I would like to know if there are some known results/litterature about the eigenfunctions of the power mean kernel :
K(x,y;m) = \left(\frac{x^{-m} + y^{-m}}{2}\right)^{-\frac{1}{m}}
which is positive semi-definite for m>0 and deiined for x,y \in [0,1].

if not for any m, at least for some special cases :
K(x,y;0) = \sqrt{xy}, geometric mean

K(x,y;1) = \frac{2xy}{x+y}, harmonic mean

K(x,y;\infty) = \min(x,y)

with K(x,y;0) and K(x,y;\infty) standing respectively for \lim_{m\rightarrow 0} K(x,y;m) and \lim_{m\rightarrow \infty} K(x,y;m).

For short, eigenfuctions of a kernel k(x,y) are the set of functions \phi_i(x) such that~:
\int_{\mathcal D(y)} k(x,y) \phi_i(y) dy = \lambda_i \phi_i(x)

These eigenfunctions must verify some properties that I won't list here.

For instance K(x,y;0) has one eigenfunction \sqrt{x}, since :

\int_0^1 \sqrt{xy}\sqrt{y} dy = \sqrt{x} \int_0^1 y dy = \frac{\sqrt{x}}{2}