Questions tagged [integral-transforms]
For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.
307
questions
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18
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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 ...
6
votes
1
answer
213
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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
votes
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(...
0
votes
0
answers
68
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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
$$f(x)=\int_0^1k(x,y)f'''(y)dy$$, $$k(x,y)=\begin{cases}-\frac{x^2(1-y)}{2} & x\leq y\...
1
vote
1
answer
50
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Possibility of bounding one functional by another functional (under certain constraints)
Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
6
votes
1
answer
304
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Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
1
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0
answers
51
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How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?
We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...
3
votes
0
answers
113
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How fast can the Mellin transform of a twist $\eta(t) e(\alpha t)$ decay?
Let $\eta:[0,\infty)\to [0,\infty)$. Consider the Mellin transform $F_{\alpha}$ of $\eta(x) e(\alpha x)$, and examine its behavior on a vertical line, such as $\Re s = 1/2$.
If $\alpha$ is close to $0$...
4
votes
0
answers
147
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Mellin transform of $(1-x)^k 1_{[0,1]}(x) e(\alpha x)$?
Let $f:[0,\infty)\to [0,\infty)$ be given by $$f(x) = \begin{cases} (1-x)^k e(\alpha x) &\text{for $0\leq x\leq 1$}\\ 0&\text{for $x>1$,}\end{cases}$$ where $e(t) = e^{2\pi i t}$ and $k\geq ...
1
vote
1
answer
87
views
Scale convolution decomposition of a density
Is it possible to decompose the density:
$$p(x) = \frac{8}{\,\pi^2} \frac{x^3\tanh(x)}{\cosh^2(x)},\quad x>0$$
into a scale convolution of two non-negative densities: $p(x) = \int_0^{\infty} \xi^{-...
3
votes
1
answer
268
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How to find the inverse of a product of two integral equations
Problem
I am trying to invert an equation of the form:
$R(l_0)=(\int_{0}^{l_0} \rho(x) \, dx)(\int_{l_0}^{l} \rho(x) \, dx)$
where $0\leq l_0 \leq l$
I.e. I want to find $\rho(x)$ given $R(l_0)$ via ...
0
votes
1
answer
161
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A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
Is it possible to express $$\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[\...
1
vote
0
answers
102
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PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
1
vote
0
answers
97
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Contour integral with two essential singularity
I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating ...
0
votes
1
answer
123
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Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$
I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$
I've tried Mathematica, but it does not converge to a solution....
2
votes
1
answer
144
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Numerical methods for integral eigenvalue equation
I have an integral equation which is not exactly an eigenvalue type equation, but similar:
$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$
Here $\lambda$ can be thought of as an eigenvalue, so it is ...
18
votes
0
answers
697
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Are these continued fractions of integrals known?
Simplified repost of Are these continued fractions of integrals known? on MSE
EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
1
vote
0
answers
67
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Expressing a double Riemann Sum as a definite integral
I am reading a paper where the authors rewrite the following expression (for a continious function $\alpha$) into an integral:
$$\lim_{n\to \infty} \left(\min_{1\leq j \leq n}\left(\sum_{k=1}^{j-1} \...
6
votes
1
answer
404
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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
votes
0
answers
96
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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\...
1
vote
1
answer
241
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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 (...
0
votes
0
answers
30
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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 $\delta>0$:
$$f(h) = \frac{1}{h} \int_0^{\delta} e^{-s/h} a(s)\: ds +...
8
votes
3
answers
420
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A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
3
votes
0
answers
288
views
Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
3
votes
0
answers
140
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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 $(0,1)$?
$$\small f(x)...
6
votes
1
answer
260
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A geometric interpretation of the fractional Fourier transform
I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18):
Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
2
votes
0
answers
122
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Sonin inversion formula, equivalence of two solutions of an integral equation
Let me first specify the problem I am facing, and then below explain where it arises. Given a function $f(x)$ on the interval $0<x<1$ and a real number $s\in(-1,1)$ I consider the integral ...
3
votes
1
answer
311
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Integral of a function changing sign
By some numerical tests, we can see that the following function is negative on $(0,1)$:
$$\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}-...
0
votes
0
answers
69
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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
votes
0
answers
43
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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},\...
2
votes
1
answer
244
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 ...
3
votes
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
$$\text{Im}\lim_{\epsilon\to ...
2
votes
0
answers
125
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Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
9
votes
0
answers
304
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Best smoothing for the Prime Number Theorem?
There are plenty of proofs of the Prime Number Theorem with explicit error terms - it actually looks like a rather competitive field (see Remark 1.4 in https://arxiv.org/pdf/2204.02588.pdf). Several ...
5
votes
2
answers
474
views
Optimizing a smoothing function with the Prime Number Theorem in mind
Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
0
votes
1
answer
72
views
A solution satisfying an integral inequality is bounded [closed]
Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality
\begin{equation}
y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2}
\end{equation}
...
3
votes
0
answers
67
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Characterizing image of integral transform applied to sections of a fiber bundle
Geometry is not my area, and so, I am not sure the title accurately captures what I am interested in exactly... I hope the tags are appropriate.
For any vector $v$, denote it's $i$-th component by $v_{...
2
votes
0
answers
93
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Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$
I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights,
$$
\int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...
1
vote
0
answers
93
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
237
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 ...
3
votes
1
answer
352
views
Fourier series of $e^{(\cos(\pi x) - m)^2}$
I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes
$$
f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2}
$$
It is a real even 2-periodic function, so its Fourier ...
1
vote
1
answer
221
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{\...
2
votes
0
answers
62
views
Fourier transform of the hyperboloid
Equip $\mathbb{R}^{d+1}$ with the Lorentzian form $\langle x, y\rangle=-x^0y^0+{\bf x}\cdot{\bf y}$ where $x=(x^0,{\bf x})$ and $\cdot$ is the usual Euclidean dot product. We define the hyperboloid $\...
2
votes
2
answers
257
views
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 ...
2
votes
1
answer
138
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 $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
1
vote
0
answers
248
views
Operators for norm for some classes of integral operators
Let $T:L^2(\mathbb{R}^n,\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)$ be a linear operator of the form: for all $x\in \mathbb{R}^n$
$$
T_{\kappa}(f)(x):=\int_{u\in\mathbb{R}^n} \, f(u)\...
4
votes
0
answers
170
views
Mellin transform of the Bessel function $Y_n$ of order $n \geq 2$
The Mellin transform of the function $h$, locally integrable on $(0,\infty)$, is defined by
$$M[h,z] = \int_0^\infty t^{z-1} h(t) dt \tag{1}$$
For some functions $h$ the above integral is not ...
2
votes
1
answer
131
views
Mellin transform (of sequences)
Is it possible to define the Mellin transform for sequences of real numbers or even for tuples? Is there any book treating this argument?
Any idea or suggestion will be greatly appreciated
Since the ...
1
vote
0
answers
123
views
Zeroes of Mellin transform
There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible ...
3
votes
2
answers
215
views
Continuity of Radon transform w.r.t the angle
Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by
$$
R[f](w,b) := ...