---
author:
- 'U. Fidalgo and J. Olson'
title: Convergent non complete interpolatory quadrature rules
---

Introduction {#sec:1}
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Let $\mathcal{C}([-1,\,1])$ be the set of all continuous functions defined on $[-1,\,1]$. Given an $n$-tuple of nodes ${\bf x}_n=(x_{1,n},\ldots,x_{j,n})$ satisfying $-1<x_{1,n}<x_{2,n}<\cdots<x_{n,n}<1$, we consider [*integration rules*]{} $$\label{rule}
I_n[f]= \sum_{j=1}^n w_{j,n} f(x_{j,n}), \quad f \in \mathcal{C}[-1,\,1]$$ associated to the integrals $$\label{integrals}
\displaystyle I(f)=\int_{-1}^1 f(x)\, \mbox{d} \lambda_0(x), \quad \mbox{where} \quad  \frac{\mbox{d} \lambda_0(x)}{\mbox{d} x}=\frac{1}{\pi \sqrt{1-x^2}}.$$ The numbers $w_{j,n},$ $j=1,\ldots,n$ are called weights.

An integration rule $I_n[\cdot]$ is said to be interpolatory if there exists a number $m \in \left\{0,1,\ldots, 2n-1\right\}$, such that the following equality holds for every polynomial $p$ with degree $\leq m$ (we denote $p \in \Pi_m$): $$\label{interpolatoryq}
I_n[p]=\int_{-1}^1 p(x)\, \mbox{d} \lambda_0(x).$$ When the equality (\[interpolatoryq\]) holds for certain $m$, and is not extendable for all polynomials with degree $m+1$, we say that $I_n[\cdot]$ is an interpolatory quadrature rule with $m$-degree of exactness. When $m=2n-1$ $I_n$ is the Gaussian quadrature rule.

Consider a sequence of interpolatory quadratures $\displaystyle \left\{I_n\right\}_{n \in \Lambda}$ constructed with the following schemes of nodes and weights $$\label{nodessequence}
  {\bf x}=\left\{{\bf x}_n=(x_{1,n},\ldots,x_{n,n})\right\}_{n\in \Lambda} \quad \mbox{and} \quad {\bf w}=\left\{{\bf w}_n=(w_{1,n},\ldots,w_{n,n})\right\}_{n\in \Lambda},$$ respectively.

We say that $\displaystyle \left\{I_n\right\}_{n \in \Lambda}$ is convergent if 