The integral-transforms tag has no wiki summary.

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### Calculate Inverse Fourier Transform [on hold]

How to calculate the Inverse Fourier Transform of the following functions:
$\dfrac{1}{-1+2\pi i x}$
$\dfrac{1}{(2\pi ix)^2-2 \pi ix +1}$
I don't know how to evaluate the integrals.

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**1**answer

115 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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22 views

### Is the Mellin transform of a measure nongrowing at imaginary infinity everywhere, or just on the fundamental strip?

Let $\mu$ be a measure on the positive real numbers. Its Mellin transform is a complex function defined by
$$
M_\mu (s) =\int x^{s-1} d \mu(x)
$$
on the set $S_\mu$ of $s \in \mathbb{C}$ where
$$
...

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**1**answer

50 views

### Asymptotics of Fresnel integrals

It is known that $I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x$ is a bounded smooth function on $(0,\infty)$ ...

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41 views

### A fundamental lemma involving a certain exponential kernel

Let $h \in L^1(\mathbb R^n, \mathbb R)$ be a scalar field and let $\Psi_t: \mathbb R^n \to \mathbb R$ be smooth mappings, parameterized by $t \in \mathbb R$.
Suppose that we are given data $$D(v,t) = ...

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**2**answers

103 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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**1**answer

89 views

### 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 $\rho,s>1$, ...

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**1**answer

142 views

### Multivariate Maximal Hilbert Transform

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

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100 views

### 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 ...

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291 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 ...

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50 views

### Reason for the Choice of Line Parameters in the Radon Transform

Why are the lines, over which the integrals in a Radon Transform are calculated, apparently always parameterized as $L(t,\phi,\alpha) := ...

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31 views

### Multiple integral of the resolvent kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel
$$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$
should be calculated. However, it is not ...

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47 views

### Integral representation formula for convex

For $u \in \mathbb{S}^{d-1} \subset \mathbb{R}^d$, it is easy to show that:
\begin{equation}
u=c_d \int_{\mathbb{S^{d-1}}} \xi \mathbb{1}_{\left\{x \cdot u >0 \right\}}(\xi) \ ...

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123 views

### Approximate $F(\theta)=\sin(\theta)\int_{-l}^{l} e^{-ikz\cos \theta} h(z)\,dz$

$$F(\theta)=\sin(\theta)\int_{-l}^{l} e^{-ikz\cos \theta} h(z)\,dz$$
We know that $F(\theta)$ is defined on $0\le \theta \le \pi$ and $h(z)$ is defined on $|z|\le l$ and $z$ is real in this case, but ...

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64 views

### General Radon-type inverse problem

Let $f : \mathbb R^n \to \mathbb R$ be a density which is sufficiently smooth and can also be restricted to have compact support for now.
Let $t \ge 0$ and $F_t : \mathbb R^n \to \mathbb R$, i.e. ...

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27 views

### Central-Slice-Theorem Analogue for Wavelet Transforms?

The 2D Radon transform and the 2D Fourier transform are related by the so-called Central Slice Theorem (cf e.g. http://en.wikipedia.org/wiki/Projection-slice_theorem) and I would like to know, whether ...

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551 views

### 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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118 views

### 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
$$
...

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123 views

### 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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89 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 ...

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100 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

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173 views

### What is the inverse kernel of this integral transform?

I am looking for the associated inverse kernel to the integral transform $T$ defined by
$(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$
whose kernel is $K(u,t) = ...

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140 views

### Integral involving exponential and Marcum-Q function

Do you have any suggestions to solve the following integral:
$\int\limits_0^\infty {{e^{ - a{x^2}}}{Q_1}\left( {bx,cx} \right)dx}$
Thank you very much.

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160 views

### Basic Questions about Radon Transforms

I am currently working on a problem that may be interpreted as recovering an unknown function from its Radon transform.
Unfortunately I don't have any background in Radon transform, but need to ...

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**1**answer

164 views

### Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function.
Thanks!

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### Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as
$$
\mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx
$$
Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...

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**1**answer

227 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 ...

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### 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, ...

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217 views

### double integral and Hadamard finite part

Given the divergent integral
$$ \int _{0}^{\infty}dx \int_{0}^{\infty}dy \frac{x^{2}y+1}{1+x+y} $$
how can I apply Hadamard's finite part to give a finite meaning to it ?
It is just made by ...

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**1**answer

192 views

### Integral operator defined on $two$ distinct dense subspaces

I have decided to edit my post a bit heavily for clarity. I was trying to be fairly general but it's hard to see what I'm asking so I've decided to limit myself to a specific example which will ...

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175 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
...

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151 views

### tranforms that lowers the number of variables of a function

Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...

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386 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

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202 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) &=& ...

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122 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) := ...

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135 views

### Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion

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}
...

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### Does this kind of integral equations have unique solution?

Suppose $f_1$ and $f_2$ are two probability density functions on support $[0,1]$ (i.e. $f_1(x)=f_2(x)=0$ for any $x\not\in[0,1]$). Let $\varphi(x)$ denote a known probability density function on ...

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108 views

### Character of continuous series representation of GL(2)

It is wellknown that the character of an irreducible, unitary representation of $GL(n,\mathbb{C})$ uniquely determines the isomorphism classes. I fail to construct a function for $GL(2, \mathbb{C})$, ...

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88 views

### What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer.
Integral 1.8.2.4:
$$
\int_0^x x^{\nu+1} ...

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### Partial recovery from Radon transform

Let $f : \mathbb R^3 \to \mathbb R$ be an integrable function. Let $\eta$ be a one-dimensional subspace of $\mathbb R^3$. We denote $p+\eta$ the affine subspace (a line) which is obtained by ...

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222 views

### Indefinite integration of multiplication of two Bessel function

I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case?
$$\int_{a}^{\infty}J_2(bx)J_1(cx)\,dx$$
Thanks in ...

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### Generalized Radon transform (Relaxed sufficient condition for invertibility)

The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace.
To be more ...

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383 views

### $\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral:
$\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$
($\mathrm{BesselJ}[n,x]$ is ...

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### “Limited angle” in n-dimensional Radon transform?

The Radon transform in two-dimensions is well studied. It maps a sufficiently nice function $f: \mathbb R^2 \to \mathbb R$ to its line integral along a certain line $L$, i.e.
\begin{align*}
...

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145 views

### A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$,
$$\int_z^\infty ...

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**1**answer

126 views

### On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information:
$$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$
where:
$X_s$ is a wide sense stationary process with known ...

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**1**answer

221 views

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

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

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### Interpretation of the integral “with respect to a plane wave” in terms of Radon transform

This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform:
$\newcommand{\R}{\mathbb{R}}$
$$
Rf(\varphi,s)=\int_{-\infty}^\infty ...

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269 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 ...

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**1**answer

195 views

### Determining the asymptotic behavior of random matrices with vanishing ratio dimensions

Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic ...