On Lebesgue Integral Quadrature
This is an incremental advancement in numerical methods for integration, potentially benefiting researchers in fields like stochastic processes and non-Gaussian analysis.
The paper tackles the problem of numerical integration by developing Lebesgue quadrature, which groups sums by function values to produce the Lebesgue integral, and shows it is advantageous for analyzing irregular and stochastic processes, especially non-Gaussian ones, with software provided for implementation.
A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$-point discrete distribution, producing the Lebesgue integral. Mathematically, the problem is reduced to a generalized eigenvalue problem: Lebesgue quadrature value-nodes are the eigenvalues and the corresponding weights are the square of the averaged eigenvectors. A numerical estimation of an integral as the Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value-nodes) and the probability of the outcome (weight). For this reason, it is especially well-suited for the study of non-Gaussian processes. The software implementing the theory is available from the authors.