Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-20T05:17:57.582Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Riemann Derivatives for Integrable Functions

Published online by Cambridge University Press:  20 November 2018

P. L. Butzer  and
W. Kozakiewicz
P. L. Butzer
Affiliation:
McGill University
W. Kozakiewicz
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The central difference of order s of the function f(x), Δs2hf(x), corresponding to a number h > 0, is defined inductively by the relations

.

If the limit of the difference quotient

exists at the point x, it is called the sth Riemann derivative or the generalized sth derivative of fix) at the point x.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 572 - 581
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Anghelutza, Th., Sur une équation fonctionelle caractérisant les polynomes, Mathematica (Cluj), 6 (1932), 1–7.Google Scholar
2. Anghelutza, Th., Sur une propriété des polynomes, Bull. Sri. Math. (2), 63 (1939), 239–46.Google Scholar
3. Brouwer, L. E. J., Over differentiequotienten en differ entialquotienten, Verh. Nederl, Akad. Wetensch. Afd. Natuurk. (Amsterdam), 17 (1908), 38–45.Google Scholar
4. Mandelbrojt, S., Analytic functions and classes of infinitely differentidble functions, Rice Institute Pamphlet, no. 1 29 (1942).Google Scholar
5. Marchaud, A., Sur les dérivées et sur les différences des fonctions de variable réelles, J. de Math. (9), 6 (1927), 337–425.Google Scholar
6. Popoviciu, T., Sur les solutions bornées et les solutions mesurables de certaines équations fonctionelles, Mathematica (Cluj), 14 (1938), 47–106.Google Scholar
7. Popoviciu, T., Les fonctions convexes (Paris, 1945).Google Scholar
8. Reid, W. T., Integral criteria for solutions of linear differential equations, Duke Math. J., 12 (1945), 685–694.Google Scholar
9. Saks, S., On the generalized derivatives, J. Lond. Math. Soc, 7 (1932), 247–251.Google Scholar
10. de la Vallée-Poussin, Ch., Cours d'analyse infinitésimale (New York, 1946).Google Scholar
11. Verblunsky, S., The generalized fourth derivative, J. Lond. Math. Soc, 6 (1931), 82–84.Google Scholar
12. Zygmund, A., Trigonometrical series (Warszawa, 1935).Google Scholar