Consider a kernel $k: X \times X \to \mathbb{R}$ on a topological space $X$, and let $H_k$ denote the associated Reproducing Kernel Hilbert Space (RKHS). It is well-established that the span of functions $\{ k_x(\cdot) : x \in X \}$ is dense in $H_k$.
My inquiry is as follows: For a given function $f \in H_k$, is it possible to ensure the existence of a monotone approximating sequence $(k_{x_n})_{n\geq 1}$? In other words, can we find a sequence such that $||f - k_{x_n}||_{H_k} \to 0$, and $k_{x_{n+1}}(y) \leq k_{x_n}(y)$ holds for all $y \in X$?
If it simplifies the discussion, feel free to assume that $X$ is compact or that we are dealing with a compact subset $K \subset X$. I would appreciate any insights.