Quantum symmetric derivatives

Authors:
J. Marshall Ash and Stefan Catoiu

Journal:
Trans. Amer. Math. Soc. **360** (2008), 959-987

MSC (2000):
Primary 26A24; Secondary 26A27

DOI:
https://doi.org/10.1090/S0002-9947-07-04249-3

Published electronically:
June 25, 2007

MathSciNet review:
2346479

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Abstract | References | Similar Articles | Additional Information

Abstract: For , 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 , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the 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:
https://doi.org/10.1090/S0002-9947-07-04249-3

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.