Mercer's Theorem For The (Noncompact) Domain [0,Infty]

Let $T_K$ be a compact self-adjoint integral covariance operator on $L^2 [0, \infty)$ $$(T_K f) (z) = \int_0^{\infty} K (z, w) f (w) dw$$ defined by kernel $K$: $$K (x, y) = \sum_{k = 0}^{\infty} \lambda_k \phi_k (x) \phi_k (y)$$ where $\{\phi_n \}_{n = 0}^{\infty}$ is a sequence of orthonormal eigenfunctions in $L^2 [0, \infty)$ and $\{\lambda_n \}_{n = 0}^{\infty}$ the corresponding eigenvalues ordered such that $$| \lambda_{n + 1} | < | \lambda_n | \forall n$$ Let $T_{K_N}$ be the truncated operator with kernel $$K_N (z, w) = \sum_{n = 0}^N \lambda_n \phi_n (z) \phi_n (w)$$ then: $$\|T_K - T_{K_N} \| \leq | \lambda_{N + 1} |$$

Proof:

Let $E_N$ be the difference $T_K - T_{K_N}$. For any $f \in L^2 [0, \infty)$: Let $f = g + h$ where $g \in \text{span} \{\phi_k \}_{k \leq N}$ and $h \in \text{span} \{\phi_k \}_{k > N}$ so that $$g (x) = \sum_{k = 0}^N \langle f, \phi_k \rangle \phi_k (x)$$ and $$h (x) = \sum_{k = N + 1}^{\infty} \langle f, \phi_k \rangle \phi_k (x)$$ where by orthogonality of $g$ and $h$ $$\langle g, h \rangle = \int_0^{\infty} g (x) h (x) dx = 0$$ we have $$\|E_N f\|^2 = \langle E_N f, E_N f \rangle = \langle E_N h, h \rangle$$ because $E_N g = 0$ by construction and since $h$ is orthogonal to the first N eigenfunctions and $$| \lambda_k | \leq | \lambda_{N + 1} | \forall k > N$$ we have $$| \langle E_N h, h \rangle | \leq | \lambda_{N + 1} | \|h\|^2 \leq | \lambda_{N + 1} | \|f\|^2$$ Therefore: $$\|E_N \| \leq | \lambda_{N + 1} |$$

Remark:

This is an extension of Mercer's theorem to unbounded domains with uniform convergence.