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Questions tagged [integral-transforms]

For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.

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automorphisms and mellin transforms

If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
geocalc33's user avatar
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Discretization of oscillating integral

Suppose I am interested in computing $$ I \equiv \int_0^B dx \, g(x) f(x) $$ where $B$ is a known upper bound for the integral, $g(x)$ is a known oscillating function and $f(x)$ is a smooth function ...
knuth's user avatar
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Mellin transform of the volume form of a probability zonoid and its fundamental strip

Let $ L^n_+$ be the set of all $n$-dimensional nonnegative random vectors $\mathbf X = (X_1, X_2,\cdot\cdot\cdot,X_n)^⊤$ with finite and positive marginal expectations, and let $\mathbf Ψ^{(n)}$ be ...
53Demonslayer's user avatar
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1 answer
88 views

Radial Fourier transform vs Hankel transform

I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...
Chris Jones's user avatar
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115 views

Counterexamples in Laplace transforms

Of all the examples I know of (bilateral) Laplace transforms $F$ defined on their maximal vertical strips $V_{a,b}=\{ z \in \mathbf{C} : a < \operatorname{Re}(z) < b \}$ with $-\infty < a <...
proofromthebook's user avatar
6 votes
1 answer
233 views

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 ...
ThomasJr's user avatar
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47 views

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(...
Rosy's user avatar
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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\...
Hao Huang's user avatar
1 vote
1 answer
53 views

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 ...
Fei Cao's user avatar
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6 votes
1 answer
306 views

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 ...
MathLearner's user avatar
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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}...
Igor Kotelnikov's user avatar
3 votes
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113 views

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 $0$...
H A Helfgott's user avatar
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4 votes
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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 ...
H A Helfgott's user avatar
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1 answer
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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^{-...
japalmer's user avatar
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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 ...
Connor B's user avatar
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1 answer
166 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[\...
Mirar's user avatar
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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}{\...
53Demonslayer's user avatar
1 vote
0 answers
103 views

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 ...
K.K.McDonald's user avatar
0 votes
1 answer
125 views

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....
Felipe Augusto de Figueiredo's user avatar
2 votes
1 answer
165 views

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 ...
Fetchinson0234's user avatar
18 votes
0 answers
702 views

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 $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
TheSimpliFire's user avatar
1 vote
0 answers
67 views

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} \...
AspiringMat's user avatar
6 votes
1 answer
405 views

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 $...
Ali's user avatar
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3 votes
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97 views

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\...
Jiyuan Zhang's user avatar
1 vote
1 answer
241 views

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 (...
53Demonslayer's user avatar
8 votes
3 answers
423 views

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\...
Ali's user avatar
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3 votes
0 answers
291 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}\...
Dispersion's user avatar
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0 answers
140 views

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)...
Migalobe's user avatar
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6 votes
1 answer
267 views

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 ...
Luke's user avatar
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2 votes
0 answers
126 views

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 ...
Carlo Beenakker's user avatar
3 votes
1 answer
312 views

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}-...
Migalobe's user avatar
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0 votes
0 answers
69 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 ...
Ft insat's user avatar
2 votes
0 answers
43 views

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},\...
Annie's user avatar
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2 votes
1 answer
248 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 ...
Jacob Helwig's user avatar
3 votes
4 answers
1k views

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 ...
Carlo Beenakker's user avatar
2 votes
0 answers
128 views

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 $\lambda\in \mathbb{R}$, $$H(h')(t) = \lambda h(t)$$ ...
H A Helfgott's user avatar
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9 votes
0 answers
304 views

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 ...
H A Helfgott's user avatar
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5 votes
2 answers
475 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 ...
H A Helfgott's user avatar
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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} ...
Mee Na's user avatar
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3 votes
0 answers
67 views

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_{...
Atom Vayalinkal's user avatar
2 votes
0 answers
93 views

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\...
David's user avatar
  • 21
1 vote
0 answers
96 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 ...
Mirar's user avatar
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3 votes
0 answers
243 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 ...
Mirar's user avatar
  • 350
3 votes
1 answer
365 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 ...
gaspardb's user avatar
1 vote
1 answer
224 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{\...
Mirar's user avatar
  • 350
2 votes
0 answers
64 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 $\...
J_P's user avatar
  • 439
2 votes
2 answers
263 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 ...
Alex's user avatar
  • 73
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)...
zoran  Vicovic's user avatar
1 vote
0 answers
252 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)\...
ABIM's user avatar
  • 4,969
4 votes
0 answers
174 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 ...
user1029664's user avatar

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