Hi,

I would like to know if there are some known results/litterature about the eigenfunctions of the power mean kernel :

$\displaystyle K(x,y;m) = \left(\frac{x^{-m} + y^{-m}}{2}\right)^{-\frac{1}{m}}$

which is positive semi-definite for $\displaystyle m>0$ and deiined for $\displaystyle x,y \in [0,1]$.

if not for any $\displaystyle m$, at least for some special cases :

$\displaystyle K(x,y;0) = \sqrt{xy}$, geometric mean

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

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

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

For short, eigenfuctions of a kernel $\displaystyle k(x,y)$ are the set of functions $\displaystyle \phi_i(x)$ such that~:

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

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