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.

Filter by
Sorted by
Tagged with
1
vote
0answers
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
vote
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 ...
0
votes
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
vote
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
votes
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 \...
1
vote
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, Birkhäuser)'' on page 526 (Problem 31): Solve (...
0
votes
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
vote
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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*} ...
0
votes
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 ...
0
votes
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
votes
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
vote
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 ...
4
votes
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
vote
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
vote
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
votes
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
votes
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 + \...
0
votes
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
votes
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
vote
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
vote
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
votes
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{\frac2π}\int_0^{\infty}\frac k{k^2+1}\cdot\...
1
vote
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
votes
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
votes
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
votes
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
vote
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 CT reconstruction, the inverse Radon transformation $R^{-1}$ is realized using "Fourier slice theorem/Projection slice theorem" and is covered in virtually every CT 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
votes
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}}...

1
2 3 4 5
13