Marchaud derivatives allow us to calculate fractional-order derivatives of sufficiently regular but not necessarily differentiable functions. Then, it is natural to ask what kind of continuous functions are annihilated by such derivatives.
Hi, its seems to me that the Marchaud derivative applies to uniformly continuous fonctions and more precisely to Holder functions.
Yeah, that is effectively true, functions being B-Holder have A
Hi, I have just found the following link to a book:
https://books.google.fr/books?id=zAXaBwAAQBAJ&pg=PA27&lpg=PA27&dq=kernel+of+marchaud&source=bl&ots=eDIR11UUWO&sig=GjrLk10VIAYpqAKDisIgLuPHzs0&hl=fr&sa=X&ved=0ahUKEwjkxYi5nKbSAhWKfxoKHU6BBTkQ6AEIKzAD#v=onepage&q=kernel%20of%20marchaud&f=false
The best book for such derivative is Samko + Kilbas + Marichev's. I never use such derivative, but I can help you. Send me your mail address
Dear Prof. Manuel Duarte Ortigueira,
I am an admirer of your work on fundamental and inspiring applications and developments of Fractional Calculus. I have know the very very Book of Samko and its papers, but such a question is not answer in this such good references, at least in those I have read. It is needed some advanced mathematical analysis, I have a manuscript on it, but I am not sure that such question is or not an open problem.
Muito obrigado, cumprimentos
Add your answer
Dear Aldo,
I would say that the question is an open problem. If the question is considered in the scale of Lp-spaces, then it follows from the identity (see Theorem 6.1 in Samko-Kilbas-Marichev, as you obviously know)
D^\alpha I^\alpha f=f
that the kernel is trivial in the image of Lp, 1
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics