Abstract
In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics.
Highlights
One century ago, while we are writing, in 1917, a paper by Hermann Weyl, Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung, appeared, [1]
While we are writing, in 1917, a paper by Hermann Weyl, Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung, appeared, [1]. It dealt with the definition of a fractional derivative in a weaker sense with respect to the approach classically known at that time with the name of Riemann–Liouville derivative
In my opinion, the name Marchaud is not so popular even among the mathematicians dealing with fractional calculus, in particular among scientists coming from
Summary
While we are writing, in 1917, a paper by Hermann Weyl, Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung, appeared, [1]. As a consequence of this research, we realized in particular that Marchaud derivative and Weyl derivative have been, in a sense, perhaps a little put aside in the last time, especially considering the great development and the large popularity that research about nonlocal operators has recently had This last remark is essentially based on the popularity of other fractional derivatives, for instance the Riemann–Liouville derivative or the Caputo derivative (see [7,18,19]) for a modern approach to these operators. We want to analyze the definition of fractional derivative given by Weyl and Marchaud, concentrating on those aspects that, in perspective, seem to be more flexible for generalizing to other situations the notion of nonlocal operator We remark that, from a historical point of view, it would be interesting to deepen our knowledge of these two characters of the mathematical world, especially considering the influence and the role of the respective mathematical schools compared to the other mathematicians of their time and their scientific legacy
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