A new proof of the GGR conjecture
For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ginchev, Guerragio, and Rocca states that the existence of the $n$th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of all $n(n+1)/2$ generalized Riemann derivatives, \[ D_{k,-j}f(x)=\lim _{h\rightarrow...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.413/ |
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| Summary: | For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ginchev, Guerragio, and Rocca states that the existence of the $n$th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of all $n(n+1)/2$ generalized Riemann derivatives,
\[ D_{k,-j}f(x)=\lim _{h\rightarrow 0}\frac{1}{h^{n}}\sum _{i=0}^k(-1)^i\binom{k}{i}f(x+(k-i-j)h), \]
for $j,k$ with $0\le j |
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| ISSN: | 1778-3569 |