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• W85281963 abstract "This paper analyzes the existence of regularizations of integrals that apply to functions with a nonintegrable singularity at an endpoint of integration. This is a problem arising naturally in many contexts including solution of PDEs and singular ODEs. Regularizations, such as the classical Hadamard finite part ($p.f.$), share two fundamental properties: (i) the regularized integral of a function on an interval only depends on the values of the function in that interval, and of course, (ii) the regularized integral is an antiderivative. Sufficient conditions for existence of regularizations are well known, and they require various types of smoothness. The following is a natural apparently open question: do regularizations exist beyond their known domain? We show that there exist regularizations of integrals satisfying the properties of $p.f.$ which apply to singular functions without any conditions on the type or strength of the singularity. However, the very existence of a regularization beyond $p.f.$ satisfying only (i) and (ii) is independent of ZF (the usual ZFC axioms for mathematics without the Axiom of Choice AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). This is established in S 7.2, where we also show that such extensions cannot be naturally given - even if we are using the full axiom of choice. In particular, we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such a regularization. In a precise sense, the classical domain of $p.f.$ is optimal. Such results for a variety of spaces of functions are precisely formulated, and proved using methods from mathematical logic, descriptive set theory and analysis." @default.
• W85281963 created "2016-06-24" @default.
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• W85281963 date "2014-01-27" @default.
• W85281963 modified "2023-09-27" @default.
• W85281963 title "Foundational aspects of singular integrals and initial value problems" @default.
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• W85281963 hasPublicationYear "2014" @default.
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