Convergence

Pointwise Convergence:
In an RKHS, a sequence of functions ${f_n}$ converges pointwise to a function $f$ if, for each $x$ in the domain and for every $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, $|f_n(x) - f(x)| < \epsilon$. The RKHS's reproducing kernel guarantees that the evaluation at each point is a continuous linear functional, thus ensuring pointwise convergence.

Mean-Square Convergence:
A sequence ${f_n}$ in an RKHS converges in mean-square to $f$ if $\int |f_n - f|^2 \to 0$ as $n \to \infty$. This is a precise measure that does not imply pointwise convergence.

Absolute Convergence:
While absolute convergence typically relates to series, if we consider a sequence of functions ${f_n}$ in an RKHS that are also integrable, the sequence converges absolutely to $f$ if $\int |f_n - f| \to 0$ as $n \to infinity$.

Uniform Convergence:
A sequence ${f_n}$ in an RKHS converges uniformly to $f$ if, for every $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$ and for all $x$ in the domain, $|f_n(x) - f(x)| < \epsilon$. Uniform convergence implies pointwise convergence and is a sufficient condition for the convergence of integrals and the preservation of continuity.

Conditions for Uniform Convergence in an RKHS:

1. Boundedness of the Sequence: If a sequence ${f_n}$ is uniformly bounded in the RKHS norm, i.e., there exists a $M > 0$ such that $|f_n|_{RKHS} < M$ for all $n$, and it converges pointwise to $f$, then it converges uniformly to $f$.

2. Compact Operators: If the evaluation functionals are compact operators, meaning that for any bounded sequence ${f_n}$ in the RKHS, the image sequence ${T_x(f_n)}$ has a convergent subsequence in the space of continuous functions, then the original sequence converges uniformly to $f$.

3. Arzelà-Ascoli Theorem: A sequence ${f_n}$ in an RKHS is uniformly convergent if it is pointwise bounded and equicontinuous, which means for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x, y$ in the domain with $|x - y| < \delta$, $|f_n(x) - f_n(y)| < \epsilon$ for all $n$.

4. Montel's Theorem: In an RKHS of holomorphic functions, every bounded sequence ${f_n}$, where boundedness means there exists a $M > 0$ such that $|f_n|_{RKHS} < M$ for all $n$, contains a subsequence that converges uniformly on every compact subset of the domain.

5. Kernel Properties: The kernel must produce functions that are both bounded and uniformly continuous. A kernel $K$ allows for uniform convergence if, for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x, y$ in the domain with $|x - y| < \delta$, $|K(x, \cdot) - K(y, \cdot)|_{RKHS} < \epsilon$.

It is important to note that these conditions are sufficient, but not necessary for uniform convergence. There may be sequences of functions in an RKHS that converge uniformly without satisfying all these conditions. Nonetheless, these conditions are widely used in practice to verify uniform convergence.

Uniform convergence is a desirable property for sequences of functions in an RKHS because it implies pointwise convergence and ensures the convergence of integrals and the preservation of continuity.