# Questions tagged [integral-transforms]

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196
questions

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30 views

### 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**

votes

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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 $a\in (1,2)$.

**0**

votes

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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 ...

**0**

votes

**1**answer

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: {\...

**0**

votes

**0**answers

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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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^...

**0**

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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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votes

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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**

votes

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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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**2**answers

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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votes

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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**

vote

**1**answer

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**

votes

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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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

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 $f(x)$ ...

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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**

votes

**1**answer

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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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**

votes

**3**answers

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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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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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**

vote

**1**answer

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**

votes

**1**answer

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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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 ...

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votes

**1**answer

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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**1**answer

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 ...

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**0**answers

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**

votes

**1**answer

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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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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**2**answers

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 ?

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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 ...

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**1**answer

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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**1**answer

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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**3**answers

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**

votes

**2**answers

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}\...

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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 $k : x \mapsto \mathbb{R}^N$
$y$ ...

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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\...

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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

**1**answer

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**

vote

**1**answer

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)= \...