Numerical Integration as a Finite Matrix Approximation to Multiplication Operator
For numerical analysts, this provides a new theoretical framework for numerical integration, though it is incremental as it reframes existing methods.
The paper reformulates numerical integration as a finite matrix approximation of the multiplication operator, showing Gaussian quadrature as a special case, and analyzes node placement and convergence.
In this article, numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite-dimensional basis. The idea is to approximate the continuous spectral representation of a multiplication operator on a Hilbert space with a discrete spectral representation of a Hermitian matrix. The Gaussian quadrature is shown to be a special case of the new method. The placement of the nodes of numerical integration and convergence of the new method are studied.