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 ...

learn more… | top users | synonyms

3
votes
0answers
29 views

Can inversion integral of characteristic functions on a finte interval be bounded?

For a real-valued uni-variate r.v. $X$, with pdf $f(x)$ and absolute integrable cf $\varphi(t)$, we have the following transform:$$2\pi f(x)=\int_{-\infty}^{\infty}e^{-itx}\varphi(t)\,dt.$$ However, I ...
0
votes
1answer
11 views

Transformation of spherical means

I just have a small question: It is always stated that $$ \frac{1}{n \alpha (n)r^{n-1}}\int_{\partial B(0,r)} u(x) \text{d}x = \frac{1}{n \alpha (n)}\int_{\partial B(0,1)} u(rx) \text{d}x $$ Now I ...
0
votes
0answers
26 views

Combining two integral equation. Hankel transform

I have this two integral equations which are an Hankel transform pair. I gotta combine them to find $A(k)$ $$ R^{n+{1}/{4}} \, \Sigma(R,t=0) \, = \, \int_0^\infty [A(k)\, k^{-1}] \, J_l(ky) \, k \, ...
4
votes
2answers
51 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
34 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 ...
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
19 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
41 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
30 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 ...
0
votes
1answer
40 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
25 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
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 ...
2
votes
1answer
37 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.) ...
0
votes
0answers
35 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 ...
0
votes
0answers
13 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)...
2
votes
1answer
79 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
21 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
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
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 ...
0
votes
1answer
21 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 ...
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 ...
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: $...
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
12 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 ...
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)^...
2
votes
1answer
110 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
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
2answers
48 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 [...
0
votes
0answers
10 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)}$...
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 ...
2
votes
0answers
36 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 ...
0
votes
0answers
36 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 u}{...
3
votes
0answers
31 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 ...
1
vote
0answers
21 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 ...
2
votes
0answers
49 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. ...
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 ...
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
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 $$\theta=\int^V\...
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 ...