# 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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### Inverse laplace transform excercise

I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$ I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$ I am just wondering is it possible ...
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### How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
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### Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) \end{...
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### Fourier transform dependent upon a parameter and $L^2$ convergence

Suppose I know the Fourier transform of a function depending upon a parameter, call it $f_\epsilon(x)$, and that I want to know the Fourier transform of a function $f(x)$. Furthermore, suppose I know ...
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### Integral transforms and scaling properties

It's well-known that the Fourier transform plays nicely with scaling. Particularly if we define, for $\alpha >0$, $D_{\alpha}$ by $D_{\alpha}f(x) = \alpha^{-1/2} f(x/\alpha)$, then (for suitable ...
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### What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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### Fourier Transform-1

I am trying to solve a Fourier transform problem and I am stuck. The problem is: $$f(t)= \frac{\sin(2t)}{e^{|t|}}.$$ I have used integration, but the answer that I come up with is different than ...
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### Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where $A$...
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### Integral transforms and uncertainty products

Heisenberg's uncertainty principle is well-studied and has become a bit of a pop science phenomenon due to its widespread implications in quantum mechanics. (Though interpretations are often ...
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### Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
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### Why do you need an integral to invert the Z-Transform?

With integral transforms both the transform and its inverse are integrals. In the case of the Z-Transform the transform is a sum. My question Why do you need an integral (instead of another sum) to ...
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### Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
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