# Tagged Questions

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 ...

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### What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
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### Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
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### Solving String Vibration Using Integral Transform

$$U_{tt} - c^2 U_{xx}= -g$$ where BC: $U_{x}(0,t)=a\sin(ωt)$ IC: $U(x,0)=0$, $U_{t}(x,0)=0$ where $c, g, A$ and $ω$ are positive constants Normally I wouldn't post for help here but I am ...
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### Laplace Transform of $L^{2}$ Function.

I know the Fourier transform is an isometry of $L^2$ functions. I've read that the Laplace Transform of an $L^2$ function is $L^2$ but cannot prove it nor can I find a proof. Does anyone know of a ...
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### Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition ...
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### Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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### Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
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### Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: \$\xi(\alpha,\tilde{\alpha},\beta)=\frac{J_{1}\left(\varrho_{0}k_{0}n\sqrt{\sin^{2}\alpha+\sin^{2}\tilde{\alpha}-2\sin\alpha\,\sin\tilde{\alpha}\cos\beta}\...