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Constructing Positive Interpolatory Cubature Formulas

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arxiv 2009.11981 v1 pith:32C7HHBJ submitted 2020-09-24 math.NA cs.NAmath.CAmath.OC

Jan Glaubitz This is my paper · ORCID
classification math.NAcs.NAmath.CAmath.OC
keywords cubaturepositiveformulasfunctionsgeneralintegrationinterpolatoryvector
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Positive interpolatory cubature formulas (CFs) are constructed for quite general integration domains and weight functions. These CFs are exact for general vector spaces of continuous real-valued functions that contain constants. At the same time, the number of data points -- all of which lie inside the domain of integration -- and cubature weights -- all positive -- is less or equal to the dimension of that vector space. The existence of such CFs has been ensured by Tchakaloff in 1957. Yet, to the best of the author's knowledge, this work is the first to provide a procedure to successfully construct them.

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  1. Efficient and Robust Carath\'{e}odory-Steinitz Pruning of Positive Discrete Measures

    math.NA 2025-10 accept novelty 6.0 of 10

    GSCSP prunes positive discrete measures to N-point moment-preserving rules in O(N^2) memory and O(MN^2+N^3) time, with a local total-variation Lipschitz stability theorem.

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