The classification of generalized Riemann derivatives
HTML articles powered by AMS MathViewer
- by J. Marshall Ash, Stefan Catoiu and William Chin PDF
- Proc. Amer. Math. Soc. 146 (2018), 3847-3862 Request permission
Abstract:
A generalized $n$th Riemann derivative of a real function $f$ at $x$ is given by \[ \lim _{h\rightarrow 0}\frac 1{h^n} \sum _{i=1}^{m}A_{i}f(x+a_{i}h). \] The above sum $\Delta _{\mathcal {A}}$ is called an $n$th generalized Riemann difference. The data vector $\mathcal {A}=\{A_1,\ldots ,A_m;a_1,\ldots ,a_m\}$ satisfies suitable conditions that make the limit agree with $f^{(n)}(x)$ whenever this exists. We explain the underlying reason for a surprising relationship between certain generalized $n$th Riemann derivatives recently discovered by Ash, Catoiu, and Csörnyei. We characterize all pairs $(\Delta _{\mathcal {A}},\Delta _{\mathcal {B}})$ of generalized Riemann differences of any orders for which $\mathcal {A}$-differentiability implies $\mathcal {B}$-differentiability. Two generalized Riemann derivatives $\mathcal {A}$ and $\mathcal {B}$ are equivalent if a function has a derivative in the sense of $\mathcal {A}$ at a real number $x$ if and only if it has a derivative in the sense of $\mathcal {B}$ at $x$. We determine the equivalence classes for this equivalence relation. The classification of these by now classical objects of real analysis was made possible by using a less known and less studied notion from algebra, the group algebra of the multiplicative group $\mathbb {R}^{+}$ of the positive reals over the field $\mathbb {R}$ of real numbers.References
- Arlene Ash, J. Marshall Ash, and Stefan Catoiu, New definitions of continuity, Real Anal. Exchange 40 (2014/15), no. 2, 403–420. MR 3499773
- 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, Remarks on various generalized derivatives, Special functions, partial differential equations, and harmonic analysis, Springer Proc. Math. Stat., vol. 108, Springer, Cham, 2014, pp. 25–39. MR 3297652, DOI 10.1007/978-3-319-10545-1_{5}
- J. M. Ash, S. Catoiu and W. Chin, The classification of complex generalized Riemann derivatives, preprint.
- J. Marshall Ash, Stefan Catoiu, and Marianna Csörnyei, Generalized vs. ordinary differentiation, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1553–1565. MR 3601547, DOI 10.1090/proc/13224
- 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. M. Ash and R. L. Jones, Mean value theorems for generalized Riemann derivatives, Proc. Amer. Math. Soc. 101 (1987), no. 2, 263–271. MR 902539, DOI 10.1090/S0002-9939-1987-0902539-2
- 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. Denjoy, Sur l’intégration des coefficients différentiels d’ordre supérieur, Fund. Math. 25 (1935), 273–326.
- H. Fejzić, C. Freiling, and D. Rinne, A mean value theorem for generalized Riemann derivatives, Proc. Amer. Math. Soc. 136 (2008), no. 2, 569–576. MR 2358497, DOI 10.1090/S0002-9939-07-08976-9
- P. D. Humke and M. Laczkovich, Convexity theorems for generalized Riemann derivatives, Real Anal. Exchange 15 (1989/90), no. 2, 652–674. MR 1059427
- P. D. Humke and M. Laczkovich, Monotonicity theorems for generalized Riemann derivatives, Rend. Circ. Mat. Palermo (2) 38 (1989), no. 3, 437–454 (1990). MR 1053383, DOI 10.1007/BF02850026
- A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund. Math. 9 (1927), 212–279.
- J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trigonometric series, Fund. Math. 26 (1936), 1–43. Also in J. Marcinkiewicz, Collected papers, Edited by Antoni Zygmund. With the collaboration of Stanislaw Lojasiewicz, Julian Musielak, Kazimierz Urbanik and Antoni Wiweger. Instytut Matematyczny Polskiej Akademii Nauk Państwowe Wydawnictwo Naukowe, Warsaw 1964 viii+673pp. MR0168434
- Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
- Sorin Rădulescu, Petruş Alexandrescu, and Diana-Olimpia Alexandrescu, Generalized Riemann derivative, Electron. J. Differential Equations (2013), No. 74, 19. MR 3040651
- Sorin Rădulescu, Petruş Alexandrescu, and Diana-Olimpia Alexandrescu, The role of Riemann generalized derivative in the study of qualitative properties of functions, Electron. J. Differential Equations (2013), No. 187, 14. MR 3104963
- Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
- B. S. Thomson, Monotonicity theorems, Proc. Amer. Math. Soc. 83 (1981), no. 3, 547–552. MR 627688, DOI 10.1090/S0002-9939-1981-0627688-2
- B. S. Thomson, Monotonicity theorems, Real Anal. Exchange 6 (1980/81), no. 2, 209–234. MR 623052
- Clifford E. Weil, Monotonicity, convexity and symmetric derivates, Trans. Amer. Math. Soc. 221 (1976), no. 1, 225–237. MR 401994, DOI 10.1090/S0002-9947-1976-0401994-1
Additional Information
- J. Marshall Ash
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- MR Author ID: 27660
- Email: mash@depaul.edu
- Stefan Catoiu
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- MR Author ID: 632038
- Email: scatoiu@depaul.edu
- William Chin
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- Email: wchin@depaul.edu
- Received by editor(s): October 5, 2017
- Published electronically: June 11, 2018
- Communicated by: Alexander Iosevich
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3847-3862
- MSC (2010): Primary 26A24; Secondary 16S34, 26A27
- DOI: https://doi.org/10.1090/proc/14139
- MathSciNet review: 3825839