§ 2.3. Point symmetry.
If the manifold is a regular polyhedron, we can examine its point symmetries which preserve the center.
For the interval we can ask for its point group symmetry operations preserving the center . The only non-trivial operation of this type is the reflection . With respect to functions on the interval we can ask for a basis of functions on invariant under this reflection. Clearly we must replace the exponential functions by the combinations , and the expansion eq.11 must be replaced by
These basis functions have both periodicity and reflection symmetry. The restriction to reflection symmetry reduces the set of basis of the Fourier series roughly by a factor two.
In Fig.13 we illustrate the effect of 4fold point symmetry of the square in the Euclidean plane, which is the universal covering of the torus. The domain of definition of functions is restricted from the full square A by 4fold point symmetry to the sector shown in B. It is easy to find the corresponding restriction on the basis functions eq. 13 for the functional analysis.