Symmetric and self-reciprocal kernels

Remarks

Example: $K(x, y) = \sqrt{2/\pi}\, \cos(xy)$ is symmetric, since $\cos(xy) = \cos(yx)$. It is self-reciprocal on $L^2(0, \infty)$: this is the inversion theorem for the Fourier cosine transform, which follows from the Fourier inversion theorem applied to the even extension of $f$. The sine kernel $\sqrt{2/\pi}\, \sin(xy)$ and the Hankel kernel $\sqrt{xy}\, J_\nu(xy)$ for $\nu \ge -1/2$, with $J_\nu$ the Bessel function of the first kind, are also symmetric and self-reciprocal. Non-example: the Laplace kernel $e^{-xy}$ is symmetric but not self-reciprocal. For $f(y) = e^{-y}$, $(Tf)(x) = 1/(1 + x)$, and $T(Tf)(x) = \int_0^\infty e^{-xy}/(1 + y)\, dy$ tends to $+\infty$ as $x \to 0^+$, while $f(x) \to 1$.