Zero-mean functional equation on hyper-parallelepiped

Abstract

Let . Given a basis for over and a function , we determine the general solution of the functional equation for all and for all . Geometrically, the above equation says that the arithmetic mean of the values of taken at the vertices of any hyper-parallelepiped obtained from translations and dilations of a fixed hyper-parallelepiped, whose sides are parallel to , equals zero. So, we will call the above equation a zero-mean functional equation on hyper-parallelepiped.