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

Published electronically:
June 25, 2007

MathSciNet review:
2346479

Full-text PDF Free Access

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.

**[A]**J. Marshall Ash,*Generalizations of the Riemann derivative*, Trans. Amer. Math. Soc.**126**(1967), 181–199. MR**0204583**, 10.1090/S0002-9947-1967-0204583-1**[A1]**J. Marshall Ash,*Symmetric and quantum symmetric derivatives of Lipschitz functions*, J. Math. Anal. Appl.**288**(2003), no. 2, 717–721. MR**2020192**, 10.1016/j.jmaa.2003.09.021**[A2]**J. Marshall Ash,*An 𝐿^{𝑝} differentiable non-differentiable function*, Real Anal. Exchange**30**(2004/05), no. 2, 747–754. MR**2177431****[ACR]**J. Marshall Ash, Stefan Catoiu, and Ricardo Ríos-Collantes-De-Terán,*On the 𝑛th quantum derivative*, J. London Math. Soc. (2)**66**(2002), no. 1, 114–130. MR**1911224**, 10.1112/S0024610702003198**[AJ]**J. Marshall Ash and Roger L. Jones,*Optimal numerical differentiation using three function evaluations*, Math. Comp.**37**(1981), no. 155, 159–167. MR**616368**, 10.1090/S0025-5718-1981-0616368-3**[AJJ]**J. Marshall Ash, S. Janson, and R. L. Jones,*Optimal numerical differentiation using 𝑁 function evaluations*, Calcolo**21**(1984), no. 2, 151–169. MR**799618**, 10.1007/BF02575911**[CZ]**A.-P. Calderón and A. Zygmund,*Local properties of solutions of elliptic partial differential equations*, Studia Math.**20**(1961), 171–225. MR**0136849****[GR]**George Gasper and Mizan Rahman,*Basic hypergeometric series*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR**2128719****[MZ]**J. Marcinkiewicz and A. Zygmund,*On the differentiability of functions and summability of trigonometric series,*Fund. Math.**26**(1936), 1-43.**[R]**R. Ríos-Collantes-de-Terán,*Conjuntos de unicidad de sistemas de funciones independientes. Quantum derivadas.,*Thesis, Departamento de Análisis Matemático de la Universidad de Sevilla, 2001.**[W]**M. Weiss,*On symmetric derivatives in 𝐿^{𝑝}*, Studia Math.**24**(1964), 89–100. MR**0162094**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
26A24,
26A27

Retrieve articles in all journals with MSC (2000): 26A24, 26A27

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

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.