Questions tagged [integral-transforms]

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Radon transform range theorem and radial functions

In dimension 2, the Radon transform range theorem states that a function $g(t,\theta)$ can be represented as a Radon transform of some function $f(x,y)$ (i.e. $g=R[f]$) if and only if for all integers ...
8
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0answers
174 views

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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33 views

A functional that occurs in Vlasov-Poisson equation

Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...
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63 views

How to integrate an exponential function of a rational function?

Can anyone help me to calculate the following integral? \begin{align} \int\limits_0^t {{{(x - t)}^2}} x\,{e^{ - \left(x + \frac{a}{{bx + 1}}\right)}}\mathrm{d}x \end{align} where $a$ and $b$ are ...
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1answer
91 views

Is this integral transform related to the Laplace transform?

The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by $${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$ Let $\varphi: {\...
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0answers
50 views

Integral with 4 Bessel functions and an exponential

I would like to solve the following integral $$ \int_0^\infty e^{-a k^2} J_{3/2}(b k) J_{3/2}(c k) J_{3/2}(f k) J_{1/2}(r k) k^{-3} dk, $$ where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...
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0answers
91 views

A close formula for a Fourier transform

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define $$ I(\alpha, x)=\int_\mathbb R \cos(xt) e^...
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0answers
63 views

Integral transformation, Laplace-like

Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)? $$ \int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt $$ It resembles somewhat the Laplace transformation. ...
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85 views

What kernel function yields power law eigenfunctions

Suppose I have a kernel function $K(x, y)$. I can then define an integral transform as follows: $$K[f] = \int_0^\infty K(x, y) f(x) dx$$. Is there any kernel function where the eigenfunctions $f(x) =...
3
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76 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
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2answers
456 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 ...
4
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1answer
88 views

Notion of a “smooth function of the order two” (Yakubovich, “Index Transforms”)

In the book "Index Transforms" by S.B. Yakubovich (World Scientific, 1996), I encountered the notion of a "smooth function of the order two" with compact support on $\mathbb R_+$, denoted $C^{(2)}(\...
2
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1answer
160 views

Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
1
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2answers
66 views

Opial type inequalities

Let $x(t)\in C^1[0,h]$ be such that $x(0)=x(h)=0$ and $x(t)$ in (0,h) ,then the following inequality $ \int^h_0 |x(t)x^{'}(t)|dt \leq \frac{h}{4}\int^h_0(x^{'}(t))^2dt$ my question: I would like ...
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0answers
104 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 ...
1
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1answer
223 views

Is this relation between divergent intergals justifiable?

Graf's book on hyperfunction theory says (page $36$) that $$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$ while the table of Fourier transforms ...
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71 views

Is it possible to define a product of two divergent integrals that would have the following properties?

Here I introduce an algebra of divergent integrals and series. But the theory is currently lacking an important element: there is no algorithm of construction of a divergent integral that would be ...
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175 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): $$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ $$...
2
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66 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 $$\varphi(s)=\left[T\left\lbrace f\right\rbrace\right](s)=\int_a^bK(x,s)f(x)dx$$ ...
7
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1answer
195 views

Injectivity of a class of integral operators

Given a probability measure $\mu$ on the interval $[0,1]$, the linear operator $$ T_\mu \! f(y) := \int_0^1 f(yx) \, d\mu(x) $$ takes the space of continuous functions $f: [0, \infty) \rightarrow \...
0
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1answer
106 views

“Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇”

This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
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1answer
121 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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46 views

self-dual integral transform

Is it possible to describe all integral transforms where the inverse transform is implemented by the same formula (with maybe a sign flipped somewhere). Fourier is such an example obviously. I am ...
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58 views

Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
4
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1answer
231 views

Pushing Cuckoo Eggs under Inverse Radon Transforms

Essentially the inverse of the Radon transforms $Rf(L)=\int_L{f(x)|dx|}$ has the ability to reconstruct $f(x)$ from the integrals over all lines; or, expressed differently to (re)construct $f(x)$ ...
2
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0answers
43 views

Integral equation with kernel defined in a rectangle

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$ Observe that the kernel is not defined on a square. My question: Can I apply the classical ...
5
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1answer
249 views

$L\log L$ 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 ...
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0answers
45 views

Y-transforms of products of Struve functions and exponential functions?

In the Bateman Manuscript Project's Table of Integral Transforms, there are several identities for Y-transforming (or similarly but more familiarly Hankel-transforming) certain special functions with ...
5
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3answers
141 views

Reconstructing a curve in $S^2$ from intersections with great circles

Take $S^2$ with its standard metric. The space of great circles in $S^2$ can be identified with the real projective plane $\mathbb{R}P^2$. Let $X$ be an embedded circle in $S^2$; associate to it a ...
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0answers
54 views

Kernel of Radon transform in $\mathbb{R}^3$

Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$: $$(Rf)(H):=\int_{l\subset ...
5
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2answers
318 views

An integral involving three Bessel functions

I am looking for a closed form for the following integral $$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$ which can be thought of as a particular case of the more general integral ...
1
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1answer
334 views

I want to disprove an equality involving a double integral

I want to show that the following equality does not hold: \begin{equation}\label{at3} \frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
0
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1answer
75 views

A System of generalized Abel's integral equation

Is there a method for solving the following system of generalized Abel's integral equation:? $(x^2 -1)\int_0^x \frac{u(t)}{(x-t)^{\frac{1}{2}}}\; dt + x\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{3}}}\; dt ...
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0answers
29 views

Calculating the Radon Transform from the Hartley Transform

It is wellknown, that the Radon Transform can be calculated from the Fourier Transform via the Central Slice Theorem. The Hartley Transform can be seen as a "purely real" version of the Fourier ...
-1
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1answer
72 views

transformation of two measures on different space

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$. Let $\Sigma_U$ be the smallest $\sigma-$ field ...
4
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1answer
152 views

Extended convolution theorem for Laplace transform

Let $(f*g)(t):=\int_0^t f(s) g(t-s)ds.$ Then the Laplace transform $L$ satisfies $L(f*g)(t)=L(f)(t)L(g)(t).$ This is known as the convolution theorem. I would like to know whether something similar ...
2
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0answers
90 views

Complex integral transforms

Is there an invertible (for some specified classes of functions) integral transform performed as a contour integral over $\mathbb{C}$, for example $F[f](w)=\oint_\Gamma K(w|z)f(z)\mathrm{d}z$ (let say,...
2
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1answer
159 views

CTRW: solve a renewal equation

Let X(t) be a continuous time random walk, with exponentially distributed waiting times of pdf $f_T(t)= k e^{-k t}\; t\geq 0$ and a jump sizes pdf $f_J(x)$. Suppose the initial distribution $\rho_{X(0)...
0
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2answers
348 views

Why Mellin transform is omitting infinite terms?

For instance, Mellin transform of function $f(x)=x$ $$\int_0^\infty f(x) x^{s-1} dx$$ returns the result $\delta(s)$ which is completely strange to me. Why only at $s=0$ the result is infinite? Why ...
12
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2answers
458 views

Relationship between the Radon transform and Twistor spaces

I have often heard that the theory of Twistor spaces is ``a complex analogue" of the Radon transform. What is the precise connection ?
2
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0answers
337 views

Hankel transform

The Hankel transform of $f=h(a-r)$ and $g=1/\sqrt{r^2+b^2}$ are $a/s\,J_{1}(as)$ and $e^{-bs}/s$ respectively. Here, h is a Heaviside side function that is 0 if $r>a$ and $1$ otherwise. a and b are ...
6
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1answer
566 views

volume over a hypercube, over simplex: twist by Euler numbers

Let $\square_n=\{(x_1,\dots,x_n): 0\leq x_i\leq1,\, \forall i\}$ be an $n$-dimensional unit hypercube, and let $\Delta_n=\{(u_1,\dots,u_n):u_1+\cdots+u_n\leq\frac{\pi}2,\, u_i\geq0,\, \forall i\}$ be $...
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1answer
494 views

Fourier transform with cubic exponential

Please give references for the integral transform of the next kind: $$ F_3(f(x))(t)=\int_{-\infty}^{\infty} \exp(Q_3(x,t)) f(t)\,dt , $$ with $Q_3(x,t)$ - a cubic polynomial of its arguments. Special ...
5
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3answers
531 views

Asymptotic behavior of integral with gamma functions

Consider the following function defined for complex numbers $z\in\mathbb{C}$ with $\Re(z)\geq \frac{1}{2}$: $$F(z)=\frac{1}{5^{\Re(z)}}\int_0^\infty \left| \frac{\Gamma(z+ix)\Gamma(z-ix)}{\Gamma(z)^2}...
4
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2answers
221 views

An integral identity relate to the Gamma function or the Beta function

I encountered the following identity in a paper on number theory, $$\int_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\...
3
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0answers
59 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 $k : x \mapsto \mathbb{R}^N$ $y$ ...
1
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0answers
103 views

Kontorovich Lebedev transform

By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has $$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...
2
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0answers
208 views

Hilbert transform

Let $\mathscr H$ be the Hilbert transform, that is the Fourier multiplier by $\text{sign } \xi$: $$ (\mathscr H u)(x)=\int_{\mathbb R} e^{2iπ x\xi }(\text{sign } \xi) \hat u(\xi) d\xi. $$ The Hilbert ...
2
votes
1answer
328 views

Fourier sine and cosine transforms and Laguerre polynomials

Let $S$ and $C$ denote the Fourier sine transform and the Fourier cosine transform, respectively, i.e., \begin{align*} S f(k) &= \sqrt{\frac2\pi} \, \int_0^\infty f(x) \sin(kx)\,dx, \\ C f(k)...
1
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1answer
103 views

Representation of Hilbert transform by a singular integral

Hilbert transform defines as follow: $$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$ $$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$ Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...