Szegő quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane. However, often it is difficult to determine the quadrature error. Recently, Spalević introduced generalized averaged Gauss quadrature rules for estimating the quadrature error obtained when applying Gauss quadrature over an interval on the real axis. We describe analogous quadrat...

Szegő quadrature rules are discretization methods for approximating integrals of the form ∫ π −π f(e it)dμ(t). This paper presents a new class of discretization methods, which we refer to as anti-Szegő quadrature rules. AntiSzegő rules can be used to estimate the error in Szegő quadrature rules: under suitable conditions, pairs of associated Szegő and anti-Szegő quadrature rules provide upper a...

In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal algebraic degree of exactness is given. The natural generalization of such rules are quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces that are different from the space of algebraic polynomial. For that purpose we present a generaliz...

in this paper‎, ‎based on cas wavelets we present quadrature rules for numerical solution‎ ‎of double and triple integrals with variable limits of integration‎. ‎to construct new method‎, ‎first‎, ‎we approximate the unknown function by cas wavelets‎. ‎then by using suitable collocation points‎, ‎we obtain the cas wavelet coefficients that these coefficients are applied in approximating the unk...

4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.2. 4.3. Gaussian quadrature with preassigned nodes Christoffel's work and related developments Kronrod's extension of quadrature rules Gaussian quadrature with multiple nodes The quadrature formula of Turan Arbitrary multiplicities and preassigned nodes Power-orthogonal polynomials Constructive aspects and applications Further miscellaneous extensions Product-type...

In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...