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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3
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1answer
22 views

Inverting a Particular Integral Operator

Consider trying to find a function $f \in L^2(0,1)$ satisfying $$a_n = \int_0^1 f(x)x^n dx$$ Where $n$ is a nonnegative integer. Is there any method to go about doing this in general for any ...
0
votes
1answer
32 views

Definite integral of trigonometric functions with complicated arguments

I came across beautiful integral (where $n$ is integer) $I(n, z) = \int_0^{\pi} \cos(nx) \sin(z \cos(x) ) \mathrm{d}x $ According to Gradshteyn and Ryzhik (p 414, Sec. 3.715, Eq. 13), solution is ...
-2
votes
0answers
25 views

Find the function whose Fourier sine transform is $\frac{s}{1+s^2}$? (Please check my solution)

There needs to a $\sqrt{\frac{2}{\pi}}$ before that integral, which I have left by mistake. I have used this result : $$\int_{0}^{\infty}\frac{\sin{a x}}{x}dx = \frac{\pi}{2}$$ where $a > 0$. My ...
3
votes
1answer
518 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 v(t)B_\...
1
vote
1answer
31 views

Which functions arise from a probability measure in this way?

Given a probability measure $\mathbf{P}$ on the interval $I=[0,1]$, we get a corresponding function $f:(\mathbb{R}_{>0})^2 \rightarrow \mathbb{R}_{>0}$ as follows: $$f(x,y) = \int_\mathbf{P} x^q ...
1
vote
0answers
18 views

Fourier transform spherically symmetric function with complex constant

In Gradshteyn's section 17.24 on Fourier transform pairs for spherically symmetric functions, the third entry relates $\frac{e^{-ar}}{r}$ and $\sqrt{\frac{2}{\pi}}\frac{1}{(a^2 + k^2)^2}$. I think ...
0
votes
0answers
39 views

Injective Integral Operator on $L^2[0,1]$ or $C[0,1]$?

Consider an arbitrary $f \in L^2 [0,1]^+ $ where $L^2[0,1]^+$ is the function space of square integrable non negative functions. We say $T$ is an Integral Operator if $T$ is of the form , $$ T(f) = ...
0
votes
1answer
32 views

Intuition behind chosing coordinates

Once given the right coordinates (polar, sphere, cylindric) I am able to determine the value of a given integral. But how do I know, if the coordinates are not explicitly given, which coordinates to ...
0
votes
0answers
29 views

How to prove this identity? Transformation theorem

Let $A\in\mathbb R^n$ be a measurable set with finite measure. For a fixed vector $p\in\mathbb R^{n+1}$ define a cone with basis $A$ and peak $p$ as $$K(A,p)=\{tp+(1-t)q \in\mathbb R^{n+1} \,| \, q \...
0
votes
1answer
49 views

Computing integral by using variable transformation

Let $I := \int_{(0,1)^2}\frac{1}{1-xy}\, d\lambda^2 (x,y)$. Can someone help me to determine $I$ only buy using the transformations $u=\frac{1}{2} (y+x)$ and $v=\frac{1}{2} (y-x)$? I don't know how ...
1
vote
2answers
33 views

Transformation of a sphere and computing an integral by using sphere coordinates

Let $V \subset\mathbb R^3$be the ellipsoid $$9x^2+4y^2+z^2≤36.$$ How can I express $V$ as a transformation of a sphere and how can I compute the sphere $$\int_v x^2\,d\lambda^3(x,y,z)$$ with sphere ...
0
votes
1answer
38 views

Fourier inverse/convolution problem

I'm struggling to do part (b) of this problem. I do not know how to start: I'm trying to use the inversion formula, but I don't know what to do with the $e^{|s|}$ part (the other one is the laplace ...
0
votes
0answers
23 views

Asymptotic behavior of inverse laplace transform [duplicate]

My question may be quite rough. Let $F(\lambda)$ be the Laplace transform of some function $f(t)$, $$ F(\lambda)= \int_0^\infty e^{-\lambda t}f(t) dt. $$ If I have knowledge about $F(\lambda)=O(\...
1
vote
1answer
34 views

Is it possible to represent the derivative operator as an integral transform?

Apparently, the Schwartz kernel theorem states that all linear operators can be represented as integral transforms (but only if you use generalized functions such as the dirac delta as kernels.) ...
2
votes
1answer
628 views

What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
0
votes
1answer
429 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 p(x,y)}{\...
14
votes
2answers
843 views

Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function $f\...
0
votes
0answers
33 views

Multi integral change of variables?

Sketch the domain D bounded by $y=x^2$, $y=1/2x^2$, and $y=2x$. Use a change of variables with the map $x=uv$, $y=u^2$ to calculate: $∫∫y^{-1}dxdy$ Ok so I found the Jacobian to be $-2u$, and ...
73
votes
6answers
13k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
0
votes
0answers
12 views

Hankel transform of oscillating function

I'm looking for the Hankel transform of the following function $f(x,\rho_0)= \left[\frac{4 \sinh ^2(x)}{\cosh (2 \rho_0)+\cosh (2 x ))}-1\right]\left[\frac{1}{\cosh(x-\rho_0)}+\frac{1}{\cosh(x+\rho_0)...
1
vote
1answer
14 views

Why does count of Z -Transform of sequence change?

I was looking at the reference [1] below and noted the author defined the Z-transform for [1, 2, 3] as $$[6, \frac{11}{4}, 2]$$ I worked it out as follows: $$X[z]=\sum_{n=0}^2x[n]z^{-n}$$ $$=x[0]z^0+\...
61
votes
4answers
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 ...
2
votes
1answer
76 views

Exercise 2 , chapter 5 , Stein & Shakarchi real analysis

Consider the Mellin transform defined initially for continuous function $f$ of compact support in $R^+=${$t\in R:t>0$} and $x\in R$ by $Mf(x)=\int_0^\infty f(t)t^{ix-1}dt$ Prove that ($2\pi$)$^{-...
0
votes
0answers
64 views

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

Is there some procedure for figuring this out or am I venturing into unsolved territory here?
0
votes
0answers
18 views

non square transformation of random variables

Let $x_0$ and $w_0$ be independent random variables and let $x_1$ be related to them by $x_1 = f(x_0, w_0)$. I want to find the joint density of $x_1, x_0, w_0$. The transformation I am interested ...
0
votes
0answers
6 views

How can I obtain the inverse transform?

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to Q1: $...
0
votes
1answer
26 views

Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
0
votes
1answer
19 views

Expressing the Weierstrass transform in terms of the unilateral Laplace transform

I was looking for a way to express the Weiestrass transform of $f(t)$, $$\mathcal{W}\{f(t)\}(s)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(t)\exp\left(-\frac{(s-t)^2}{4}\right)dt$$ in terms of the ...
0
votes
1answer
22 views

Weierstrass transform on the Riemannian manifold

I've read on this Wikipedia article that Weierstrass transform (WT) can be defined on any Riemannian manifold $(M,g)$, but it seems a bit complicated to me. I'm not sure but I guess one can write the ...
3
votes
2answers
234 views

Kernel of an integral operator with Gaussian kernel function

Suppose we have the integral operator $T$ defined by $$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$ where $f$ is assumed to be continuous and of polynomial growth at most (just to ...
1
vote
0answers
28 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
81 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
27 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 estimates/...
0
votes
0answers
12 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
58 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
19 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
109 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 ...
0
votes
2answers
47 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
29 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 $$\left(i\lambda\right)^{4}F\left(\lambda\right)+3\left(i\lambda\right)^...
0
votes
1answer
56 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 ...
1
vote
0answers
57 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 $$\gamma(x+y)=\gamma(x)\gamma(y)\\|\gamma(x)|=1$...
0
votes
0answers
35 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 \phi(w)}$...
1
vote
1answer
85 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 M_c^{-1}...
0
votes
0answers
26 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
73 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
219 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 ...
2
votes
0answers
35 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 ...