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### Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem.
For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...

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265 views

### How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& c_k\...

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### Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...

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165 views

### A question about multidimensional integral

Consider the function
$$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$
Is there a necessary condition on the $\Omega_i$'...

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142 views

### Orthogonality of Pfaffian polynomials in $SO(2m)$

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem.
Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = \...

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92 views

### Integration of Bessel Function of the first kind

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows:
$$\int_{0}^\infty F(x)[x^3*B*J_0(xy)+x^4*J_1(xy)]dx=G(y)$$
where $B$ is a ...

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381 views

### What is the Fourier transform of this function?

Consider the function
$$
f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.
$$
It is known that $f(x_1,x_2)\in L^2(\...

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72 views

### Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers):
$$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$
I could not find it in ...

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93 views

### Evaluating an integral of a periodic function. It's positive?

My purpose is to show that this integral
\begin{equation}
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,...

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35 views

### Integral transforms involving square roots

I am considering the following integral equation
$\frac{1}{y} = \int_a^{\infty} g(x,y) x^{-1/2} dx$,
where $g(x,y)$ is to-be-determined and $a$ is a positive constant (if it is instructive, it can ...

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64 views

### Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform,
$$
f(s)=\int_0^\infty e^...

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109 views

### Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$.
If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$,
$$
A_wf(y)=\int_y^1\frac{f(x)w(x,y)}{\sqrt{x^2-...

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668 views

### 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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185 views

### Definite integral with modified Bessel functions, trigonometric function and a power

I require the following integral involving the modified Bessel functions of the first and second kinds of order one
$$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x, \...

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59 views

### Integration involving modified bessel function, exponential and power

I need to find the following integration.
$$
\int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N
$$
where
$$
a>0, \quad N \geq 1
$$
Any help will be much appreciated.
BR
Frank

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320 views

### Multivariate Maximal Hilbert Transform

One way to define the maximal Hilbert transform of a function, $f$, is by
$$\mathcal{H}[f](x):=\sup_{\varepsilon>0} \left| \int_{|x-t|\geq\varepsilon} \frac{f(t)}{x-t} \, dt\right|, \quad x\in\...

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265 views

### A system of non-linear equations that is decomposable as a product — uniqueness of solution?

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 + ...

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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)...

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### Weak continuity of the Hilbert transform

Is there a simple direct way to prove that the Hilbert transform sends $L^1(\mathbb R)$ into $L^1_w(\mathbb R)$? The Hilbert transform is the convolution by $pv(1/x)$ which is the (distribution) ...

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117 views

### What function is a Gaussian integral

Let $g(u,\delta)=E[f(x)]$ where the expectation is over $N(u,\delta^2)$.
Is there a characterization what function $g(u,\delta)$ can be produced this way? Is there a procedure solve the inverse ...

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341 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} f(t,x)=\frac{f(t,...

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136 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-...

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### Looking for some “nontrivial” examples of pseudodifferential operators/symbols

I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial.
Obviously, the first example of a symbol that most people talk about is just a polynomial in $\...

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### Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?

I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...

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### Physical interpretation of the mellin transform variable?

I shall keep this to the point: Given a time domain signal say microphone recording of a conversation:
Laplace tranfrom of x is some function X(s) say defined in the complex plane. I like to think ...

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### Conditions for Mellin inversion

Under which conditions is the function
$$
g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R}
$$
the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...

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### Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet.
Link:http://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded
...

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### $L^2$-boundedness of integral operator

Let $a:{\bf R}^d\to M^{d\times d}$ semi-definite matrix consisted of smooth functions i.e.
$$
\langle a(x) \xi,\xi \rangle=\sum\limits_{k,j=1}^d a_{kj}(x)\xi_j \xi_k \geq 0, \ \ x\in {\bf R}^d, \ \ \...

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### 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 ...

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### Relation between the eigenvalue density and the resolvent?

Disclaimer: This is a cross-post from math.stackexchange. Given that there is little activity on the subject (random-matrice) on the aformentioned site, and given that many interesting discussion on ...

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### A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations

EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference.
I keep stumbling on the ...

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### What am I missing in this highly oscillatory integral? [closed]

I want to numerically integrate this equation (in python):
$\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k) $,
where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist kind....

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### Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)

Consider the following Volterra integral equation
$g(t)=∫_0^tK_n(t,s)w_n(s)ds$
where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...

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### Wave front set from the FBI or Segal-Bergman transform (and a motivation)

In the André Martinez's notes "Introduction to microlocal and semiclassical analysis" the Wave Front Set is defined as the complement of the set of points having neighborhoods where the FBI transform ...

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### 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\...

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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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### Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:
$y[n] = x[n] + y[n-1]$
$Y(z) = X(z) + Y(z) \...

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### How to evaluate the following integral related to exponential distribution

I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that
$$
\int_0^{\...

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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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### Using Mellin transform for a certain function

In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where
$$
D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( \sqrt{1-4z+4z^{p+...

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### Transformation of kernel

I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if $R(y_1,...

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247 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^{-...

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146 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 ...

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188 views

### Kernel of the Radon transform

Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density (...

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154 views

### Motivating the Bessel translation operator

In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ...

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591 views

### 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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### Selecting Rays for Simulated Radon Transform

I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because ...

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### Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared:
$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$
where $\rho,s>1$, $Q^{\...

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### Is it possible to get an equation with two exponentials and a bessel function in closed form?

Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...