The integral-transforms tag has no usage guidance.

**-2**

votes

**0**answers

16 views

### If derivative of f(x) is f'(x), then what is the integral of (f'(x))^2 in terms of f(x)? [migrated]

Is there some procedure for figuring this out or am I venturing into unsolved territory here?

**3**

votes

**1**answer

76 views

### 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 ...

**2**

votes

**0**answers

158 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 ...

**1**

vote

**0**answers

89 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 ...

**1**

vote

**1**answer

70 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 ...

**4**

votes

**0**answers

89 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 ...

**0**

votes

**1**answer

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 ...

**1**

vote

**0**answers

63 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 ...

**2**

votes

**1**answer

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 $A_wf:I\to\mathbb R$,
$$
...

**1**

vote

**0**answers

49 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

**3**

votes

**1**answer

170 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, ...

**0**

votes

**0**answers

50 views

### 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) ...

**2**

votes

**1**answer

115 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 ...

**6**

votes

**2**answers

264 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 + ...

**6**

votes

**1**answer

135 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
...

**1**

vote

**0**answers

102 views

### 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 ...

**1**

vote

**0**answers

46 views

### 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 ...

**8**

votes

**1**answer

188 views

### 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 ...

**1**

vote

**0**answers

55 views

### 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 ...

**3**

votes

**1**answer

58 views

### 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
...

**0**

votes

**0**answers

57 views

### $L^2$-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, \ \ ...

**6**

votes

**1**answer

132 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 ...

**3**

votes

**1**answer

386 views

### 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 ...

**2**

votes

**0**answers

123 views

### 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 ...

**0**

votes

**0**answers

53 views

### 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 ...

**2**

votes

**1**answer

109 views

### 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 ...

**4**

votes

**1**answer

125 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 ...

**2**

votes

**0**answers

61 views

### 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) ...

**3**

votes

**0**answers

59 views

### 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( ...

**1**

vote

**0**answers

104 views

### 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 ...

**4**

votes

**1**answer

245 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} ...

**2**

votes

**1**answer

183 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 ...

**5**

votes

**1**answer

145 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 ...

**1**

vote

**2**answers

152 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 ...

**0**

votes

**1**answer

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

**7**

votes

**1**answer

319 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 ...

**3**

votes

**1**answer

151 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 ...

**2**

votes

**0**answers

380 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 ...

**1**

vote

**0**answers

58 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) := ...

**0**

votes

**0**answers

54 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) \ ...

**3**

votes

**1**answer

76 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. ...

**2**

votes

**0**answers

47 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 ...

**19**

votes

**1**answer

742 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 ...

**0**

votes

**0**answers

164 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
$$
...

**3**

votes

**1**answer

129 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 ...

**3**

votes

**0**answers

112 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 ...

**2**

votes

**0**answers

132 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 ...

**3**

votes

**0**answers

225 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) = ...

**1**

vote

**2**answers

337 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.

**3**

votes

**3**answers

340 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 ...