Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new $(2\ell +1)$ -point quadrature rule $\widehat G_{2\ell +1}$ , referred to as a generalized averaged Gaussian quadrature rule. This rule has $2\ell +1$ nodes and the nodes of the corresponding Gauss rule $G_\ell$ with $\ell$ nodes form a subset. This is similar to the situation for the $(2\ell +1)$ -point Gauss-Kronrod rule $H_{2\ell +1}$ associated with $G_\ell$ . An attractive feature of $\widehat G_{2\ell +1}$ is that it exists also when $H_{2\ell +1}$ does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of $\widehat G_{2\ell +1}$ is simpler than the construction of $H_{2\ell +1}$ . A disadvantage might be that the algebraic degree of precision of $\widehat G_{2\ell +1}$ is $2\ell +2$ , while the one of $H_{2\ell +1}$ is $3\ell +1$ . Consider a (nonnegative) measure $d\sigma$ with support in the bounded interval $[a,b]$ such that the respective orthogonal polynomials, above a specific index $r$ , satisfy a three-term recurrence relation with constant coefficients. For $\ell \ge 2r-1$ , we show that $\widehat G_{2\ell +1}$ has algebraic degree of precision at least $3\ell +1$ , and therefore it is in fact H 2 ℓ ...

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286].

We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378–391] and we extend their results. Construction of new Gaussian quadrature formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives, is considered. We prove the existence and uniqueness of Kronrod extensions with multiple nodes of standard Gaussian quadrature formulas with multiple nodes for several weight functions, in order to construct some new generalizations of quadrature formulas for the Fourier coefficients. For the quadrature formulas for the Fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct Kronrod extensions with multiple nodes and highest algebraic degree of precision. For this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas.