Let $k(a,b) = \frac{\gcd(a,b)}{a+b}$. Then one can prove that this is a positidive definite, symmetric function on $\mathbb{N} \times \mathbb{N}$, hence a "kernel" in the sense of functinal analysis and one can apply the Moore-Aronszajn theorem:
Let $\phi(a): b \mapsto k(a,b)$ and the reproducing kernel Hilbert space is given by ( https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space#Moore%E2%80%93Aronszajn_theorem )
$$H = \{ \sum_{i=1}^\infty a_i \phi(x_i) | \lim_{n \rightarrow \infty} \sup_{p \ge 0} \sqrt{\sum_{i=n}^{n+p} a_i^2} = 0\}$$
I want to try to get a feeling for the elements $f \in H$, but for this I need to understand the expression
$$\lim_{n \rightarrow \infty} \sup_{p \ge 0} \sqrt{\sum_{i=n}^{n+p} a_i^2} = 0$$
better. Can you give examples of sequences $(a_i)_{i \in \mathbb{N}}$ which
a) satisfy the expression above
b) do not satisfy the expression above
?
Is the last equation equivalent to
$$ \sum_{i=1}^\infty a_i^2 < \infty$$ ?
Thanks for your help!
I don't know too much about reproducing kernel Hilbert spaces, so I'll just look at your limit in isolation. As you say, it is equivalent to $\sum_{i=1}^\infty a_i^2<\infty$. First, we can get rid of the square root: $$ \lim_{n\to\infty}\sup_{p\ge 0}\sum_{i=n}^{n+p}a_i^2=0 $$ Since we are summing positive terms, the supremum is just a limit, so: $$ \lim_{n\to\infty}\sum_{i=n}^{\infty}a_i^2=0 $$ This is just the tails of the series $\sum_{i=1}^\infty a_i^2$, and it is easy to prove that the series converges if and only if the tails go to $0$.
These square-summable sequences are exactly the sequences in $\ell^2$ space. For example, $\langle \frac1n\rangle_{n\in\mathbb N}$ is in $\ell^2$ (i.e. satisfies your condition), while $\langle \frac{1}{\sqrt n}\rangle_{n\in\mathbb N}$ is not.