On Lebesgue Integral Quadrature
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A new type of quadrature is developed. For a given measure Gauss quadrature finds optimal values of function argument (nodes) and corresponding weights. Developed in this paper Lebesgue quadrature instead find optimal values of function values (value-nodes) and corresponding weights. Gauss quadrature group sums by function argument, it can be viewed as n-point discrete measure, producing Riemann integral. Lebesgue quadrature group sums by function value, it can be viewed as n-point discrete distribution, producing Lebesgue integral. Mathematically the problem is reduced to generalized eigenvalues problem, Lebesgue quadrature value-nodes are the eigenvalues and corresponding weights are the square of averaged eigenvector. Numerical estimation of integral as Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value-nodes) and the probability of outcome (weight), for this reason it is especially well suited for non-Gaussian processes study. Implementing the theory software is available from authors.
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