Title: On Lebesgue Integral Quadrature,
Abstract: A new type of quadrature is developed. The Gauss 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 Gauss 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.