All Questions
Tagged with integration fractional-calculus
13 questions
4
votes
1
answer
966
views
Fractional moments of multivariate normal distributions
Is there an analytic formula for fractional moments of multivariate normal distribution?
$E(\prod_{i=1}^k x_i^{\nu_i})={?}$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and ...
- 41
3
votes
0
answers
231
views
Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?
The well-known integral expression for the entire function:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...
- 4,169
2
votes
1
answer
166
views
Integral inequality for Schwartz function
Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that:
$$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$
...
- 339
2
votes
1
answer
324
views
Uniform estimation of an integral involving a Hölder-continuous function
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\...
- 339
2
votes
1
answer
291
views
An inequality involving fractional Laplacian
I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$
...
- 339
1
vote
0
answers
210
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
1
vote
0
answers
47
views
Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
- 339
0
votes
1
answer
176
views
Symmetry of fractional laplacian
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that:
$$\int_{\...
- 339
0
votes
1
answer
124
views
Uniform estimation of an integral
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
- 339
0
votes
1
answer
64
views
Laplace transforms of fractional equation
is there a finite expression of the Laplace transforms of the function
\begin{align}
L\left[ {\frac{{{x^n}}}{{{{(1 + x)}^m}}}} \right]\quad n \ge m
\end{align}
- 377
0
votes
0
answers
145
views
Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms
Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
0
votes
0
answers
62
views
Fractional laplacian on $H^s(\mathbb{R}^n)$ and symmetry
Let $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n),$ i define the fractional laplacian of $u$ in the following way:
$$(-\Delta)^su(x)=C(n,s)P.V.\int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\quad\...
- 339
0
votes
0
answers
82
views
Integral equality involving fractional laplacian
Let $s\in(0,1)$, let $u\in H^s(\mathbb{R}^n)$. For all $\psi\in\mathcal{S}(\mathbb{R}^n)$, let:
$$ (-\Delta)^s\psi(x)=c(n,s)\lim_{\epsilon\to0^+}
\int_{\mathbb{R}^n\setminus B_\epsilon(0)}\frac{\psi(...
- 339