Quantum symmetric $L^{p}$ derivatives
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- by J. Marshall Ash and Stefan Catoiu PDF
- Trans. Amer. Math. Soc. 360 (2008), 959-987 Request permission
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.References
- J. Marshall Ash, Generalizations of the Riemann derivative, Trans. Amer. Math. Soc. 126 (1967), 181–199. MR 204583, DOI 10.1090/S0002-9947-1967-0204583-1
- J. Marshall Ash, Symmetric and quantum symmetric derivatives of Lipschitz functions, J. Math. Anal. Appl. 288 (2003), no. 2, 717–721. MR 2020192, DOI 10.1016/j.jmaa.2003.09.021
- J. Marshall Ash, An $L^p$ differentiable non-differentiable function, Real Anal. Exchange 30 (2004/05), no. 2, 747–754. MR 2177431, DOI 10.14321/realanalexch.30.2.0747
- J. Marshall Ash, Stefan Catoiu, and Ricardo Ríos-Collantes-De-Terán, On the $n$th quantum derivative, J. London Math. Soc. (2) 66 (2002), no. 1, 114–130. MR 1911224, DOI 10.1112/S0024610702003198
- J. Marshall Ash and Roger L. Jones, Optimal numerical differentiation using three function evaluations, Math. Comp. 37 (1981), no. 155, 159–167. MR 616368, DOI 10.1090/S0025-5718-1981-0616368-3
- 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, DOI 10.1007/BF02575911
- A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225
- 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, DOI 10.1017/CBO9780511526251
- J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trigonometric series, Fund. Math. 26(1936), 1–43.
- 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.
- M. Weiss, On symmetric derivatives in $L^{p}$, Studia Math. 24 (1964), 89–100. MR 162094, DOI 10.4064/sm-24-1-89-100
Additional Information
- J. Marshall Ash
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- MR Author ID: 27660
- Email: mash@math.depaul.edu
- Stefan Catoiu
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- MR Author ID: 632038
- Email: scatoiu@math.depaul.edu
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2346479