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.

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The paired domains of integral transforms. Is there a moniker?

Use the Fourier transform example. For several years I thought of Fourier transform pairs as functions in two domains which were paired by a term "conjugate domains" and significantly the dimensions ...
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Hankel Transform Property/Inverse For a Function

I don't know much about Hankel transform properties, but I do know it is its own inverse. However, I would like to know that if the arguments were swapped, i.e.on the left hand side, $r\to k$, would ...
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Unitary change of variables in real integral

I am interested in the solution of the folliwng multivariate Gaussian integral: \begin{equation} I=\int_{\mathbb R^N} \mathrm d x\; e^{-\frac 12 x^T\Omega x} \end{equation} where $\Omega$ is a complex ...
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Using orthogonal transform in set of SDE's

If I have a set of SDE's in Ito form \begin{align} d\phi(z) &= \sin(\psi(z))dW_1(z)+\cos(\psi(z))d\tilde{W}_1(z) \\ d\psi(z) &= -\bigg(\sin(\psi(z))dW_1(z)+\cos(\psi(z))d\tilde{W}_1(z)\bigg) ...
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Laplace Transform Integral

$$\mathscr{L}\left ( \int_0^t e^{t-\tau}\cos(t-\tau)e^{-\tau}d\tau \right )$$ Hi all I am trying to solve this Laplace transform but I dont know if it's correct, please tell me. Here is my attempt:...
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Solution of Hilbert type Fredholm integral equation

Is there a formula similar to Hilbert inverse relation that would solve following integral equation in general form for $\phi$ as an integral? $$f(x) = \int_1^\infty \frac{\phi(y)dy}{x-y}$$ Provided ...
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Difficult numerical Integral with Besselfunctions: transformation of variables?

For a physics problem that I'm trying to study I would like to exand an eigenproblem in the eigenfunctions of the laplacian over a unit disk with neuman boundry conditions. To do this I need to ...
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Inverse Laplace transform of $s^c/log(s)$

I'm trying to solve for the inverse Laplace transform of $\frac{s^c}{\log(s)}$ where $c$ is some constant. Mathematica is apparently unable to solve it, and while I know there's a running joke that ...
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24 views

Is there a broader definition of the convolution operation?

The convolution operator is defined as $${\displaystyle (f*g)(t)\triangleq \ \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} $$ where it shares a relationship with the Laplace transform such ...
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Jacobi function

Im reading a material for discrete and contionus Jacobi transform by E.Y. Deeba and E.L Koh. I can't understand a proof of lemma 2.2. To be more precise i can't understand how did they bound absolute ...
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27 views

Change of argument in Laplace Transform

The Laplace transform of a function $f=f(x)$ has the following definition: $$\mathcal{L}({f(x)})=\int^{\infty}_0e^{-sx}f(x)dx \tag{1}$$ However, when $f=f(ax-c)$, where $a$ and $c$ are arbitrary ...
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application of infinite Hilbert transform to an integral equation

Using infinite Hilbert transform pair, find the solution to the integral equation $$\frac{1}{1+x^2}=\int_{-\infty}^\infty\frac{Y(t)}{x-t}dt$$ Modifying the RHS a bit we have $$\frac{1}{\pi(1+x^2)}=\...
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how to obtain the general solution using separation of variables.

Consider the Laplace equation $$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0,\quad\quad-\infty<x<\infty,\quad 0<y<\infty$$ with the boundary conditions $...
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Unboundedness of complex $R^{-1}$ implies unboundedness of real $R^{-1}$?

Define the Radon transformation as $Rf(\varphi,s) = \int_{x \in L(\varphi,s)} f(x) dx_L = \int_{t=-\infty}^{\infty} f(s\theta + t\theta^\bot)dt$ where $\theta = \theta(\varphi) = (\cos \varphi, \sin \...
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$\int_{s = - \infty}^{\infty} Rf(\varphi,s)h(s) ds = \int_{x \in \mathbb{R}^2} f(x) h(\langle \theta, x \rangle) dx$

I want to show the following: Let $f \in L^1(\mathbb{R^2})$ and let $\varphi \in [0,2\pi]$. Let $h \in L^{\infty}(\mathbb R)$. Then $\int_{s = - \infty}^{\infty} Rf(\varphi,s)h(s) ds = \int_{x \in \...
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Validating solution to PDE using integral transforms

I'm trying to obtain the analytical solution of a Fokker-Planck PDE, which the solution is a probability density function, and then use this to find the mean of some quantity in the paper. The paper ...
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Reference request: why integral transforms?

I understand there have been previous questions on why/how exactly integral transforms arise, but I am here asking specifically for reference requests to sources other than full-length books. My ...
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How do Integral Transforms work II (About their Kernels)

This is kind of a second part of this question: How do Integral Transforms work. There, I replied a comment asking about how can I find kernel for the inverse integral transform and then I was ...
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Operational Calculus

Recently, I have been looking at operational calculus, integral transforms and so on. If you check my profile, you will see lots of questions related to these topics and no satisfatory answer or good ...
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Hilbert Transform singularities

It is widely known that the Hilbert Transform (HT) of a discontinuos function like a square signal has "problems" around the discontinuity points. Also, the Fourier transform of the HT of real ...
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Definite integral transform $\int_{x=a}^{x=b}f(x)dx=\int_{t=0}^{exp(-a)}f(-ln(t))\frac{dt}{t}$

I found that transform by Chebyshev: But it doesn't work with simple function in Mathcad test: What's wrong with this ...
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Solve this Fourier Transform

I want to solve the following integral: $$ f_{X}(x) = \int_{-\infty}^{\infty} \exp\left( \frac{-j (N-1) \alpha t - (N+1) \alpha \beta t^{2}} {(1- j \beta t) (1+j \beta t)}...
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about the solution method of a non-homogeneous heat equation

Solve the following problem $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial u}{\partial t}-2x$$ subject to $u(0,t)=0$, $u(1,t)=0$ and $u(x,0)=x(1-x)$ where $0<x<1$ and $t>0$. I have two ...
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an application of convolution in a mixed BVP

Show that the solution to the problem $$\frac{\partial^2 u}{\partial x^2}=a^{-2}\frac{\partial u}{\partial t}$$ subject to $u_x(0,t)=-f(t)$ and $u(x,t)\to 0$ as $x\to \infty$ and $u(x,0)=0$ where $...
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75 views

Inverse Laplace Transform of $\frac{1}{s+1}$ using Mellin's Inverse Formula

I am trying to compute the inverse Laplace Transform of $F(s)=\frac{1}{s+1}.$ I know that it is supposed to be $f(t)=e^{-t}$, but I am specifically trying to get that result using Mellin's Inverse ...
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solving an integral equation using Fourier transform of convolution type [duplicate]

Solve the integral equation : $$\int_{-\infty}^\infty\frac{u(t)}{(x-t)^2+a^2}dt=\frac{1}{x^2+b^2}$$ I used the convolution theorem for Fourier transform w.r.t. $x$ as follows $$\mathcal{F}\bigg(\int_{...
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solving a steady state temperature distribution problem using Hankel transform

Show that the steady-state temperature distribution problem $$\frac{\partial^2u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{\partial^2u}{\partial z^2}=0$$ subject to $\...
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solving a singular integral equation using Mellin transform

Solve the following singular integral equation using suitable integral transform : $$\int_0^\infty u(t)\cos(xt)dt=e^{-x}$$ One easy method is if I use fourier cosine transform. But instead I chose ...
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Integrals of Radon transforms

I was looking at the following lemma from the paper "The weak Harnack inequality for the Boltzmann equation without cut-off" by Cyril Imbert and Luis Silvestre, I am struggling to prove this lemma. ...
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Is there an integral transform kernel that satisfies the following property?

Let $K(k, u)$ be an integral transform kernel that transforms $$\tilde{p}(k) = \int du K(k,u) p(u),$$ where all the variables are real. For a given function $g(u_1, u_2)$, is there a way to find out ...
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Mellin Transform of $h(x-a)$

The question is from Integral Transform For Engineers By Andrews and Shivamoggi. Evaluate the Mellin transform of the given function. When possible, use known integral results from previous ...
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another series evaluation involving cosine term using Mellin transform

Evaluate the following series using Mellin transform : $$\sum_{n=1}^\infty \frac{\cos an}{n}$$ Yesterday I had a discussion with a similar type problem in this forum, with $\displaystyle f(n)=\frac{\...
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Mellin transform of a general term of an alternating series

What is the Mellin transform of the following function $$f(n)=\frac{(-1)^{n-1}}{n^2}\cos an \ ?$$ I encountered this problem while evaluating the series $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n-...
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Use of Mellin transform for evaluation of a series

Show that $$\sum_{n=1}^\infty \frac{\sin an}{n}=\frac{\pi-a}{2} \ , \ 0<a<2\pi$$ I was asked to use Mellin transform to prove this result. So I used a formula related to the general series as ...
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Checking a bound on the Stieltjes transform from Terence Tao's notes

I'm trying to check a bound on Stieltjes transform of a probability measure, that's given in equation (2.92) on P. 170 in Terence Tao's notes "Topics in Random Matrix Theory". Denote the Stieltjes ...
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Exponential order

Help me to find out which of the following function are of exponential order ? 1.$f(t)=\sqrt{|tan t|}$ 2.$f(t)=e^{tlogt}$ This question is from the book 'Integral Transform for Engineers' by Larry ...
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Writing differentiation as an integral transform?

Define, for $f\in C^1(\mathbb R)$, the operator $D$ as $$ Df(x)=f'(x). $$ My question is: can $D$ be written in the form $$ Df(\xi)=\int_{\mathbb R} K(\xi,x)f(x)dx $$ for some function or distribution ...
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Solution of Partial Differential Equation induced from Feynman Kac Theorem.

Title : Analytical solution of Patial Differential Equation induced from Feynman Kac Theorem. In probability space $(\Omega, \mathfrak{F}, \mathbb{P} )$, Let a random variable $X$ be a solution of ...
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In what sense does the Laplace Transform give components of a signal in something called the 's-domain'?

For example, I understand how this make sense for the Fourier transform. When we do the transform, we get 'how much of each frequency' in present in the signal. We get the value of the coefficients, ...
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Integral of the product of two functions

I don't know if it is something trivial and well-known, or something that depends on more conditions. Assume $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a multivariate strictly positive function ...
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Radon transform inversion formula

I have trouble following Deans's derivation of the inverse Radon transform formula for $n=2$ on this page of his book "The Radon Transform and Some of its Applications" (see snapshot) Formulas (3.9) ...
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Explanation of the Ramp Filter

A student came to my office with a question about the Ramp filter. As I basically don't have any knowledge of the subject, I was unable to answer the question. Her question was this: Take an input ...
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Can all nonlinear groups be represented by integral transforms?

This is an extension to the infinite-dimensional case of the usual question, "are all groups linear". Given that nonlinear groups exist, can such groups always be represented as a group of integral ...
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What is the inverse transform for this integral transform?

I can't seem to figure out what the inverse of this transform would be, if there is one. Can anyone here find it, or prove there is none if that is the case? $$\int{f(t)\exp(-\frac{1}{1-x^2})dt} $...
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How do summations/integrals like Fourier, Laplace, z-transforms preserve all the information about the original signal?

In normal summations, like 2+3=5, the information about the original numbers is lost. But in infinite summations like integral transforms, no information is lost and the function can still be ...
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Looking for a rigorous treatment about Laplace transform.

I am looking for a book that deal with integral transforms in a very rigorous way, Fourier analysis I discover that functional analysis and Lebesgue Integral books cover the Fourier analysis( With a ...
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31 views

Double Fourier transform of a single variable function

So what I'd like to achieve is the following: suppose we have a time dependent signal $f(t)$ - e.g. the wave of a song. This song contains a drumbeat that (for the sake of simplicity) has a single ...
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Uniqueness of the inverse kernel of an invertible integral transform

Say we have an integral transform $T$ that maps any function $f$ to the function $\varphi=T\left\lbrace f\right\rbrace$, defined by $$\varphi(s)=\left[T\left\lbrace f\right\rbrace\right](s)=\int_a^...
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74 views

Is there a Mellin transform or an analogue on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$?

From Wikipedia, the Mellin transform is an isometry $M : L^2(\mathbb{R}^+) \mapsto L^2(\mathbb{R})$, $$\{M f\} (s) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}^+} x^{-1/2 + \mathrm{i} s} f(x) dx.$$ ...
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61 views

Integral over unit sphere

I am looking for ideas to calculate $$ J(x) = \int_{|y| = 1} f(x\cdot y) dy $$ where $x,y\in\mathbb R^n$, $|x|,x\cdot y$ are the Euclidean norm and dot products, and $f(t)$ is a real function of a ...

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