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)dsigma_{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) $K_{phi}$ ?