ALGEBRAICALLY DEGENERATE APPROXIMATIONS OF BOOLEAN FUNCTIONS

Properties of k-dimensional approximations of Boolean functions are investigated. One of main results is the theorem on the structure of k-dimensional functions whose degree equals d and whose distance from a given Boolean function of n variables is no longer than 2(n-d) (1-epsilon), 1 <= d <= k <= n, epsilon is an element of (0,1). This theorem considerably strengthens the well-known P. Gopalan result and makes it possible to considerably increase the efficiency of his algorithm for constructing all the mentioned k-dimensional Boolean functions.