Questions tagged [integral-transforms]
This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine transforms.
609
questions
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23 views
Extract root of Fourier transformed function
I am trying to work with numerical data of $g(x)$ where $g(x)=f^2(x)$. I need to extract the equivalent of the energy spectral density of $f(x)$ i.e. $|\hat{f}(k)|^2 \ \forall \ k$.
Given $\hat{g}(k)$,...
1
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1answer
40 views
Fourier Transform in polar coordinates of 1
Like in the table of transforms https://en.wikipedia.org/wiki/Fourier_transform#Distributions,_one-dimensional the FT (Fourier transform) of $\delta$ is 1 and the FT of 1 is $\delta$, but in polar ...
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0answers
19 views
Analytically Evaluate Cosine and Sine transform of a Sine composed with a Sine
I have been looking for ways to evaluate in general two integrals as follows below.
(1)
$\int \sin(f(t))\cos(\sigma t)dt$
(2)
$\int \sin(f(t))\sin(\sigma t)dt$
But I had no luck mostly. Now I stepped ...
1
vote
1answer
18 views
Proving the Integral is holomorphic
It is an integral from a note.
We concerend conformal map on $\mathbb{C}_+=\left\{ z:\text{Im}z>0 \right\} $ which satisfies $$
\text{Im}f\left( z \right) >0,\text{Im}f'\left( z \right) \ne 0\...
1
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1answer
22 views
Recurrence relation with Z-Transform
I'm revising the Z-Transform. I am looking at the book which gives an example of how to solve the recurrence relation
$$x_{k+2} - 3x_{k+1} +2x_k = 1$$
where $x_0 = 0$ and $x_1 = 1$. The book uses the ...
0
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0answers
19 views
Boundedness of integral transform
I am trying to prove that the following integral transform is a bounded linear operator. I am able to show that it is a linear operator but am unsure how to show that it is bounded.
Let $B\subset \...
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0answers
39 views
How to solve a 2nd order PDE with asymptotic boundary condition?
I came across the following diffusion problem in ''Myint-U, Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers (2007, BirkhƤuser)'' on page 526 (Problem 31):
Solve (...
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0answers
14 views
multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function
I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$
I struggle a lot of hours trying to find the trick in item C.
Can anyone help please ?
2
votes
2answers
66 views
Interpreting the logarithm as a sum of simple poles along the negative real axis
I've heard it remarked that you can basically consider $\log z$ to be a function which has simple poles everywhere on the negative real axis (with a constant "residue density" at each pole). ...
1
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1answer
51 views
The weak type (1,1) estimate for the Hilbert transform
I'm reading proofs that $H$, the Hilbert transform, is weak-$(1,1)$, so I'd like to show that there is a constant $C>0$ such that
$$| \{|Hf| > \lambda\}| \le \frac{C}{\lambda} \|f\|_{L^1}$$
for ...
1
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0answers
30 views
A Laplace-Bessel transform?
Consider the Laplace transform of a vectorial ($\in\mathbb{C}^3$) function $\vec{v}(x)\exp(i k_y y + i k_z z)$
$$
\mathcal{L}(\vec{v}(x)\exp(i k_y y + i k_z z))(s) \equiv \exp(i k_y y + i k_z z) \...
0
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1answer
32 views
Transform of a function of three random variables
Let $X$, $Y$, and $Z$ be independent random variables, where X is Bernoulli with parameter $1/3$, $Y$
is exponential with parameter $2$, and $Z$ is Poisson with parameter $3$.
(a) Consider the new ...
2
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0answers
51 views
What's inherently wrong with these Fourier transforms?
I was trying to write a formula for antidifference operator $\Delta^{-1}=(e^D-1)^{-1}$ using Fourier transforms. I obtained the formal formula:
$$\Delta^n[f](x)=\frac1{2 \pi }\int_{-\infty }^{+\infty }...
0
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1answer
15 views
Discrete differintegral using Fourier transform?
Using Fourier transform we can give a formula for a differintegral:
$$f^{(a)}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}{(-i\omega)}^{a} \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, ...
0
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1answer
27 views
can i write this $\int_{\mathbb{R}^2} \psi(\textbf{x}) \, d^{2}\textbf{x} = 2\pi \int_{0}^{\infty} \rho \, \phi(\rho)\, d\rho $?
If the function $\psi$ is isotropic, (i.e. $\psi(\textbf{x}) = \phi(|\textbf{x}|)$, where $\phi \in L^{1}(\mathbb{R})$, then can i write
\begin{equation}
\label{eq:2.1}
\int_{\mathbb{R}^2} \psi(\...
0
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0answers
87 views
What is the Laplace transform of $1/x$?
Wolfram Mathematica gives $-\ln|x|-\gamma$ but this cannot be correct because at zero Laplace transform of this function should take the value zero.
0
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0answers
39 views
Advantages of Laplace transform over Fourier transform dealing with renewal theory
I've just started reading "Renewal Theory" by D. R. Cox. I've been struck by the use of the Laplace transform instead of the usual charasterisc function to treat the subject. The main ...
2
votes
1answer
68 views
Looking for a characterization of the image of continuous probability densities supported in $[0,1]$ via the operator $f\mapsto (f*I)f$
Let
\begin{equation*}
U = \big\{f\in C(\mathbb{R}) \mid f=f\mathbb{I}_{[0,1]}\big\},
\end{equation*}
where $\mathbb{I}_{[0,1]}$ is the indicator function of the interval $[0,1]$.
Let
\begin{equation*}
...
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0answers
14 views
What are the standard ways of deriving and verifying the formulas for integral transforms where the formal formula for the transform diverges?
There are multiple formulas for integral transforms of various functions in the tables of integral transforms, but in many cases the integral, formally representing the transform diverges.
What is the ...
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0answers
20 views
Is it possible to simplify and improve this transform formula?
This formula
$$\mathcal{F}_s^{-1}\left[\frac{\left(e^{-i s}-1\right)} {(-i s)^2}\mathcal{F}_t[f(t)](s)\right](\omega )-2 i \mathcal{F}_s^{-1}\left[\frac1s\mathcal{F}_t[f(t)](s)\right](0)$$
is ...
0
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0answers
13 views
Probability distributions that are also well-known integral transforms, when integrated over their support?
I have some basic training in statistics from school.
While learning the basics about integral transforms on my own time, and probability distributions at school, I wondered if there are any examples ...
1
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1answer
37 views
Integral transform reduced?
I discovered that the following integrals are equal:
$$ \int_0^1sx^{s-1}\exp\bigg(\frac{t}{\log(x)}\bigg)~dx=\int_0^1\exp\bigg(\frac{st}{\log(x)}\bigg)~dx $$
Let $f^s(x)=x^s,$ then the LHS can be ...
0
votes
0answers
27 views
Can this integral be solved directly, transform method or is it unsolvable?
I have the following integral to solve:
$ f(x) = \int_{0,x_0}^{t,x(t)} e^{2t}\text{sin}(x(t)) \frac{dx(t)}{dt}$
How to approach this problem? Is it a transform or some other form of the problem?
I ...
1
vote
0answers
15 views
Multi-dimensional Wigner distribution
I would like to perform a Wigner transform of an object that depends on 4-different coordinates, and in addition, might satisfy a periodicity condition like $A(x_{1}+X,x_{2}+X,x_{3}+X,x_{4}+X)=A(x_{1},...
0
votes
0answers
43 views
Fast Fourier Inversion: Functions of a Complex Argument $f:\mathbb{C} \rightarrow \mathbb{R}$
I'm interested in functions $f: \mathbb{C} \rightarrow \mathbb{R}$ with associated Fourier decompositions $$ f(a + ib) = \int_{-\infty}^{\infty} F(\lambda) \ e^{i \lambda (a + ib)} \ d\lambda.$$
We ...
2
votes
0answers
53 views
Mellin transform yields Bessel function?
Consider the Mellin transform on bounded support $(0,1).$ I computed the following:
$$\mathscr M[f;s]=F(s)=\int_0^1 x^{s-1}e^{\frac{1}{\log(x)}}~dx=\frac{2K_1(2\sqrt{s})}{\sqrt{s}} $$
Where $K_1$ is ...
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0answers
56 views
How to extend the Radon transform to $L^2(\mathbb{R}^2)$?
The (2D) Radon transform $R$ is usually defined for functions in the Schwartz space $S(\mathbb{R}^2)$ or bump functions $C_c^\infty(\mathbb{R}^2)$ by
\begin{align*}
R\colon C_c^\infty(\mathbb{R}^2)&...
1
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1answer
58 views
Finding an inverse integral transform
I have the following integral transform:
$$ f(y) = \frac{1}{\sqrt{4\pi y}} \int_0^\infty x \exp\left(-\frac{x^2}{4y}\right) g(x) dx$$
where $g(x)$ is an even polynomial in $x$. Does somebody know the ...
1
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1answer
83 views
A Few Conceptual Questions About Laplace Transforms and Moment Generating Functions
I have a few quick questions designed to understand Laplace Transforms and Moment Generating Functions better.
Is the formulaic way to go from a Moment Generating Function to a Probability Density ...
2
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0answers
41 views
Boundness of Hilbert Transform - Finding an absolute constant for $0<p<1$
I am doing a course in Harmonic analysis and we are looking into the Hilbert Transform. Boundness properties for $L^p$ when $p \in (1, +\infty)$ is not unfamilar. We have gone through the classical ...
0
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2answers
56 views
Sum of a random variable and its own square
So I have the random variable $X$ in the following function:
$$ g(X) = \frac{X}{T}\left[ 1 + (X-1)\text{sinc}^{2}(fT) \right] $$
Expanding, gives us:
$$ g(X) = \frac{1 - \text{sinc}^{2}(fT)}{T}X + \...
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0answers
23 views
Composition of Hankel transform and translation operator.
For $\alpha \geq -\frac{1}{2}$ consider the measure $$d\gamma_\alpha(t)=\dfrac{t^{2\alpha+1}}{2^\alpha\Gamma(\alpha+1)}dt. $$ For $f\in L^1([0,\infty),\gamma_\alpha),$ we define its $\alpha$'th order ...
0
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0answers
16 views
Set of functions given integral conditions
Imagine I have the following set of equations which need to be satisfied for a set of complex functions $f_{1},f_{2}$:
\begin{eqnarray}
\int_{-\infty}^{\infty} dx e^{i(p-p' + (n-m-q)T)x}f_{1n(2n)}^{*}...
1
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0answers
27 views
Calculate kernel on generalised linear transformation
I have been lastly working with transformations of the following type:
\begin{eqnarray}
f(t)=\sum_{k}\int_{-a}^{a}dx e^{ixt+ikb}p_{k}(x,t)g_{k}(x)
\end{eqnarray}
By looking around, I discovered ...
1
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0answers
16 views
Inversion of a modified Abel transform with higher order on denominator
I was doing a research about retrive element densities from emission lines intensity observed by spacecraft. Follow the symbols of https://en.wikipedia.org/wiki/Abel_transform, suppose that intensity ...
1
vote
1answer
39 views
Expansion of nonlinear functions with damping properties in exponential series
I am working on solving nonlinear differential equations and found such a solution with exponential properties.
$\frac{dx}{dt}=\frac{d}{dx}(sech(x)^2)$
The solution of which is:
$x(t) = \sinh ^{-1}\...
0
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0answers
77 views
Find $f(x)$ if Fourier sine transform of $f(x)$ is $\frac k{k^2+1},\,k$ being the transform variable.
We are given that: $\mathscr{F}_s\{f(x)\}=F_s(k)=\frac k{k^2+1}.$ We need to find: $f(x)=\mathscr{F}_s^{-1}\{F_s(k)\}.$
My work so far:
We have:
$$f(x)=\sqrt{\frac2Ļ}\int_0^{\infty}\frac k{k^2+1}\cdot\...
1
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0answers
28 views
Mellin “Convolution” Theorem
The Laplace convolution theorem states that
$$\mathcal{L}f\cdot\mathcal{L}g = \mathcal{L}(f*g),$$
where $f*g := \int_0^tf(\tau)g(t-\tau)\mathrm{d}\tau$.
My question is: Is there a function $K$ that ...
0
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0answers
39 views
Solving a System of Coupled Partial Differential Equation - Integral Transforms
I'm trying to solve a system of coupled partial differential equations for a system I'm modeling. I've taken the simplest case for my system below:
$$\frac{\partial C_A}{\partial t}=\frac{\partial^...
2
votes
0answers
57 views
Inverse Mellin Transform
We all know that the Inverse Mellin Transform is
$$\left\{\mathcal{M}^{-1}\varphi\right\}(x) = f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, \mathrm{d}s.$$
So what is my ...
0
votes
1answer
39 views
Finding the Laplace Transform of $\frac{|x-a|}{x-a}$
I need to find de Laplace transform of $$f(x)=\frac{|x-a|}{x-a}$$ for $a>0$. So, I proposed $f(x)$ such that
$$f(x)= \left\{ \begin{matrix} -1, & \mbox{$0<x<a$} \\ 1, & \mbox{$x>a$}...
2
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0answers
40 views
Resolving an integral equal to the exponential generating function involving the Riemann zeta function
It is well-known that
$$-\gamma-\psi\left(1-x\right)=\sum_{n=1}^{\infty}\zeta\left(n+1\right)x^{n}$$
Using the OGF to EGF integral transformation, then
$$\frac{1}{2\pi}\int_{-\pi}^\pi (-\gamma-\psi(1-...
0
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0answers
23 views
This transform is similar to the Hankel transform. Does it have an inverse?
I am trying to determine whether the following integral can be inverted to obtain $f(r)$:
$$ \int_{0}^\infty r f(r)J_n((ia+\rho)r)dr = F(\rho), $$
where $J_n(x)$ is a Bessel function of the first kind ...
1
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0answers
41 views
Convolution using Integral Transforms
Returning to the question:
Approximation of the convolution operator
And new discussion:
Convolution using the Laplace integral transform of certain functions
$f(t) = e^{-t}$
$g(t) = e^{-(e^{-t})^2}$
...
1
vote
1answer
41 views
Basic properties of the Radon transform
This is a page from Evan's PDEs where the Radon transform is defined.
I have three brief questions:
(1) It says the integrals
$\int_{\Pi(s,\omega)}\nabla u\cdot b_{i}\,dS$ vanish because the vectors ...
0
votes
0answers
17 views
Inverse Laplace transform with branch cut at positive reals
I want to Laplace invert a function F(s) with simple poles at
$s = 0, -3$
and a branch cut from 0 to Infinity.
How to define the Bromwich integral properly?
$f(t)=\frac1{2\pi\mathrm i}\int_{\gamma-\...
2
votes
0answers
51 views
Inverse Radon transform approximation and natural spaces of Fourier transformation
In a CTĀ reconstruction, the inverse Radon transformation $R^{-1}$ is realized using "Fourier slice theorem/Projection slice theorem" and is covered in virtually every CTĀ book or course. We ...
2
votes
1answer
36 views
Convergence of hankel transform for polynomials
The hankel transform is related to fourier transforms that have some kind of spherical symmetry. The simplest is related to the 2D radially symmetric fourier transform. This transform $F(k)$ of $f(r)$ ...
3
votes
1answer
49 views
Proving integral transform using banach fixed-point theorem
I'm currently working on the following problem:
Let $K: [0,1]^2 \to \mathbb{R}$ be continuous with $|K(x,y)| < 1$ for all $(x,y) \in [0,1]^2$. Prove the existence of a function $f \in C([0,1])$ ...
0
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0answers
23 views
Question from Arfken and Weber 7th edition on 3d Fourier transform
The form factor $F(\textbf{k})$ and the charge distribution $\rho (\textbf{r})$ are 3D Fourier transforms of each other:
$F(\textbf{k}) = (2\pi)^{-3/2} \int \rho(\textbf{r}) e^{i\textbf{k}.\textbf{r}}...
