First-order Borel-Perron Integration: Properties and Applications
DOI:
https://doi.org/10.26713/jims.v18i1.3534Keywords:
Borel derivative, Perron integral, Borel-Perron integral, Generalized derivativesAbstract
We study a Perron-type integral built from Borel derivatives. Using first and second order Borel derivatives we introduce the first-order Borel-Perron integral (denoted by B\(_1\)P) via pre-majorants and pre-minorants. We establish basic properties such as monotonicity, linearity, and additivity, develop the indefinite B\(_1\)P-integral and prove a fundamental theorem of B\(_1\)P-calculus. We also prove two stability features in the spirit of classical Perron theory: B\(_1\)P-integrable functions are finite almost everywhere and the integral is invariant under modification on null sets. In addition, we obtain an integration-by-parts formula against functions of bounded variation. Finally, we sketch applications to a Taylor formula with B\(_1\)P-integral remainder and to a Poisson integral for B\(_1\)P-integrable boundary data.
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[1] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, Vol. 4, American Mathematical Society, Providence, RI, (1994), DOI: 10.1090/gsm/004.
[2] S. Ray and A. Garai, On Borel derivative, Bulletin of the Calcutta Mathematical Society 106(4) (2014), 273 – 280.
[3] S. Ray and S. Ghosh, On the higher order Borel derivative, Proceedings of IMBIC 5 (2016), 94 – 99.
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