3.20 and Proposition 3.35 of [@fol95]. Equation  states that $\pi$ is equivalent to the sub-representation of the left-regular representation (on $L_2(G)$) having $K$ as a cyclic vector. Conversely, if a bounded continuous function $K$ satisfies ,  and , then there exists a unique (up to a unitary equivalence) reproducing representation $\pi$ whose reproducing kernel is $K$.

For the remainder of the paper, we will always impose the following basic assumption:

\[assume:KernelAlmostIntegrable\] We assume $K \in \mathcal{T}_w$, [*i.e*]{}, $$\label{eq:KernelAlmostIntegrable}
    K \in L_{p,w}(G)
    \text{ for all } 1 < p < \infty.$$

We add some remarks.

i)  Since $w(x)\geq 1$, Assumption  implies that $K \in L_p(G)$ for all $p > 1$. If $\pi$ is irreducible, this last fact gives that $V$ is an isometry up to a constant, so that $\pi$ is always a reproducing representation. If $\pi$ is reducible, condition  is not sufficient to ensure that $\pi$ is reproducing; however if $K \ast K = K$, then $\pi$ is always reproducing.

ii) If $w^{-1}$ belongs to $L_q(G)$ for some $1 < q < \infty$, then Hölder’s inequality shows $K \in L_1(G)$, but in general $K \notin L_{1,w}(G)$. However in many interesting examples $w$ is independent of one or more variables, so 