30
votes
Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
This is basically a lightly transformed version of Euler's cotangent partial fraction expansion
$$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{n=1}^\infty \frac{1}{z-n} + \frac{1}{z+n}$$
(the log derivative ...
15
votes
Do distance functionals separate probability measures?
No. Suppose that $\Omega$ consists of four points arranged in a square, where adjacent points have distance 1 between them and opposite points have distance 2. Specifically, if the points are labeled ...
12
votes
Accepted
Representation of the Dirac delta function
As usual in such examples, there is no need to integrate against a test function. One can simply use the fact that if a sequence (or net) of distributions converges in the distributional sense, then ...
11
votes
Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
The op $$T_x = \frac{D_x}{e^{D_x}-1} = e^{b.D_x},$$
where $(b.)^n = b_n$ are the Bernoulli numbers, is (mod signs) often referred to as the Todd operator (maybe originally given that name by ...
11
votes
Accepted
On an asymptotic integral decay
I'll change $[-1,0]$ to $[0,1]$ (so $t\to -t$), which seems easier on the brain.
$f(t)=e^{-1/t}$ is a counterexample: For $0<t<1/\sqrt{\lambda}$, we have $f\le e^{-\sqrt{\lambda}}$, so this part ...
10
votes
volume over a hypercube, over simplex: twist by Euler numbers
This is only a partial answer. The Beukers-Kolk-Calabi change of variables
$$x_1=\frac{\sin{u_1}}{\cos{u_2}},\;\;x_2=\frac{\sin{u_2}}{\cos{u_3}},\ldots,
\;x_{n-1}=\frac{\sin{u_{n-1}}}{\cos{u_n}},\;\;...
8
votes
A system of non-linear equations that is decomposable as a product -- uniqueness of solution?
The linear-in-$g$ system is uniquely solvable whenever $f_0\ne0$. therefore the solution set is parametrized by $f$ in the form
$$g=f_0^{-10}P(f;a),$$
where $P$ is linear in $a$, polynomial in $f$ ...
7
votes
Accepted
Asymptotic behaviour of an integral
With the aid of Mathematica I found that
$$g(\theta)=\textrm{EllipticK}(a^2)-\textrm{EllipticF}(t,a^2).$$
I get the first terms of the asymptotic expansion
$$\frac{\sqrt{\pi}}{2\sqrt{k}}-\frac{\sqrt{\...
7
votes
Accepted
Physical interpretation of the mellin transform variable?
Physical interpretation: To develop a physical intuition, this article might be informative:
The power spectrum of the Mellin transformation with applications to scaling of physical quantities
The ...
7
votes
Accepted
Asymptotic behavior of integral with gamma functions
If I just insert the large-$z$ asymptotics of $\Gamma(z)\rightarrow \sqrt{2 \pi } e^{-z} z^{z-\frac{1}{2}}$, and take $z>1/2$ real for simplicity, I find
$$5^z\,F(z)\rightarrow \int_0^\infty \left(...
7
votes
Do distance functionals separate probability measures?
On the positive side, the answer is affirmative if $\Omega$ is the unit interval $[0,1]$ with its standard distance. In this case $\phi_\mu$ is a convex $1$-Lipschitz function (in fact, it is also ...
6
votes
Uncertainty principle for Mellin transform
There is something the above posts missed (perhaps because the wording didn't make it explicit): the condition on the function $f(x)$ is one sided (i.e., it assumes something on the decay as $x\to \...
6
votes
Radon transform between affine grassmannians
Yes, it is known.
See Boris Rubin, Radon transforms on affine Grassmannians, 2004, in particular theorem 4.2.
A free full text pdf is available at the linked page.
The reconstruction formula is not ...
6
votes
Why Mellin transform is omitting infinite terms?
This question has not been well received, but I am intrigued by the delta function Mellin transform and would like to respond. I found this 2004 paper by Norbert Südland and Gerd Baumann instructive. ...
6
votes
Accepted
Injectivity of a class of integral operators
If the support of $\mu$ is contained in $[0, b]$ for some $b \in (0,1)$, then $T_\mu f = 0$ for any $f$ that is $0$ on $[0,b]$, so it is not injective.
EDIT: Another example, where the support of $\...
6
votes
Integral with 4 Bessel functions and an exponential
Let's consider the second integral, which can be written in the following form:
$$
I(p, q, i, j, k, l; a, b, c, d)
:=
\int_0^\infty
dt\,
\exp(-p t^2)
t^q
j_i(a t) j_j(b t) j_k(c t) j_l(d t)
$$
where ...
6
votes
Representation of the Dirac delta function
$\newcommand\ep\epsilon\newcommand\de\delta\newcommand\R{\mathbb R}$Consider any family of nonnegative measurable kernels $K_\ep\colon(0,\infty)\to\mathbb R$ for $\ep\in(0,\infty)$ such that $\int_0^a ...
5
votes
Accepted
Infinite dimensional version of a simple fact on certain singular matrices
For the first question, the answer is not necessarily.
Very rough idea: The rank-nullity theorem doesn't always hold on infinite dimensional spaces.
Rough idea: Let the operator $A$ be defined on $...
5
votes
Accepted
An integral identity relate to the Gamma function or the Beta function
To evaluate $\int_{-\infty}^\infty {dx\over (1+ix)^a\,(1-ix)^b}$, view this as $\int_{-\infty}^\infty \widehat{f_a}(x)\,\overline{\widehat{f_{\overline{b}}}(x)}\,dx$ where $f_c$ are functions whose ...
5
votes
Relationship between the Radon transform and Twistor spaces
The correspondence between the Radon transform (from a space of real-valued functions on $\mathbb{R}^2$ to the space of functions on the manifold of straight lines in $\mathbb{R}^2$) and the Penrose ...
5
votes
2D Fourier transform of log function
If $v(r)=\log r$, then $\Delta v=\delta$ and $\hat {(\Delta v)}=1$, that is $-(\xi^2+\eta^2)\hat {v}(\xi, \eta)=1$. However this yields $-\hat{v}=1/(\xi^2+\eta^2)$ which has no sense since this ...
5
votes
Accepted
Fast computation of convolution integral of a gaussian function
Convolution with a Gaussian kernel of an $n$-point function has $n^2$ complexity, while Fourier transformation (FFT), multiplication, and inverse Fourier transformation is only of complexity $n\log n$....
5
votes
Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$
Maple does not know a symbolic answer for this. The very special case $a=2,b=0,c=1,d=2$ is evaluated by Maple in terms of the Whittaker M function or a $\;{}_1F_1$ hypergeometric function:
$$
\int_{0}...
5
votes
Accepted
Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?
With some effort (the lower integration limit requires care) I found this answer for the definite integral:
$$I(x)=\int\limits_{\delta}^{x} \left(y-\delta\right)^{-\frac{1}{ \epsilon+3}} y^{-\frac{\...
5
votes
How to solve the following $0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i t} dt, \forall \omega \in \mathbb{R}$
It may be helpful to rewrite this in a way that avoids the principal value:
$$0=\int_{-\infty}^\infty e^{-(bt+\omega)^2/2} f(t+\omega) \frac{2}{i t} dt=\int_{-\infty}^\infty dt \int_{-\infty}^{\infty}...
5
votes
A solution satisfying an integral inequality is bounded
Apparently, the question should be interpreted as follows:
Let $y\colon[0,\infty)\to[0,\infty)$ be a measurable function satisfying the inequality
\begin{equation}
y(t)+\int_0^t y(s)\,ds\le c_1\int_0^...
5
votes
Accepted
Integral of a function changing sign
In fact, this conjecture fails to hold for $x$ in a right neighborhood of $0$.
Indeed, let $g(x,s)$ denote the integrand. Note that $g(x,s)$ is continuous in $x\in[0,1)$ for each real $s>0$, and
$$...
5
votes
A density claim
This is not an answer to the question, but a proof of something much weaker: there is a sequence $c_m\to0$ such that if $\lVert g_m\rVert\leq c_m$ for all $m$, then the result from the question is ...
5
votes
A density claim
Following Jochen Wengenroth suggestion, I sketch the proof of the result from a comment.
The claim follows from the following
Lemma. If $\mu$ is a measure supported on $[0,X]$, where $X>1$ such ...
4
votes
Wave front set from the FBI or Segal-Bergman transform (and a motivation)
For your first question, have a look at chapter 1 of the book of Jean-Marc Delort, "F.B.I. Transformation - Second Microlocalization and Semilinear Caustics", Springer Lecture Notes in Mathematics ...
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