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### Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
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### Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...
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### When is an integral transfrom trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
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### MacWilliams Identity for Asymptotic Weight Spectrum of a Code

Introduction Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]}$ is called the ...
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### Properties of a matrix-valued generalization of the $\Gamma$ function

I am interested in the following function (Mellin transform of matrix exponential): $$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$ Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$....
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### Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
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### Can elliptic integral singular values generate cubic polynomials with integer coefficients?

For the elliptic integral of first kind, $K(m)=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-m^2sin^2\theta}}$, it is well-known that $K(m)$ can be expressed in what Chowla and Selberg call "finite terms" (i....
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I have a system of non-linear equations $a_1=f_0 g_1$ $a_2=f_1 g_1 + f_0 g_2$ $a_3=f_2 g_1 + f_6 g_2 + f_0 g_3$ $a_4=f_3 g_1 + f_7 g_2 + f_6 g_3 + f_0 g_4$ $a_5=f_4 g_1 + f_8 g_2 + f_7 g_3 + ... 1answer 140 views ### Asymptotic behaviour of an integral For$k\in\mathbb{N}_{0}$and$x\in\mathbb{R}$, define $$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$ where $$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1-... 1answer 133 views ### Radon transform between affine grassmannians Let \overline{GR}(n,k) be the manifold of all affine k-dimensional subspaces in R^n, and let R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l)), 0\le k<l\le n-1, be the ... 0answers 195 views ### Trace class norms of special integral operators Let \mu be a finite compactly supported Borel measure on the real line. On the space L^2(\mu) consider the integral operators$$ (K_a f)(x)=\int k_a(x, y)f(y)d\mu(y) $$with$$ k_a(x, y)=\frac{a\... 0answers 130 views ### Norms and distributions Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions)$\mu$on the$n$-sphere$S^n \subset \mathbb{R}^{n+1}\$ for which the function $$F(v) := \... 0answers 203 views ### Regularity class of certain diffeomorphisms of the real line. I care about the following class of homeomorphisms of \mathbb R, which I'll call \mathcal C^?. For simplicity, let us restrict attention to compactly supported homeomorphisms (a homeomorphism \... 1answer 196 views ### Forms satisfying the zero-energy condition on the projective plane Theorem (Michel). A 1-form on the projective plane is exact if and only if its integral over any projective line is equal to zero. Is there a simple proof of this result due, I think, to R. Michel ?... 1answer 147 views ### Asymptotics of Fresnel integrals It is known that \begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*} is a bounded ... 1answer 282 views ### Is this inverted integral transform valid? I have the following transform:$$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$with the following conditions: f(x) and F(y) must ... 1answer 490 views ### Numerically finding a Mercer expansion for a given covariance kernel Let c(r) be a nice, continuous function with compact support. For example, c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big) for r \in [0,1], and c(r) = 0 otherwise. On ... 0answers 162 views ### On the multidimensional Mellin transform of measures Consider an integral transform of Borel measures supported on \mathbb{R}^n_+ given by$$ f(z) =\int\limits_{\mathbb{R}^n_+} x^{z}\frac{\mu(dx)}{x} $$where z = (z_1,...,z_n) \in \mathbb{C}^n, x^... 2answers 483 views ### Elaborating Mercer's theorem (RKHS) on Cameron-Martin space k(x,y)=\min(x,y) Hi, I'm trying to employ Mercer's theorem on the kernel k(x,y)=\min(x,y). It is known (and easy to verify) that this is a nonnegative-definite kernel over [0,T] for any T>0. Fix T>0. ... 4answers 4k views ### Does the inverse Laplace transform of the square root exist? Does the inverse Laplace transform, defined by the integral, $$F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds$$ ... 1answer 254 views ### Variations on the Mellin and Dirichlet transforms There are a number of variations on the Laplace transform that turn up all over math. Some examples: \int_{-\infty}^{\infty} f(t)e^{-st} dt - The Laplace transform \sum_{-\infty}^{\infty} f(t)z^{-... 1answer 183 views ### Continuity of integral Assume that f:[0,2\pi]\to [0,2\pi] is a continuous function such that f(0)=f(2\pi) and define the function$$g(s)=\int_{-\pi}^\pi \frac{\sin f(t+s)-\sin f(s)}{\sin t/2} dt.Is g continuous or ... 1answer 128 views ### Injectivity of the Funk transform for nonsmooth functions Let S^{n-1} be the unit sphere in \mathbb R^n and \Gamma_n the collection of great circles on it. Assume n\geq3. The Funk transform of a function f:S^{n-1}\to\mathbb R is a map Ff:\Gamma_n\... 2answers 180 views ### Reconstructing set of points from one-dimensional images Consider a set of N points in n-dimensional space, i.e. \begin{align*} \{x_1, \dots, x_N\} \subset \mathbb R^n. \end{align*} Let us be given a finite family of non-injective matrices \begin{... 1answer 324 views ### All solutions to a set of integral equations I would like a better understanding of the set of pairs (f_1,f_2) of functions [0,1] \times [0,1] \to [0,1] which satisfy the following conditions: For all y \in [0,1], f_1(x,y) \geq f_1(x',y)... 2answers 281 views ### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform Question: Suppose a(x,y)\in C^\infty([0,1]\times [0,1]) and suppose\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$Is a(x,0)=0, \forall x\in[0,1]? ... 0answers 141 views ### Calculate an integral For x\in \mathbb R_+, let us define$$ I_\lambda(x)=\frac2π\int_0^{+\infty}\frac{\sin t}{t\sqrt{1+t^2x^2}}\cos(\lambda \arctan (xt)) \,dt,\quad \lambda\in 1+2\mathbb N. $$We see that I_\lambda(0)=... 0answers 98 views ### Evaluating an integral of a periodic function. It's positive? My purpose is to show that this integral I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du\,\,,\,\,x,... 0answers 116 views ### Estimating singular values of integral operators I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on L^2({\mathbb R},dx) the integral operator$$(Tf)(x)=\int_{\mathbb R}...
Is there a way to solve analytically the Fredholm integral equation of the second kind $$\int_0^{100} K(s, t) f(s) ds = \lambda f(t)$$ where the kernel has the piecewise 'linear' form \begin{align} ...