# Tag Info

### 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 ...

### 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 ...

### 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 ...

### 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 ...

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 $x\to \... 6 votes ### When is an integral transform trace class? A remark on (Q3): There is this famous example of T.Carleman (1916 Acta Math link) where he constructs a (normal ) operator with a continuous kernel such that it belongs to all Schatten p-classes if ... 6 votes ### Asymptotics of Fresnel integrals This is a classical, and very rich subject. A few years ago I advised a senior thesis on this subject. It came out nicely. I think that this thesis is a good place to start. It also has ... 6 votes Accepted ### Geodesics in finite groups First, my apologies for this late answer, I only found the question today. Below, I probably recall too many things, but I felt it could put some context around the short answer to question 1 saying: ... 6 votes ### Radon transform between affine grassmannians Yes, it is known. See Boris Rubin, Radon transforms on affine Grassmannians, 2004, in particular theorem 4.2. A free full text pdf is available at the linked page. The reconstruction formula is not ... 6 votes Accepted ### Asymptotic behavior of integral with gamma functions If I just insert the large-$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 $$5^z\,F(z)\rightarrow \int_0^\infty \left(... 6 votes ### Why Mellin transform is omitting infinite terms? This question has not been well received, but I am intrigued by the delta function Mellin transform and would like to respond. I found this 2004 paper by Norbert Südland and Gerd Baumann instructive. ... 6 votes Accepted ### Injectivity of a class of integral operators If the support of \mu is contained in [0, b] for some b \in (0,1), then T_\mu f = 0 for any f that is 0 on [0,b], so it is not injective. EDIT: Another example, where the support of \... 6 votes ### Integral with 4 Bessel functions and an exponential Let's consider the second integral, which can be written in the following form:$$ 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) $$where ... 5 votes Accepted ### Variations on the Mellin and Dirichlet transforms I know that the discrete Mellin transform was defined by V.S.Ryko: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=5138&option_lang=rus English reference: Soviet ... 5 votes Accepted ### Infinite dimensional version of a simple fact on certain singular matrices For the first question, the answer is not necessarily. Very rough idea: The rank-nullity theorem doesn't always hold on infinite dimensional spaces. Rough idea: Let the operator A be defined on ... 5 votes Accepted ### An integral identity relate to the Gamma function or the Beta function To evaluate \int_{-\infty}^\infty {dx\over (1+ix)^a\,(1-ix)^b}, view this as \int_{-\infty}^\infty \widehat{f_a}(x)\,\overline{\widehat{f_{\overline{b}}}(x)}\,dx where f_c are functions whose ... 5 votes ### Relationship between the Radon transform and Twistor spaces The correspondence between the Radon transform (from a space of real-valued functions on \mathbb{R}^2 to the space of functions on the manifold of straight lines in \mathbb{R}^2) and the Penrose ... 5 votes ### 2D Fourier transform of log function If v(r)=\log r, then \Delta v=\delta and \hat {(\Delta v)}=1, that is -(\xi^2+\eta^2)\hat {v}(\xi, \eta)=1. However this yields -\hat{v}=1/(\xi^2+\eta^2) which has no sense since this ... 5 votes Accepted ### Fast computation of convolution integral of a gaussian function Convolution with a Gaussian kernel of an n-point function has n^2 complexity, while Fourier transformation (FFT), multiplication, and inverse Fourier transformation is only of complexity n\log n.... 5 votes ### Solution to \int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx Maple does not know a symbolic answer for this. The very special case a=2,b=0,c=1,d=2 is evaluated by Maple in terms of the Whittaker M function or a \;{}_1F_1 hypergeometric function:$$ \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}... 4 votes Accepted ### Interpretation of the integral "with respect to a plane wave" in terms of Radon transform Sure you can't - but somehow you can. Obviously,$x\mapsto h(\theta\cdot x)$is not an integrable function (if not$\equiv 0$) since it is constant along lines perpendicular to$\theta$. However, if$...
Maybe you can accept the following result as a closed form. Transforming the variable of integration in your integral  I_{n}(a,b):= \frac{1}{2 a}\ e^{-a/2}\int_{0}^{\infty} dx\ x^{(b-2)/4} e^{-x/2}(...