# 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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1 vote
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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)\\ +&...
1 vote
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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 ...
1 vote
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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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### 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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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}}{\...