  The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in
[0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and
setting some of the variables to zero, this method provides a tool to calculate
all non-zero sums sum_{i+j=k; (i,j) in (P cap Z^2)} (a_i b_j) in a rectangle P
with perimeter p in O(p log p) time.
  This paper extends this geometric interpretation in order to allow arbitrary
convex polygons P with k vertices and perimeter p. Also, this extended
algorithm only needs O(k + p(log p)^2 log k) time.
  Additionally, this paper presents fast algorithms for counting sub-cadences
and cadences with 3 elements using this extended method.
