Summary
Various generalized derivatives are defined and related. Some of these are the Peano derivatives, the symmetric (Peano) derivatives, the symmetric Riemann derivatives, a generalized derivative from numerical analysis, the very large family of A derivatives, symmetric quantum derivatives, and quantum symmetric Riemann derivatives. Additionally, L p, 1 ≤ p < ∞ versions of many of these derivatives are considered. Relations between some of these derivatives are mentioned. Some counterexamples showing that other relations are not true are also given.
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Notes
- 1.
When I gave this talk in the fall of 2012, I knew that many, many cases of A 1 had manifested irreversibility. This led me to speculate to the audience that, except for the trivial case, irreversibility of the fourth implication when n = 1 always occurred. My black swan moment (see [Ta]) arrived with my discovery in [ACC] of a small family of cases of A 1 that do imply ordinary differentiability. All such cases are classified in [ACC].
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Acknowledgements
The author’s research was partially supported by a Summer Research Grant, College of Science and Health, DePaul University.
This paper is dedicated to Calixto Calderón.
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Ash, J.M. (2014). Remarks on Various Generalized Derivatives. In: Georgakis, C., Stokolos, A., Urbina, W. (eds) Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-319-10545-1_5
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