I just read the following result due to Loss-Sloane published in J. Funct. Anal. this year [here]. This is just a lemma in their paper that I found very interesting.
Let be any function in . Consider the inversion and set
Proof. For fixed consider the regions
By changing variables and we find that
which, by symmetry under exchange of and ,
The second integral can be written as
and by changing the variable in the last integral we find that this sum vanishes. Letting yields the desired equality.