Questions tagged [integral-transforms]
For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.
308
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Hölder continuity of Radon transform of smooth function
Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
0
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0
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17
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Recovering the population eigenvalue distribution from the Marchenko-Pastur distribution
Question: If we know the value of $y>0$ and the Marchenko-Pastur distribution $\nu$ (and thus also $m_\nu$), can we reconstruct the distribution $H$ from equality (1) below?
Background on the ...
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0
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101
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PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
6
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1
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Mellin-Barnes integral representation of the exponential function with a non-real argument
I have been studying a definite integral that I found out to be a particular (and possibly one of the simplest) case(s) of the arcane Mellin-Barnes integral. Solving this problem would lead to a ...
0
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0
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48
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Inverse kernel of a sine kernel [migrated]
I’m not a mathematician and I’m working with some transforms in physical chemistry.
I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and ...
0
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0
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45
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Computing the Laplace transform of an expression
I would like to find the Laplace transform of the following expression with respect to the Laplace parameter s
$ \int_{z=u}^{\infty} e^{-az/c} g^{'}(\dfrac{z-u}{c}) \int_{x=0}^{\infty} \varphi(z-x)dF(...
2
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2
answers
254
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Definite integral of Bessel function of the first kind times $x^{3/2}$
I am looking for preferably a closed form (or series solution if not possible) for the following integral:
$$\int_0^a x^{3/2} J_\nu (bx) dx$$
where $\nu$ is an integer. This 1D integral appears when ...
0
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1
answer
123
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Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$
I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$
I've tried Mathematica, but it does not converge to a solution....
9
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2
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The log transform turns scalar multiplication into addition. Is there an analogous transformation for matrix-vector multiplication?
Napier's method of logarithms and corresponding tables of logarithms provided a important tool to simplify hand computation by converting multiplication and division to equivalent problems of addition ...
20
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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 $a\in (1,2)$.
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0
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66
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the design of kernel function and integral transform
I read a solution of an integral inequality.
The solution uses condition $$f(1)=f(0)=f'(0)=0$$ to derive that
$$f(x)=\int_0^1k(x,y)f'''(y)dy$$, $$k(x,y)=\begin{cases}-\frac{x^2(1-y)}{2} & x\leq y\...
18
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Are these continued fractions of integrals known?
Simplified repost of Are these continued fractions of integrals known? on MSE
EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
1
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Possibility of bounding one functional by another functional (under certain constraints)
Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
6
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Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
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$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
1
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0
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How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?
We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...
3
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0
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113
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How fast can the Mellin transform of a twist $\eta(t) e(\alpha t)$ decay?
Let $\eta:[0,\infty)\to [0,\infty)$. Consider the Mellin transform $F_{\alpha}$ of $\eta(x) e(\alpha x)$, and examine its behavior on a vertical line, such as $\Re s = 1/2$.
If $\alpha$ is close to $0$...
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Mellin transform of $(1-x)^k 1_{[0,1]}(x) e(\alpha x)$?
Let $f:[0,\infty)\to [0,\infty)$ be given by $$f(x) = \begin{cases} (1-x)^k e(\alpha x) &\text{for $0\leq x\leq 1$}\\ 0&\text{for $x>1$,}\end{cases}$$ where $e(t) = e^{2\pi i t}$ and $k\geq ...
1
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1
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Transporting a Cauchy foliation of Minkowski space
Consider a spacetime $(\zeta^{3,1},g)$
where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (...
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Scale convolution decomposition of a density
Is it possible to decompose the density:
$$p(x) = \frac{8}{\,\pi^2} \frac{x^3\tanh(x)}{\cosh^2(x)},\quad x>0$$
into a scale convolution of two non-negative densities: $p(x) = \int_0^{\infty} \xi^{-...
3
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1
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How to find the inverse of a product of two integral equations
Problem
I am trying to invert an equation of the form:
$R(l_0)=(\int_{0}^{l_0} \rho(x) \, dx)(\int_{l_0}^{l} \rho(x) \, dx)$
where $0\leq l_0 \leq l$
I.e. I want to find $\rho(x)$ given $R(l_0)$ via ...
0
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161
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A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
Is it possible to express $$\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[\...
21
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4
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When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
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0
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Contour integral with two essential singularity
I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating ...
2
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139
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Numerical methods for integral eigenvalue equation
I have an integral equation which is not exactly an eigenvalue type equation, but similar:
$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$
Here $\lambda$ can be thought of as an eigenvalue, so it is ...
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1
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Find an integral kernel for the solution of a partial differential equation: an initial value problem
Consider the following partial differential equation with an initial condition $u(x,0)=f(x)$:
\begin{equation}
\frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\...
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0
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Expressing a double Riemann Sum as a definite integral
I am reading a paper where the authors rewrite the following expression (for a continious function $\alpha$) into an integral:
$$\lim_{n\to \infty} \left(\min_{1\leq j \leq n}\left(\sum_{k=1}^{j-1} \...
6
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On an asymptotic integral decay
Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $...
3
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A generalisation of Cauchy-Stieltjes transform
For a nice function $\nu$ (say smooth and compactly supported), its Cauchy-Stieltjes transform is defined as
$$\int_\mathbb R \frac{\nu(s)}{z-s}\mathrm{d}s$$
which is holomorphic in $\mathbb C\...
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Range of a Laplace-type transform
I'm interested in germs of functions $f(h)$ for $h\geq 0$ small. My question is, for what functions $f(h)$ can I write, for some $\delta>0$:
$$f(h) = \frac{1}{h} \int_0^{\delta} e^{-s/h} a(s)\: ds +...
3
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0
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Bounding an integral transform ouside a circle (or inside a strip)
Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call
$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$
the dispersion function of $g$.
Now, one can ...
3
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1
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125
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Injectivity of the Mellin Transform and discontinuities
I am reading Stopple's A Primer of Analytic Number Theory. On page 234, the Mellin Transform $\mathcal{M} f$ of a function $f$ is defined as
$$\mathcal{M} f (s) = \int_1^{\infty} f(x) x^{-s - 1} dx$$
...
14
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Do distance functionals separate probability measures?
Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
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A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
3
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Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
3
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Monotonicity of a function defined by an integral
The question below is motivated by the related question Integral of a function changing sign and the associated answer:
Can we study the monotonicity of the following function on $(0,1)$?
$$\small f(x)...
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Sonin inversion formula, equivalence of two solutions of an integral equation
Let me first specify the problem I am facing, and then below explain where it arises. Given a function $f(x)$ on the interval $0<x<1$ and a real number $s\in(-1,1)$ I consider the integral ...
6
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A geometric interpretation of the fractional Fourier transform
I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18):
Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
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0
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247
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Operators for norm for some classes of integral operators
Let $T:L^2(\mathbb{R}^n,\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)$ be a linear operator of the form: for all $x\in \mathbb{R}^n$
$$
T_{\kappa}(f)(x):=\int_{u\in\mathbb{R}^n} \, f(u)\...
3
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1
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Integral of a function changing sign
By some numerical tests, we can see that the following function is negative on $(0,1)$:
$$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
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Fourier transform of an exponential function with radical argument divided by a radical
I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
2
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Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space
Problem Statement
Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
2
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0
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When can convolutional integral operators be sampled
Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form
$$
f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt,
$$
for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...
3
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4
answers
1k
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Representation of the Dirac delta function
The Dirac delta function appears in the Sokhotsky formula,
$$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$
to be understood in the integral sense
$$\text{Im}\lim_{\epsilon\to ...
6
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1
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410
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On the multidimensional Mellin transform of measures
Consider an integral transform of Borel measures supported on $\mathbb{R}^n_+$ given by
$$
f(z) =\int\limits_{\mathbb{R}^n_+} x^{z}\frac{\mu(dx)}{x}
$$
where $z = (z_1,...,z_n) \in \mathbb{C}^n$, $x^...
9
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2
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2k
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Steinmetz, Laplace and Fourier transforms
I am looking for references on Steinmetz Transform and its relation with Laplace and Fourier transforms. There is an Italian Wikipedia page about this topic but with no references.
2
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0
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125
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Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
14
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1
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What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$
My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...
5
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2
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473
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Optimizing a smoothing function with the Prime Number Theorem in mind
Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
9
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0
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304
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Best smoothing for the Prime Number Theorem?
There are plenty of proofs of the Prime Number Theorem with explicit error terms - it actually looks like a rather competitive field (see Remark 1.4 in https://arxiv.org/pdf/2204.02588.pdf). Several ...