Parity symmetry with respect to both x=0 and x=L requires periodicity with period 4L: Connections between computer graphics, group theory and spectral methods for solving partial differential equations

摘要

A function is symmetric with respect to a point x = L if f(x+L)=f(-x+L) for all x and similarly is antisymmetric if f(x+L)=-f(-x+L). A function which is either symmetric or antisymmetric is said to be of “definite parity” with respect to L. The sines and cosines of a Fourier series have definite parity with respect to two points; all cosines are symmetric with respect to the origin while all sines are antisymmetric with respect to x=0;cos(2nx) and sin([2n+1]x) for integral n are also symmetric with respect to x=π/2 while all other Fourier basis functions are antisymmetric with respect to the same point. Such symmetries can be exploited in numerical calculations; for example, computing the angular Mathieu functions using N basis functions can be split into four uncoupled eigenproblems each of dimension N/4. It is natural to ask: Are there other classes of functions with similar symmetries? Using concepts from computer graphics, we prove that all functions which are symmetric with respect to two points separated by a distance L must be spatially periodic with period 4L. We also prove that the only function which is of definite parity with respect to three distinct points must be a constant. These theorems define parity in the usual sense of a global property such that even parity with respect to the origin means f(x)=f(-x) for all x∈[-∞,∞]. We construct counterexamples to both theorems that are functions with local parity, that is, symmetry which applies only for a finite interval in x.