Inequalities between the probability of a subspace and the probabilities of its cosets

We consider an
n
-dimensional vector space over
GF(q)
which has a probability distribution defined on it. The sum of the probabilities over a proper
k
-dimensional subspace is compared to a sum over a coset of this subspace. The difference of these set probabilities is related to a sum of the Fourier transforms of the distribution over a subset of the domain of the transforms. We demonstrate the existence of a coset and both an upper and a lower bound on the difference associated with this coset. The bounds depend on the maximum and nonzero minimum of the transforms as defined on a special subset of the transform domain. Two examples from coding theory are presented. The first deals with a
q
-ary symmetric channel while the second is concerned with a binary compound channel.