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 ...
Terry Tao's user avatar
  • 104k
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 ...
George Lowther's user avatar
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 ...
klempner's user avatar
  • 206
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 ...
Tom Copeland's user avatar
  • 9,377
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 ...
Christian Remling's user avatar
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}},\;\;...
Zurab Silagadze's user avatar
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 $f$ ...
Denis Serre's user avatar
  • 50.2k
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{\...
juan's user avatar
  • 6,881
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 ...
Carlo Beenakker's user avatar
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(...
Carlo Beenakker's user avatar
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 ...
Pietro Majer's user avatar
  • 54.3k
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 \...
H A Helfgott's user avatar
  • 18.9k
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 ...
Joonas Ilmavirta's user avatar
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. ...
Carlo Beenakker's user avatar
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 $\...
Robert Israel's user avatar
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 ...
JCGoran's user avatar
  • 159
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 ...
Iosif Pinelis's user avatar
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 $...
Willie Wong's user avatar
  • 34.6k
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 ...
paul garrett's user avatar
  • 22.3k
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 ...
Carlo Beenakker's user avatar
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 ...
Giorgio Metafune's user avatar
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 $n\log n$....
Carlo Beenakker's user avatar
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}...
Gerald Edgar's user avatar
  • 39.9k
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{\...
Carlo Beenakker's user avatar
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}...
Carlo Beenakker's user avatar
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^...
Iosif Pinelis's user avatar
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 $$...
Iosif Pinelis's user avatar
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 ...
Saúl RM's user avatar
  • 7,224
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 ...
Oleg Eroshkin's user avatar
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 ...
Pedro Lauridsen Ribeiro's user avatar

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