\frac{1-a}{\kappa} n \in \mathbb{N}$. Let $\displaystyle {\bf x}=\left\{{\bf x}_n=\left(x_{1,n}, \ldots, x_{n,n}\right)\right\}_{n\in \Lambda}$ be a scheme of nodes. If for each $j=1,\ldots n,$ $n\in \Lambda$ there are two constants $A\geq 0$ and $\ell>0$ satisfying $$\label{maincond}
\left|(1-a)\pi \int^1_{x_{j,n}} \mathrm{d} \widetilde{\sigma}(t) -a \arccos x_{j,n}-\frac{2j-1}{2n}\pi\right|\leq A  e^{-\ell n},$$ then there always exist weights $\displaystyle {\bf w}=\left\{{\bf w}_n=(w_{1,n}, \ldots, w_{n,n})\right\}_{n\in \mathbb{N}}$, where $\displaystyle \left\{I_n\right\}_{n\in \Lambda}$ corresponding to ${\bf x}$ and ${\bf w}$ is convergent.

In Section \[examples\] we give some explicit schemes that satisfy the relation (\[maincond\]). The statement of Theorem \[main\] is proved in Section \[proofmaintheorem\]. In such proof we use results coming from the orthogonal polynomials theory that are analyzed in Section \[sec:2\] and Section \[asymptotic analysis\]. In Section \[sec:2\] we study algebraic properties of families of orthogonal polynomials and their connections with convergent conditions of non-complete interpolatory quadrature rules. In Section \[asymptotic analysis\] we describe the strong asymptotic behavior of an appropriated family of orthogonal polynomials with respect to a varying measure.

Some explicit convergent schemes of nodes {#examples}
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We consider three particular cases where the inequality (\[maincond\]) holds. In the three situations the measure $\sigma=\delta_{\zeta}$ corresponds to a Dirac delta supported on a point belonging to the real line $\zeta>2$. 