x}=\left\{{\bf x}_n=\left(x_{1,n}, \ldots, x_{n,n}\right)\right\}_{n \in \mathbb{N}}$ an admissible scheme of nodes for a measure $\mu$, and take a corresponding family of orthogonal weights $\left\{\rho_n\right\}_{n \in \mathbb{N}}$. For each $n$, $\mu_n$ denotes the measure with differential form $\mbox{d} \mu(x)= \rho_n(x) \mbox{d} \mu(x)$, and introduce its family of orthonormal polynomials $\displaystyle \left\{p_{\mu_n,j}\right\}_{j\in \mathbb{Z}_+}$. This means that $\displaystyle p_{\mu_n,j}\equiv q_{\mu_n,j}/ \left|\left|q_{\mu_n,j}\right|\right|_{2, \mu_n},$ $j\in \mathbb{Z}_+$ where $||\cdot||_{2,\mu_n}$ denotes the $\mbox{L}_2$ norm corresponding to the measure $\mu_n$.

Given a function $f \in \mbox{L}_{2,\mu_n}$ and $j\in \mathbb{Z}_+$ we consider the $j$-th partial sum of the Fourier series corresponding to $f/\rho_n$ on the bases $\displaystyle \left\{p_{\mu_n,j}\right\}_{j\in \mathbb{Z}_+}$: $$S_{f, \mu_n, j}=\sum_{k=0}^{j-1} f_k p_{\mu_n,k}(x), \quad f_k=\int f(x) p_{\mu_n, k}(x) \mbox{d} \mu_n(x), \quad k=0,\ldots, j-1.$$ Using the Christoffel-Darboux identity (see [@Sze Theorem 3.2.2]) we can deduce $$\label{CDidentity}
S_{f,\mu_n,j}(x)=\int \frac{q_{\mu_n,j}(x)q_{\mu_n,j-1}(t)-q_{\mu_n,j}(t)q_{\mu_n,j-1}(x)}{\left|\left|q_{\mu_n,j-1}\right|\right|_{2,\mu_n}^2(x-t)} f(t) \mbox{d} \mu_n(t).$$ The following result is an extension of [@Sze Theorem 15.2.4 (equality 15.2.7)]

\[mainsec\] Let $\displaystyle (x_{1},\ldots,x_{n})$ be an $\mu$ admissible system of nodes. Given a polynomial $\displaystyle q_{m(n)}$ take the system of weights $\displaystyle (w_{1,n},\ldots,w_{n,n})$ whose elements $w_{j,n}$, $j=1, \ldots,n$, are constructed using (\[lambda\]). Then there always exists a weight $\rho_n$ such that $$\label{identityknown}
\displaystyle \frac{w_{j,n}}{q_{m(n)}(x_{j})}=\frac{\left|\left|q_{\tau_n,n-1}\right|\right|_{2,\tau_n}^2S_{1/\rho_n, \tau_n,n}(x_{j})}{q_{\tau_n,n-1}(x_{j})q_{\tau_n,n}^{\prime}(x_{j})}=-\frac{\left|\left|q_{\tau_n,n}\right|\right|_{2,\tau_n}^2S_{1/\rho_n, \tau_n,n+1}(x_{j})}{q_{\tau_n,n+1}(x_{j})q_{\tau_n,n}^{\prime}(x_{j})},$$ where the measure $\tau_n$ is such that $\displaystyle \frac{\displaystyle \mbox{d} 