Let $\phi:[-1,1] \to \mathbb R$ be a function such that
- $\phi$ is $\mathcal C^\infty$ on $(-1,1)$.
- $\phi$ is continuous at $\pm 1$.
For concreteness, and if it helps, In my specific problem I have $\phi(t) := t \cdot (\pi - \arccos(t)) + \sqrt{1-t^2}$.
Now, given a $k \times d$ matrix $U$ with linearly independent rows, consider the $k \times k$ positive-semidefinite matrix $C_U=(c_{i,j})$ defined by $c_{i,j} := K_{\phi}(u_i,u_j)$, where
$$ K_\phi(x,y) := \|x\|\|y\|\phi(\frac{x^\top y}{\|x\|\|x\|}) $$
Question. How express the eigenvalues of $C$ in terms of $U$ and $\phi$ ?
I'm ultimated interested in lower-bounding $\lambda_{\min}(C_U)$ in terms of some norm of $U$ (e.g spectral norm or Frobenius norm).
Let $X$ be the $(d-1)$-dimensional unit-sphere in $\mathbb R^d$, equipped with its uniform measure $\sigma_{d-1}$, and consider the integral operator $T_\phi: L^{2}(X) \to L^2(X)$ defined by $$ T_{\phi}(f):x \mapsto \int K_{\phi}(x,y)f(y)d\sigma_{d-1}(y). $$ It is easy to see that $T_\phi$ is a compact positive-definite operator.
Question. Are the eigenvalues of $C_U$ be expressed as a function of (eigenvalues of) $K_{\phi}$ ?
