Symmetric and self-reciprocal kernels

Summary

A kernel $K(x, y)$ of an integral transform is symmetric when it does not change if $x$ and $y$ are swapped. It is self-reciprocal (a Fourier kernel) when the transform is its own inverse: applying it twice gives back the original function. The Fourier cosine and sine kernels and the Hankel transform kernel $\sqrt{xy}\, J_\nu(xy)$ are both symmetric and self-reciprocal. For such kernels the transform and its inverse transform are given by the same formula, which makes the theory and the tables symmetric.