Questions tagged [integral-transforms]

This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine transforms.

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How to solve this partial differential equation (heat-diffusion equation)

I'm having trouble in solving a specific partial differential equation. It writes: $$ \dfrac{\partial p}{\partial t} = c \left( \dfrac{\partial^{2} p}{\partial x_{1}^{2}} + \cos^2\left(\theta\...
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To what extent the Laplace transform of the function $x^q$ is justified when $-1<q<0$?

The (one-sided) Laplace transform of the function $x^q$ according to the tables is ${\operatorname {\Gamma } (q+1) \over s^{q+1}}$. According to the tables, it is valid for $q>-1$. Other tables ...
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Generalised Integral transforms of distributions

I want to start with Generalized Integral transforms of distributions and applications. I have gone through books of A.H. Zemanian and Debnath & Bhatta. I am working with Hankel-type transforms ...
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Necessary and sufficient condition for Radon transform of a probability density function to be bounded

Let $f:\mathbb R^n \to \mathbb R$ be a probability density function, meaning that $f \ge 0$ and $f \in L^1(\mathbb R^n)$, and define its Radon transform $R[f]$ by $$ R[f](w,b) := \int_{\mathbb R^n}\...
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Formulas of Mellin inversion theorem that involve Riemann zeta function $\zeta (s)$ and floor function $\lfloor x\rfloor$

Functions $f(x)=\lfloor x\rfloor$ and $g(s)=\frac{\zeta (s)}{s}$ are related by Mellin inversion theorem, for $c>1$, $\Re(s)>1$. $$\mathcal{M}_x(f(x))(s)=\mathcal{M}_s^{-1}(g(s))(x)$$ $$\tag{1.1}...
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The Cauchy-Stieltjes transform of the exponential distribution

I'm interested in the integral $$G(z)=\int_\mathbb{R} \frac{ce^{-cx}\mathrm{d}x}{z-x}$$ which defines the Cauchy-Stieltjes transform of the exponential distribution parameterized by some real $c>0$....
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What are the elementary properties of dirac-delta function from which every other properties of it could be deduced?

I am studying dirac-delta function first time in my undergraduate course and different books have defined this function in different ways which when graphed together contradicts each other. I want to ...
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Inverse mellin transform with reciprocal gamma and an exponential

Can somebody explain to me how to compute the inverse Mellin transform of $$s\in(\mathbb{R}^*_++i\mathbb{R})\mapsto \frac{1}{\Gamma(s)}\exp \frac{(s+D)^2}{E}$$ In other words, I want to compute the ...
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Local convexity of a certain function defined via Radon transforms

Let $f$ be a "sufficiently smooth" probability density on $\mathbb R^n$. For $\theta=(w,c) \in \Theta := \{(w,c) \in \mathbb R^{n+1} \mid w \ne 0\}$ and $t \ge 0$, define $$ F_t(\theta) := \...
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Validity of Kusznetov's trace formula (which test functions are permissable)

In Iwaniec's "Spectral methods of automorphic forms" he says (page 128) that a (smooth and of bounded variation) $f$ can be written through Kontorovitch-Lebedev inversion if $f$ satisfies \[ ...
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Solving a multivariate recurrence relation (or its dual PDE)

I have a two-variable recurrence relation of the form, \begin{align} -&[(N+1)n+N(n+1)+(M+1)m+M(m+1)]p(n,m)\\ -&\epsilon[(n+1)m+(m+1)n)]p(n,m)\\ +&(N+1)(n+1)p(n+1,m)+(M+1)(m+1)p(n,m+1)\\ +&...
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Is the following Fourier Transform in cylindrical coordinates correct?

I am trying to solve the integral $$\int_ {Cylinder}e^{-i\vec{k}\vec{r}}dV=\int_0^Rrdr\int_0^{2\pi}d\phi\int_0^Le^{-ik_zz}e^{-i(k_xx+k_yy)}dz$$ I tried to rewrite it using polar coordinates and solved ...
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Decay rates of eigenvalues of Hilbert-Schmidt integral operator

Let $\Omega \subset \mathbb{R}^n$ be bounded. Suppose we have an integral kernel $K: \Omega^2\to \Omega$ with $\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|K(x,y)|^2dxdy < \infty$. We know that the ...
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Is there an integral transform formula for $(-\nabla^2 + m^2)^{\frac{1}{2}}$ in three dimensions? What about its one sided inverse?

I have come across the following formula for the positive square root of the (negative) 3D Laplacian $$(-\nabla^2)^{\frac{1}{2}}[u](y) = C \text{ p.v. }\int_{\mathbb{R}^3}\frac{u(y)-u(x)}{\|y-x\|^4}dx$...
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Fourier transform of [something]*Gaussian

It has been 20 years since failed to understand Fourier transforms during my formal education. I am trying to solve an optical problem, and it looks as though the Fourier transform should be helpful. ...
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Beta-function integrand basis for function space

A problem I am trying to tackle seems like it could be significantly simplified if it were possible to choose a basis for the relevant function space as something like: $$ \mathcal{B}_\mathrm{B} = \...
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Which branch of math includes Integral transforms?

I looked upto Wikipedia and found out there's more transformations than Laplace and Fourier. Which branch of math actually covers all of this? Like what course should I take?
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Inverse Mellin Transform of $f(w)=\exp\big(\frac{1}{\log w}\big)?$

I noticed that the inverse Mellin Transform of the classical hyperbola $y=1/z$ is the Heaviside step function. Particularly it takes the form $\theta(1-s).$ Now I wondered what would happen if one ...
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Hankel transform of exponential involving square root argument without HT

Given integral to investigate asymptotic behaviour on: $$A=Re\int_{0}^{\infty} J_0(xs) e^{-iw{\sqrt {gx}}}xdx$$ for large $s$ and $w$ $\sqrt{2 / \pi x} cos(x-(2n+1)\pi/4)=J_n(x)$ Want to investigate ...
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Interchanging order of integration in the Mellin transform

It is possible to understand (perhaps somewhat non-rigorously) the Fourier transform through interchanging order of integration and use of the delta function, like so: $$\hat{f}(k)\equiv\int_{-\infty}^...
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Hankel transform of $exp(-a\sqrt{r^2+z^2})/\sqrt{r^2+z^2}$

We know the Hankel transform of order 0 is defined as \begin{equation} {\displaystyle F_{0}(k)=\int _{0}^{\infty }f(r)J_{0}(kr)\,r\,\mathrm {d} r}. \end{equation} In this regard, I am now trying to ...
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How to invert an autocorrelation with a kernel?

This problem came up while stuyding stochastic backgrounds of gravitational waves in the early universe. I would like to invert $$ \Omega_{\mathrm{GW}}(k) = \int_0^{1/\sqrt{3}} \int_{1/\sqrt{3}}^\...
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What is dyadic sampling in the context of a wavelet transform?

In Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges the authors introduce to the reader (page 24) the notion of wavelet transforms as a way of having multiscale representations (...
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Why do different Fourier transform conventions not make a difference in physical applications?

In physics, we are used to at least two Fourier transform conventions. These are $$\tilde{f}(k)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(x)e^{ikx}dx,\\ f(x)=\int_{-\infty}^{\infty}\tilde{f}(k)e^{-ikx}dk$...
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Inverse mellin transform of $\Gamma^3(s)$

Is the inverse mellin transform of $\Gamma^3(s)$ known? Mathematica tells that inverse mellin transform of $\Gamma^2(s)$ is $2 K_0(2\sqrt{x})$ where $K_0(x)$ is the bessel function. I would like to ...
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Computation of some Fourier/Hankel transforms

I need to compute the Fourier transform of the following functions : $$f(x_1,x_2) = \frac{1}{a + (1-|x_1|^2-|x_2|^2)^2}$$ where $a>0$ is a positive constant. I have seen that because this function ...
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How to compute the Hilbert transform of $1/(x-a)$?

Equation (5) in this paper by H. H. Chen, Y. C. Lee, and N. R. Pereira says that $$H\left(\frac{1}{x - a}\right) = \frac{i}{x - a},$$ where $a$ is a complex constant with $\mathfrak{Im}(a) < 0$. $H$...
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What are the properties of this exotic function transform, which divides Taylor series coefficients by $(n!)$?

While investigating a model in particle physics, I encountered an exotic function transform. The form that appears in our application acts on analytic functions. In its original form, it can be ...
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Is $\int_0^\infty e^{-nz}f(\theta+z)dz$ a unilateral Laplace transform?

I have learned that $$\phi(t)=\int e^{-tx} f(x)dx$$ is a unilateral Laplace transform of $f(x)$. Then while trying to prove the completeness of the smallest order statistic $T=X_{(1)}$ for $f(x)=e^{-(...
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Need some help with multiple integration regarding a Fourier transform

I am looking at a multiple integral given in equation 3.1 of this paper, which is a Fourier transform, with $b = |\vec{b}|$ and $p = |\vec{p}|$: $$ \hat{f}(b) \equiv \int d^{D-2} \vec{p} \, e^{i\vec{b}...
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Transformation of Area restricted by 3 functions and x-Axis

With the help of a suitable transformation and Fubini I want to determine the integral $$ \int_{V} x^{3} y d \lambda_{2}(x, y), $$ where $V$ is the open subset of $\mathbb{R}_{+}^{2}$ bounded by the ...
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Automorphic integral transforms: Examples for transformations that leave the domain of integration intact?

Consider the integral of a function $f$ over a domain $D \subset \mathbb{R}$, and a function (automorphism?) $g: D \rightarrow D$ with $g'(x) > 0$ for all $x \in D$. Then we may write $$ \int_D f(x)...
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Summation formulas for integral transforms other than the Fourier

It seems to me that the Fourier transform harbours multiple useful results that allow one to sum an infinite series. The Poisson Summation Formula and Parseval's theorem. Also, Plancherel's theorem ...
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Possible bug LaplaceTransform in Mathematica

Let us consider in Mathematica 13.0 on Windows 10/Linux LaplaceTransform[DiracDelta[x - 2]*Exp[-x^2], x, s] E^(-2 (2 + s)) ...
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Decomposing an intensity spectrum as a superposition of blackbody spectra

My question goes as follows. The same way any integrable function $f(x)$ can be somewhat expressed as a superposition of plane waves as $\int_{-\infty}^{+\infty} F(\lambda)e^{2\pi i x\lambda}d\lambda$,...
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Integral of time function and time dependent arbitrary function with respect to time.

I have the following integral to evaluate: $\int_{A_0,0}^{y(t),t} e^te^{y}\dot{y}dt$ where $y$ is time dependent or $y=y(t)$. Here are my two different attempts and neither gave me a correct answer, $\...
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Closed form solution for Beurling transform: a 2d integral

Is it possible to find $$I = \int_{-\infty}^{\infty}\int_{-\infty}^0 \frac{e^{i\xi}e^{\eta}}{(\alpha-\xi)^2+(\beta-\eta)^2}((\alpha-\xi)+i (\beta-\eta)) \ d\eta d \xi, $$ in closed form?
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Integrals of two modified Bessel functions of the second kind with respect to the order

I've been trying to calculate this Kontorovich-Lebedev transform which involves an integral with a product of two modified Bessel functions of the second kind with respect to their order. Does anyone ...
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Integral with four modified Bessel functions of the second kind of imaginary order

I've been trying to compute the following integral with no avail, does anyone have any ideas? All free constants are strictly positive. $K_\nu(x)$ is the usual modified Bessel function of the second ...
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Taking the Z transform and the Fourier transform of the same function

I saw that we could apply two transforms to the propagator of the Continuous-time random walk (CTRW), $$P(x,t)= \sum_{N=0}^\infty [\lambda^{*N}(x)w^{*N}(t)*\int_t^\infty d\tau w(\tau)] $$ where the ...
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How Can One Express u(xy) as a Diagonalized Transform Kernel, K(x,y)?

Consider a projection operator $P_{u}g(x)=<g(x),u(x)>$, where $u(x)$ is an eigenfunction normalized under an inner product, $<u_{m}(x),u_{n}(x)>=\delta_{m,n}$. (ASIDE: Inner products may ...
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Reconstructing function from integral transform

Physicist here, so forgive me if I'm being a bit sloppy. I was considering the integrals $$ \tau(s) = \int_{0}^{L}\frac{{\rm d}x}{\sqrt{1-f(x)/s}} $$ for all $s>\max\{f\}$, and I came to wonder ...
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Inversion Formula Gaussian Convolution

I am looking at the following 2004 paper by S. Saitoh, called "Approximate real inversion formulas of the Gaussian convolution": https://www.researchgate.net/publication/...
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Fourier Transform with a Different Bilinear Form

I was thinking about how convolutions can be thought of as a "sliding dot products," this got me wondering, what would a "sliding cosine similarity" or a "sliding [bilinear ...
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Inverse of integral transform $f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x$

Given $g(x)$ defined for positive reals, say $f(s)$ is defined as below $$f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x.$$ Is there a relationship to named integral transforms, or a generic ...
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2 votes
1 answer
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Integral of the Radon transform equals the function twice integrated

I read that for a function $f:\mathbb R^2 \to \mathbb R$ with radon transform $\mathcal Rf(r,\theta)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y)\, \delta(r-x \cos \theta - y \sin \theta ) \, ...
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Radon transform of a delta function results in a sine function

I took a screenshot from an online video explanation about the Radon transform. It is stated that the radon transform of a delta function (right graph) results in a sine function. Can someone please ...
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Using Bernstein's Theorem to conclude this equality

in a part of book of Prüss (Evolutionary integral equations and applications pg. 99) it say..."Since $b$, $c \in \mathcal{BF}$ (Bernstein functions) by Bernstein's Theorem exist a function $\beta ...
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Use of Banach Algebra to define the fourier transform, is this generalizable?

In Rudin's real and complex analysis section 9.22 there's an interesting application, at least to me, of banach algebras techniques. I am not a mathematician so I might be misunderstanding what this ...
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Attempt to formulate solution to coupled PDEs via variation of parameters

I have the following PDEs that I would like to solve for $f(x,\omega)$ and $g(x,\omega)$: \begin{align*} \dfrac{\partial^{2}f}{\partial x^{2}}-\omega^{4}f & =-\omega\dfrac{\partial{\cal B}}{\...
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