All Questions
Tagged with fourier-transform integral-transforms
32 questions
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 $a\in (1,2)$.
- 178
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":
$$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
- 10.1k
6
votes
1
answer
343
views
Integral convolution equation $\int_{B_n(R) } e^{- \| x - t\|} d\nu(t) = e^{- \|x \|^2/2}$ on $x \in B_n(R)$. Find measure $\nu$
Let $B_n(R)$ denote the $n$ ball centered at zero with radius $R$. We are interested in the following integral equation: given $R>0$ and $\lambda>0$, let
\begin{align}
\int_{B_n(R)} e^{- \...
- 671
5
votes
0
answers
254
views
Is there a practical application of natural integral or differintegral?
The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$f^...
- 10.1k
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 $\...
- 16.2k
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.
$$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho r)L*...
3
votes
1
answer
763
views
2D Fourier transform of log function
I am studying the paper found here. Halfway in the paper (Equation 6), the inverse 2D Fourier transform of $1/(k_x^2+k_y^2)$ needs to be determined. Is is stated that this is straightforward, and that ...
- 230
3
votes
3
answers
171
views
tranforms that lowers the number of variables of a function
Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...
- 183
3
votes
1
answer
162
views
Generalized Radon transform (Relaxed sufficient condition for invertibility)
The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace.
To be more ...
user45183
3
votes
0
answers
308
views
Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
- 870
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 ...
- 350
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 $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
- 521
2
votes
1
answer
334
views
Fourier transform of the indicator function of the semi-ball
I wonder if it is possible to derive an analytical form for the Fourier transform of the indicator function of a semi-ball:
$$g_1(\underline{\xi}) = \int_{\mathcal{B}_+(R)} e^{i \underline{\xi} \cdot \...
- 21
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 ...
- 53
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)...
- 634
2
votes
1
answer
190
views
Can Mellin transform be applied in this function? What's the result?
$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
- 1,305
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 $n$ ...
- 139
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(\...
- 87
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\...
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}=...
- 2,239
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{\...
- 350
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 ...
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 ...
- 350
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 ...
- 6,853
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 ...
- 63
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 ...
- 47
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
$$
Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
- 385
0
votes
1
answer
210
views
A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
Is it possible to express $$\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[\...
- 350
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 ...
- 617
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 ...
- 1
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
$$
I(\alpha, x)=\int_\mathbb R \cos(xt) e^...
- 16.2k
-4
votes
1
answer
370
views
Is delta function symmetric against real axis? [closed]
Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$?
I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis.
We can write Delta function as
$$\delta(z) = \...
- 10.1k