walsh transform and linear translation

Some new results showing the classification of Boolean functions of order n, n ¿ 5, under the Rademacher¿Walsh transform, together with the operation of spectral translation, are examined. It is found that there are 48 prototype functions, of which 21 are threshold functions. Examination of the canonic spectral forms of these threshold functions shows that a threshold function may be identified uniquely for n ¿ 5, under a sum-of-squares criterion.