Gaussian Quadrature Rule using ε-Quasiorthogonality
This work provides a computationally efficient alternative to classical Gaussian quadrature for arbitrary measures, benefiting numerical integration tasks in scientific computing.
The paper introduces approximate Gaussian quadrature (AGQ) rules using ε-quasiorthogonality for integral approximation, achieving lower complexity and fewer nodes than classical quadratures at fixed accuracy. It also demonstrates application to discretizing the Fourier transform for short exponential representations.
We introduce a new type of quadrature, known as approximate Gaussian quadrature (AGQ) rules using ε-quasiorthogonality, for the approximation of integrals of the form \int f(x)d α(x). The measure α(\cdot) can be arbitrary as long as it possesses finite moments μn for sufficiently large n. The weights and nodes associated with the quadrature can be computed in low complexity and their count is inferior to that required by classical quadratures at fixed accuracy on some families of integrands. Furthermore, we show how AGQ can be used to discretize the Fourier transform with few points in order to obtain short exponential representations of functions.