On the Threshold Order of a Boolean Function

The notion of a threshold function is generalized to a Boolean function of threshold order r. Two characterizations of a Boolean function of threshold order r are presented, which are generalizations of the results of Kaplan and Winder and of Chow, for the case r = 1. Kaplan and Winder's characterization of a threshold function by means of Chebyshev approximation is generalized to a Boolean function of threshold order r. This results in classifying any Boolean function as a threshold function of some order r less than or equal to the number of variables. Chow's theorem on the ``equivalence'' between threshold functions and statistical recognition with independent distributions is generalized to the case of a Boolean function of threshold order r and of statistical recognition with a certain kind of dependence, called dependence of order r.