Questions tagged [integral-transforms]
For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.
116
questions with no upvoted or accepted answers
18
votes
0
answers
697
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 $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
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 ...
9
votes
2
answers
3k
views
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 ...
9
votes
0
answers
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 $\...
7
votes
0
answers
284
views
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
$$
k_a(x, y)=\frac{a\...
6
votes
0
answers
157
views
Norms and distributions
Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function
$$
F(v) := \...
5
votes
0
answers
252
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):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$f^...
5
votes
0
answers
235
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Estimating singular values of integral operators
I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb R}...
5
votes
0
answers
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 $\...
5
votes
0
answers
341
views
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 ...
4
votes
0
answers
144
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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 ...
4
votes
0
answers
168
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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 ...
4
votes
0
answers
248
views
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:\...
4
votes
0
answers
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=\...
4
votes
0
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199
views
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)=...
4
votes
0
answers
381
views
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,...
4
votes
0
answers
144
views
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
$$
D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( \sqrt{1-4z+4z^{p+...
4
votes
0
answers
285
views
Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
4
votes
0
answers
362
views
inverse Laplace transform of $\delta_1(\cdot)$
Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...
3
votes
0
answers
113
views
How fast can the Mellin transform of a twist $\eta(t) e(\alpha t)$ decay?
Let $\eta:[0,\infty)\to [0,\infty)$. Consider the Mellin transform $F_{\alpha}$ of $\eta(x) e(\alpha x)$, and examine its behavior on a vertical line, such as $\Re s = 1/2$.
If $\alpha$ is close to $0$...
3
votes
0
answers
95
views
A generalisation of Cauchy-Stieltjes transform
For a nice function $\nu$ (say smooth and compactly supported), its Cauchy-Stieltjes transform is defined as
$$\int_\mathbb R \frac{\nu(s)}{z-s}\mathrm{d}s$$
which is holomorphic in $\mathbb C\...
3
votes
0
answers
285
views
Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
3
votes
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)...
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_{...
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
0
answers
114
views
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 ...
3
votes
0
answers
148
views
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
...
3
votes
0
answers
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$ ...
3
votes
0
answers
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 ...
3
votes
0
answers
1k
views
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{...
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})$, ...
3
votes
0
answers
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$$
the dispersion function of $g$.
Now, one can ...
3
votes
0
answers
466
views
Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere
Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form.
Some background may be ...
2
votes
0
answers
122
views
Sonin inversion formula, equivalence of two solutions of an integral equation
Let me first specify the problem I am facing, and then below explain where it arises. Given a function $f(x)$ on the interval $0<x<1$ and a real number $s\in(-1,1)$ I consider the integral ...
2
votes
0
answers
43
views
When can convolutional integral operators be sampled
Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form
$$
f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt,
$$
for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...
2
votes
0
answers
125
views
Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
2
votes
0
answers
93
views
Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$
I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights,
$$
\int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...
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
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 ...
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 ...
2
votes
0
answers
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 $b \in \mathbb R$, ...
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 ...
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 $\mathbb{R}^2$:
$$
\mathcal{A}f (x,y) := \...
2
votes
0
answers
31
views
finite fractional integral transform
Several integral transforms are generalized by introducing their fractional counterparts,e.g., fractional Fourier transform is a very popular one. Similarly, their discrete versions are also ...
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
2
votes
0
answers
65
views
Reference request for type of specific integral equation in two variable:
Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...
2
votes
0
answers
42
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
2
votes
0
answers
161
views
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 ...
2
votes
0
answers
162
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 $n$ ...
2
votes
0
answers
196
views
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
$$\varphi(s)=\left[T\left\lbrace f\right\rbrace\right](s)=\int_a^bK(x,s)f(x)dx$$
...