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 ...

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55
votes
3answers
2k views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
1
vote
0answers
23 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where ...
1
vote
2answers
77 views

Find “$g(x)$” knowing that “$x=\int_{0}^{\infty} g(tx) dt$”???

The entire question states what I am looking for. I'm looking for a function $g(x)$ in terms of $x$ which satisfies the condition that follows. This seems like it's related to "integral ...
2
votes
0answers
26 views

Approximate solution for integration of two Bessel Functions

I am an engineering student. I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$ which is a function of $(n-1)$. degree polynomial. My statement as follows: ...
0
votes
0answers
11 views

Abel transformation - sources

I'd like to study the Abel transformation, that is, $$Af(x) = \int\limits_x^\infty\frac{f(t)t}{\sqrt{t^2 - x^2}}\ \mathrm{dt},\quad x\in(0,\infty).$$ I'm especially interested in ...
0
votes
0answers
11 views

Transforming a PDE to an ODE by discretizing the size-axis of the integro-partial differential equations

I have this issue on modelling a process. It is a system of ODEs except for one equation related to the population balance of crystals in a mixture that is a PDE. The author of the publication ...
4
votes
1answer
53 views

Differentiation under integral sign?

I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
0
votes
0answers
17 views

Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$ A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
2
votes
1answer
84 views

Riesz transform does not preserve continuity

I've read somewhere that the Riesz operator $R_j$ defined by $$R_j f(t) := c(n) \, \text{pv} \int_{\mathbb{R}^n} \frac{x_j}{|x|^{n+1}} f(t-x) \, dx$$doesn't preserve the continuity, but I can't ...
3
votes
1answer
489 views

A problem about Laplace transform and Parseval–Plancherel theorem

I am reading a paper about fractional differential equation. One of the piece said as follow: By applying the Parseval–Plancherel theorem we may show: \begin{equation} \int_0^\infty ...
0
votes
0answers
8 views

mellin transform

Could you help me how (π) in the boundary condition where w(x,0)=2π transfer into 2T/s when we solve the equation ? It is really important.
0
votes
2answers
46 views

Is this kernel invertible ? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}$

Is the following Kernel invertible? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}, x\in[0,1],y\in [0, \infty)$ i.e. if $\int_0^1 K(x,y) f(x) dx=0 ,\forall y\in [0, \infty)$ can we conclude $f(x)=0,x\in ...
1
vote
1answer
28 views

Using Fourier Transform to solve an ODE

Consider the differential equation $$f^{iv}+3f^{''}-f=g$$ I have read that taking the Fourier Transform of both sides gives ...
0
votes
1answer
54 views

Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
0
votes
1answer
415 views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 ...
1
vote
0answers
56 views

The Fourier-Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
0
votes
0answers
34 views

Complex inversion of a function

I am trying to find the function whose laplace transform is below using the complex inversion formula: $$ f(s)= \frac{se^s}{(s-2)^3}$$ My attempt below seems to be giving me the wrong answer but I'm ...
0
votes
0answers
9 views

Transform and domain such that phase shift of the original function results in shift in transformed domain variable

It is known that the Fourier transform of a phase-shifted function results in a constant shift of the dependent variable of the phase spectrum: If $ F(x(t)) = X(w) = |A(w)| \cdot e^{-i \cdot ...
1
vote
1answer
81 views

Mellin-Barnes transform of $\frac{1}{\Gamma}$

Let $\varphi \in L_1(c+i\mathbb R)$, where $c > 0$. Then we can define the Mellin-Barnes transform (or the inverse Mellin transform) of the function $\varphi$ by the formula $$ \mathcal ...
0
votes
0answers
22 views

Why M.G.F transform is injective a.s.?

We always use the theorem that If we know a random variable's MGF, we can determine its Pdf, which means the map from Pdf to Mgf is injective almost surely. And I just wanna know why this is ture.
1
vote
2answers
72 views

Inverse Laplace Transform of $e^{\frac{1}{s}-s}$

doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$ F(s) = e^{\frac{1}{s}-s} $$ I can't find it in any table and the strong singular growth ...
6
votes
3answers
202 views

How to learn Integral Transform?

I major in Electronic Engineering when back in college. I learned the Fourier Transform, Laplace Transform, Z Transform and wavelet Transform. But I always feel a lack of thorough understanding of ...
1
vote
0answers
29 views

Solving differential equation by transform with contour integral

In my notes, I have an example for solving for the Airy function in the equation: $$\frac{d^2y}{dx^2}-xy=0$$ So he uses the contour integral to represent the solution (with $C$ as a yet unspecified ...
0
votes
1answer
118 views

Is it possible to calculate the inverse Laplace transform of the following?

I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...
1
vote
0answers
183 views

Limits in Double Integration Question

I’m really having problems getting suitable limits when using double integration. For example: Let $E$ be a region defined by $$E=\left\{(x,y): y-x \leq 2,\ x + y \geq 4,\ 2x + y \leq 8\right\}$$ ...
0
votes
0answers
34 views

Fourier sine/cosine transforms of :1) derivatives raised to power & 2)derivatives in exponential

I need help to solve the Fourier finite sine and cosine transforms: First, reciprocal of derivative: $$ F_s\left(\frac{1}{\frac{\partial u}{\partial x}}\right)= \int_{0}^{a}\frac{1}{\frac{\partial ...
3
votes
0answers
30 views

How and why an integral Transform is created?

I don't know if what I'm going to ask will make any sense, but I was just wondering about integral transforms. I am talking about, for example, Mellin Transform, or Laplace Transform or Hilbert ...
3
votes
1answer
2k views

Calculating an integral derived from the convolution of two Fourier transforms

Let $\sigma>0$ , $1<\alpha\leq 2$, and $-1\leq \beta \leq 1$. I am looking for a closed-form solution (or something near) for the following integral. $$\frac{1}{2 \pi } \text{PV}\int_{-\infty ...
1
vote
0answers
18 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
0
votes
0answers
40 views

Existence of solutions in time and Laplace domains

I have not made use of Laplace transforms for many years since my education and I am a bit rusty on applying the various theorems associated with the transform. I have an equation $f(t)=0$ and I am ...
0
votes
1answer
32 views

How do you show $F$ has a well-defined inverse function $G : (0,1) \to \mathbb{R}$, which is strictly increasing [closed]

If $X$ be a random variable with cdf $F$, and $F$ is continuous and strictly increasing on $\mathbb{R}$ how do you show $F$ has a well-defined inverse function $G : (0,1) \to \mathbb{R}$, which is ...
5
votes
1answer
76 views

What is the advantage of the Fourier Transform over the Hartley Transform?

The Hartley_transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega ...
2
votes
0answers
46 views

Integral calculation by using Mellin Transform

I want to use the Mellin Transform (MT) to calculate the integral: $\int_0^{1 } \exp(-2\rho^2) J_0(\pi \rho r)\rho \, d\rho$ in which $r>=0$ and real. I have calculated it by numerical methods. ...
10
votes
3answers
2k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
2
votes
0answers
34 views

Fourier transforms of Gaussians with rational functions

Consider functions of the form $$ f(x)= e^{- x^2/\sigma^2} \frac{a_0 + a_1 x + \cdots +a_n x^n }{b_0+ b_1 x + \cdots + b_m x^m } $$ Under which circumstances do these functions have closed forms for ...
2
votes
1answer
1k views

Inversion formula for the Abel transform

I need an inversion formula for the Abel transform $$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$ Hint: The inversion formula found on Wikipedia appears to be incorrect. The ...
21
votes
1answer
203 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...
1
vote
1answer
21 views

Does a symetric complex function $k(t,s)$ verify $\overline{k(t,s)}=k(t,s)$?

I am trying to figure out why an integral operator is self-adjoint. The operator is: $$K(f)=\int_{0}^{1} k(t,s)f(s)ds$$ From $L^2([0,1])$ to $L^2([0,1])$ and $0, \leq t,s \leq 1$ So I did a bit of ...
1
vote
1answer
71 views

How to integrate this looking simple ODE?

I meet an ODE about $V(\theta)$ $$\frac{d^2V}{d\theta^2}+\frac{1}{2V}=0.$$ But I can not figure out how to integrate it to yield $$\left(\frac{dV}{d\theta}\right)^2+logV=C_1$$ or ...
0
votes
1answer
68 views

Laplace Transform

The question I had was Find the Laplace transform of $$f(t)=10e^{-200t}u(t).$$ Would it be correct to take out the 10 because it is a constant, find the Laplace transform of $e^{-200t}$ and then ...
0
votes
0answers
26 views

Integral transforms involving logarithms?

We have the fourier-transform: $$F\{f\}(w) = \int_{-\infty}^\infty f(x)\exp(iwx)dx$$ Which has extremely many applications and interpretations throughout science and engineering. For instance since ...
2
votes
0answers
19 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...
4
votes
2answers
164 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around ...
0
votes
0answers
8 views

What are the different type of Daubechies Wavelet transform?

Like Daub4 are there others named as Daub2, daub3 or we only have daub4 , daub8, daub16? What is the order of a transform(represented usually by N)? Does this order have any resemblance with the ...
6
votes
2answers
270 views

Stieltjes Transform

Stieltjes transform for a distribution $F(x)$ is defined as $$m(z)=\int \frac{dF(x)}{x−z}$$ where z is complex with positive imaginary parts and $F(x)$ is a distribution function. Basically, I am ...
1
vote
0answers
26 views

Stieltjes Transform is injective?

Let $\mu$ a probability measure on $\mathbb{R}$, we define the Stieltjes transform by : $$ S[\mu](\lambda)=\int_\mathbb{R} \frac{d\mu(t)}{t-\lambda} $$ For all ...
0
votes
0answers
13 views

Hankel transformation

How to solve the Hankel transform of $x^2e^{-ax}$ with kernel $x(J_2(px))$ where $J_n(x)$ denotes bessel function of order $n$ of first kind?
0
votes
1answer
30 views

Gaussian quadrature with a to $[0,1]$ reference domain instead of a $[-1,1]$ reference domain?

For 1-d Gaussian quadrature with two points per element we have the following formula to transform an integral from an arbitrary domain $[a, b]$ to the reference domain $[-1,1]$ on which various ...
1
vote
1answer
94 views

Bilateral Laplace transform

My knowledge of Bilateral Laplace transform is less. Here are the few questions I need answer. What is the condition for existence of bilateral Laplace transform? How is the condition for existence ...
1
vote
0answers
36 views

Find the Fourier transform for this function

Find the Fourier transform for this function $$f(x)=e^{x-e^x}$$ My Solution:- $T[f(x)]=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx} f(x)dx$ $=\frac{1}{\sqrt{2\pi}} ...