$$\label{conv}
\lim_{n\in\Lambda} I_n[f]=\int_{-1}^1 f(x)\, \mbox{d} \lambda_0(x),\quad \mbox{for all} \quad f\in C[-1,1].$$ According to a classical result of Pólya [@DR page 130], when $\displaystyle  m(n) \to \infty$ as $n\to \infty $, the equality holds true if and only if $ \displaystyle \sup_{n\in \Lambda} \sum_{j=1}^n \left|w_{j,n}\right| < \infty.$ This condition is satisfied if the weights $w_{j,n}$ are all positive. From (\[interpolatoryq\]) we observe that $$\label{polyacond}
\sum_{j=1}^n w_{j,n}=\int_{-1}^1\, \frac{\mbox{d} x}{\pi \,\sqrt{1-x^2}}=1 < \infty.$$ In the Gaussian quadrature rule (maximum degree of exactness $m(n)=2n-1$) the weights $w_{j,n}$, $j=1,\ldots, n$ are all positive and the convergence of the rule is guarantied. However the nodes are all fixed. For each $n \in \mathbb{N}$ the points of evaluation $x_{j,n}$, $j=1,\ldots,n$ must be the roots of the $n$th orthogonal polynomial with respect to $\lambda_0$ (see for instance [@Sze]). This is the Chebyshev polynomial with degree $n$. This means that if we do not have the value of $f$ at each point $x_{j,n}$ the calculus gets stuck. It is convenient to have more flexibility in the distribution of the evaluation nodes. We study convergent interpolatory integration rules with orders of exactness $m <2n-1$.

The authors of [@BLS2] analyze a wide class of interpolatory quadrature rules with $m(n)$ degrees of 