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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quantum symmetric $ L^{p}$ derivatives


Authors: J. Marshall Ash and Stefan Catoiu
Journal: Trans. Amer. Math. Soc. 360 (2008), 959-987
MSC (2000): Primary 26A24; Secondary 26A27
Published electronically: June 25, 2007
MathSciNet review: 2346479
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Abstract: For $ 1\leq p\leq\infty$, a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For $ 1\leq p\leq\infty$, symmetrization holds, that is, whenever the $ L^{p}$ $ k$th Peano derivative exists at a point, all of these derivatives of order $ k$ also exist at that point. The main result, desymmetrization, is that conversely, for $ 1\leq p\leq\infty$, each $ L^{p}$ symmetric quantum derivative is a.e. equivalent to the $ L^{p}$ Peano derivative of the same order. For $ k=1$ and $ 2$, each $ k$th $ L^{p}$ symmetric quantum derivative coincides with both corresponding $ k$th $ L^{p}$ Riemann symmetric quantum derivatives, so, in particular, for $ k=1$ and $ 2$, both $ k$th $ L^{p}$ Riemann symmetric quantum derivatives are a.e. equivalent to the $ L^{p}$ Peano derivative.


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Additional Information

J. Marshall Ash
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
Email: mash@math.depaul.edu

Stefan Catoiu
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
Email: scatoiu@math.depaul.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04249-3
PII: S 0002-9947(07)04249-3
Keywords: Generalized derivatives, quantum derivatives, $L^{p}$ derivatives
Received by editor(s): July 22, 2005
Received by editor(s) in revised form: January 28, 2006
Published electronically: June 25, 2007
Additional Notes: The first author’s research was partially supported by NSF grant DMS 9707011 and a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.