All Questions
7 questions
7
votes
3
answers
905
views
A definition of the fractional derivative
I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it?
$$\frac{d^n}{dx^n}f(x) = \lim_{h \...
- 1,129
5
votes
1
answer
795
views
How to define transfinite derivatives of a function?
There are all manners of theories generalizing the notion of derivative. Amongst them is the fractional calculus, a rich theory which gives a sense to the derivation and integration of non-integer (i....
2
votes
1
answer
246
views
Eigenfunctions of the fractional Laplacian are smooth?
Let $\Omega\subset\mathbb{R}^n$ open, bounded with smooth boundary, let $s\in(0,1)$. I know that the fractional Laplacian has a sequence of eigenfunctions $\{e_k\}_{k\in\mathbb{N}}\subset H^s(\mathbb{...
- 339
1
vote
1
answer
191
views
What is the fractional derivative smoothness of functions from the Zygmund class?
Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional ...
- 2,715
0
votes
0
answers
52
views
References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
user537376
0
votes
0
answers
353
views
Inverse of the Riesz potential of a measure
Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$,
$$
I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy.
$$
Assuming $f$ ...
- 4,034
0
votes
0
answers
48
views
definition of functions that "weakly vanishes as $y\to\infty$" and find a proof of Theorem 9 in the article
I'm reading "Extension problem and fractional operators: semigroups and wave equations" by P.R. Stinga, i have two questions:
in Theorem 7 the author use the state "weakly vanishes as $...
- 339