# 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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### 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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### 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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### 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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### 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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I read that for a function $f:\mathbb R^2 \to \mathbb R$ with radon transform $\mathcal Rf(r,\theta)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y)\, \delta(r-x \cos \theta - y \sin \theta ) \, ... 0 votes 0 answers 206 views ### Radon transform of a delta function results in a sine function I took a screenshot from an online video explanation about the Radon transform. It is stated that the radon transform of a delta function (right graph) results in a sine function. Can someone please ... 1 vote 1 answer 61 views ### Using Bernstein's Theorem to conclude this equality in a part of book of Prüss (Evolutionary integral equations and applications pg. 99) it say..."Since$b$,$c \in \mathcal{BF}$(Bernstein functions) by Bernstein's Theorem exist a function$\beta ...
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}}{\...