# Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $$d$$ be a large positive integer and let $$S_{d-1}$$ be the unit-sphere in $$\mathbb R^d$$ and let $$K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$$ be defined by $$K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$$, for $$\gamma > 0$$. Note that $$K_\gamma$$ is positive-semidefinite in the sense that $$\sum_{i=1}^n \sum_{j=1}^n a_ia_jK_\gamma(x_i,x_j) \ge 0$$ for all $$x_1,\ldots,x_n \in S_{d-1}$$ and $$a_1,\ldots,a_n \in \mathbb R$$. Let $$L^2(S_{d-1})$$ be the (Hilbert space) of square-integrable functions $$S_{d-1} \to \mathbb R$$ w.r.t the uniform probability measure $$\tau_d$$ on $$S_{d-1}$$. Consider the induced kernel integral operator $$T_{K_\gamma}:L^2(S_{d-1}) \to L^2(S_{d-1})$$ defined by

$$f \mapsto T_{K_\gamma}f: x \mapsto \int K_\gamma(x,x')f(x')d\tau_d(x').$$

It is clear that $$T_{K_\gamma}$$ is a compact positive operator. Let $$\lambda_1 \ge \lambda_2 \ge \ldots$$ be its eigenvalues.

Question. What are good estimates (upper-bounds and lower-bounds) for the $$\lambda_j$$'s valid for large $$d$$ ?

I'm particularly interested in the cases $$\gamma = 1$$ and $$\gamma = 2$$.

• Were you able to find the answer to it? Aug 18 at 4:42