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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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
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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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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 ...
Marc Kjerland's user avatar
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264 views

Regularity class of certain diffeomorphisms of the real line

I care about the following class of homeomorphisms of $\mathbb R$, which I'll call $\mathcal C^?$. For simplicity, let us restrict attention to compactly supported homeomorphisms (a homeomorphism $\...
André Henriques's user avatar
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Trace class norms of special integral operators

Let $\mu$ be a finite compactly supported Borel measure on the real line. On the space $L^2(\mu)$ consider the integral operators $$ (K_a f)(x)=\int k_a(x, y)f(y)d\mu(y) $$ with $$ k_a(x, y)=\frac{a\...
limanac's user avatar
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Norms and distributions

Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function $$ F(v) := \...
alvarezpaiva's user avatar
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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^...
Anixx's user avatar
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Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb R}...
Gandalf Lechner's user avatar
5 votes
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230 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 $\...
Bazin's user avatar
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Using the Mellin transform to invert ill-posed problems of harmonic transforms

I will try to explain the problem using just words. Let's see how far I get! I will use the harmonic transform of the Radon transform for 2 or more dimensions as an example, but there is a large ...
William Hawkins's user avatar
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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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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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A tricky integral equation

In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.) Let $f:\...
Hans-Peter Stricker's user avatar
4 votes
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439 views

Learning from eigenvalues of Hilbert-Schmidt integral operator

Do eigenvalues of the Hilbert-Schmidt integral operator determine the underlying measure up to translation, reflection and rotation? Details: Suppose we have a measure $\mu$ on a Euclidean space $X=\...
Uzu Lim's user avatar
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Calculate an integral

For $x\in \mathbb R_+$, let us define $$ I_\lambda(x)=\frac2π\int_0^{+\infty}\frac{\sin t}{t\sqrt{1+t^2x^2}}\cos(\lambda \arctan (xt)) \,dt,\quad \lambda\in 1+2\mathbb N. $$ We see that $I_\lambda(0)=...
Bazin's user avatar
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Evaluating an integral of a periodic function. It's positive?

My purpose is to show that this integral  \begin{equation} I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du\,\,,\,\,x,...
ppooppii's user avatar
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Using Mellin transform for a certain function

In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where $$ D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( \sqrt{1-4z+4z^{p+...
john_leo's user avatar
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Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion

Is there a way to solve analytically the Fredholm integral equation of the second kind $$ \int_0^{100} K(s, t) f(s) ds = \lambda f(t) $$ where the kernel has the piecewise 'linear' form \begin{align} ...
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inverse Laplace transform of $\delta_1(\cdot)$

Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...
Fedor Petrov'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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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
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285 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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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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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
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 ...
Mirar's user avatar
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Radon transform on complex Grassmannians

Let $Gr_{i,n}$ denote the Grassmannian of complex linear $i$-dimensional subspaces in the Hermitian space $\mathbb{C}^n$. Let $1\leq i<n/2$. Consider the Radon transform between space of functions ...
asv's user avatar
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Fantappie transform(ation)s in Gelfand et al. "Generalized functions"

In the 6-volume "Generalized functions" a treatment of Fantappie transformations is promised in Vol. 1 (bottom of p.461 of the Russian edition) to come in Vol. 5. However, there is no Fantappie ...
Dima Pasechnik's user avatar
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71 views

Is there a name for the general type of operation that sweeps a kernel over a function (e.g. like convolution, morph. dilation, registration, etc)

There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where: $f$ is a function $f : x \mapsto \mathbb{R}^N$ $k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$ $y$ ...
Tasos Papastylianou's user avatar
3 votes
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137 views

Central-Slice-Theorem Analogue for Wavelet Transforms?

The 2D Radon transform and the 2D Fourier transform are related by the so-called Central Slice Theorem (cf e.g. http://en.wikipedia.org/wiki/Projection-slice_theorem) and I would like to know, whether ...
Manfred Weis's user avatar
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What is the inverse kernel of this integral transform?

I am looking for the associated inverse kernel to the integral transform $T$ defined by $(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$ whose kernel is $K(u,t) = \frac{...
hanscorelli's user avatar
3 votes
0 answers
145 views

Character of continuous series representation of GL(2)

It is wellknown that the character of an irreducible, unitary representation of $GL(n,\mathbb{C})$ uniquely determines the isomorphism classes. I fail to construct a function for $GL(2, \mathbb{C})$, ...
Marc Palm's user avatar
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3 votes
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172 views

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 ...
Peter Cudmore's user avatar
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466 views

Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. Some background may be ...
alvarezpaiva's user avatar
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2 votes
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122 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
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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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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2 votes
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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
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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 $\...
J_P's user avatar
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0 answers
144 views

The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
Tom Copeland's user avatar
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2 votes
0 answers
75 views

Most general space for the Wigner-Weyl transform

The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then ...
user avatar
2 votes
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116 views

Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation

Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
dohmatob's user avatar
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2 votes
0 answers
133 views

Are Poisson integrals uniquely determined by their radial limits?

Let $\mu$ be a complex Borel measure on the unit circle, and suppose its Poisson integral $u$ satisfies $\lim_{r\to 1-}u(re^{i\theta})=0$ for every $\theta$. Does it follow that $\mu=0$? This is of ...
apanpapan3's user avatar
2 votes
0 answers
106 views

Hilbert's fourth problem and a non-linear integral transform

The following nonlinear integral transform takes continuous functions defined on the cylinder $\mathbb{R} \times S^1$ to $C^2$ functions defined on the plane $\mathbb{R}^2$: $$ \mathcal{A}f (x,y) := \...
alvarezpaiva's user avatar
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2 votes
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31 views

finite fractional integral transform

Several integral transforms are generalized by introducing their fractional counterparts,e.g., fractional Fourier transform is a very popular one. Similarly, their discrete versions are also ...
Asma's user avatar
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0 answers
65 views

On a question relating integral equation:

I don't know if the following question qualifies as research level. If it isn't, sorry. Set the following terminology: $ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$ $\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
GSA_1's user avatar
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0 answers
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Reference request for type of specific integral equation in two variable:

Consider the following integral equation: $$\int_0^\infty K(t,y)\phi(t,x)dt=0$$ Here, $K(t,y)$ is a trigonometric kernel and $\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$). I want to find the ...
GSA_1's user avatar
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Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous

Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by $$ G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt. $...
dohmatob's user avatar
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2 votes
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Regularization of the area under hyperbola

So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
Anixx's user avatar
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2 votes
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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 $n$ ...
SarthakC's user avatar
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Uniqueness of the inverse kernel of an invertible integral transform

For any invertible integral transform $T$ of kernel $K$ that maps a function $f$ to the function $\varphi$ such that $$\varphi(s)=\left[T\left\lbrace f\right\rbrace\right](s)=\int_a^bK(x,s)f(x)dx$$ ...
Harmonic Sun's user avatar