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6 votes
1 answer
343 views

Integral convolution equation $\int_{B_n(R) } e^{- \| x - t\|} d\nu(t) = e^{- \|x \|^2/2}$ on $x \in B_n(R)$. Find measure $\nu$

Let $B_n(R)$ denote the $n$ ball centered at zero with radius $R$. We are interested in the following integral equation: given $R>0$ and $\lambda>0$, let \begin{align} \int_{B_n(R)} e^{- \...
1 vote
0 answers
69 views

Is there an generalisation of convolution theorem to integral transforms

Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
20 votes
1 answer
1k views

Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
0 votes
1 answer
210 views

A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$

Is it possible to express $$\exp\left[\mathrm{e}^{x^2/2}p\left(\frac{d}{dx}\right)\mathrm{e}^{-x^2/2}\right] f(x)$$ as an integral transform or something similar? $p(x)$ is a polynomial. $$\exp\left[\...
1 vote
1 answer
282 views

Find an integral kernel for the solution of a partial differential equation: an initial value problem

Consider the following partial differential equation with an initial condition $u(x,0)=f(x)$: \begin{equation} \frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\...
3 votes
0 answers
308 views

Question on estimate in one of Jean Bourgain's 1992 papers

The paper in question is A Remark on Schrodinger Operators. The goal of the argument is to estimate the following integral: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
0 votes
0 answers
81 views

Fourier transform of an exponential function with radical argument divided by a radical

I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
2 votes
1 answer
272 views

Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space

Problem Statement Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
1 vote
0 answers
104 views

Kernel representation of a power of (pseudo-)differential operator

Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ...
3 votes
0 answers
272 views

A generalization of Weierstrass transform

As stated in this article, the Weierstrass transform of $f(x)$ is defined as: \begin{equation} W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy \end{equation} which can be ...
2 votes
1 answer
141 views

The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$

Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
1 vote
0 answers
71 views

Hyperplanes which equalize the Radon transforms of two distributions

Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
1 vote
0 answers
173 views

Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
2 votes
1 answer
334 views

Fourier transform of the indicator function of the semi-ball

I wonder if it is possible to derive an analytical form for the Fourier transform of the indicator function of a semi-ball: $$g_1(\underline{\xi}) = \int_{\mathcal{B}_+(R)} e^{i \underline{\xi} \cdot \...
16 votes
2 answers
2k views

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
0 votes
1 answer
227 views

Laplace transform injectivity for different values of $p$

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty ).$ Assume that there exist $p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,$ $p_{0}\neq ...
1 vote
0 answers
38 views

Solving an equation containing Laplace transform

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...
2 votes
0 answers
172 views

What are the necessary/sufficient conditions for a Fourier transform to have at least $k$ roots?

Let $f(x)$ be a symmetric function from $\mathbb{R}\to \mathbb{R}$, and $\hat f(k)$ be it's Fourier transform. What are the necessary and sufficient conditions for $\hat f(k)$ to have at least $n$ ...
-4 votes
1 answer
370 views

Is delta function symmetric against real axis? [closed]

Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$? I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis. We can write Delta function as $$\delta(z) = \...
5 votes
0 answers
254 views

Is there a practical application of natural integral or differintegral?

The following formulas give natural differintegral (that is one with naturally fixed integration constant): $$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ $$f^...
3 votes
1 answer
763 views

2D Fourier transform of log function

I am studying the paper found here. Halfway in the paper (Equation 6), the inverse 2D Fourier transform of $1/(k_x^2+k_y^2)$ needs to be determined. Is is stated that this is straightforward, and that ...
0 votes
0 answers
112 views

A close formula for a Fourier transform

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define $$ I(\alpha, x)=\int_\mathbb R \cos(xt) e^...
2 votes
1 answer
190 views

Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
1 vote
1 answer
385 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
2 votes
1 answer
286 views

CTRW: solve a renewal equation

Let X(t) be a continuous time random walk, with exponentially distributed waiting times of pdf $f_T(t)= k e^{-k t}\; t\geq 0$ and a jump sizes pdf $f_J(x)$. Suppose the initial distribution $\rho_{X(0)...
2 votes
1 answer
468 views

How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let: $\begin{eqnarray} p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\ G(x,y) &=& c_k\...
2 votes
0 answers
443 views

What is the Fourier transform of this function?

Consider the function $$ f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du. $$ It is known that $f(x_1,x_2)\in L^2(\...
5 votes
0 answers
236 views

Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform $\...
1 vote
0 answers
233 views

Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$ \mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx $$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
3 votes
3 answers
171 views

tranforms that lowers the number of variables of a function

Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...
3 votes
1 answer
162 views

Generalized Radon transform (Relaxed sufficient condition for invertibility)

The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace. To be more ...
4 votes
1 answer
704 views

Integral solving request

Dear all, please help me solve the following integral. I need to solve this integral for one of my problems. $$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho r)L*...