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3 votes
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Is the Short-Time Fourier Transform an Isometry in $L_2(\mathbb{R}^d)$?

You can generalise Parseval's theorem to STFT, but for an arbitrary window $\phi$, it is not an isometry from $L^2(\mathbb R^d)$ to $L^2(\mathbb R^{2d})$, you will need to normalise $\phi$ ...
LPZ's user avatar
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3 votes
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Evaluating the convolution of $e^{-at^2}$ and $e^{-bt^2}$ via Fourier transforms

The convolution theorem tells us that $$ \mathcal{F}[f \ast g] = \mathcal{F}[f] \cdot \mathcal{F}[g] $$ (up to some minor differences dependent upon your definition of the Fourier transform, refer to ...
PrincessEev's user avatar
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2 votes
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Calculating $\int_{-\infty}^{\infty}e^{ixt}\cos(at)e^{-\frac{t^2}{2}}\,dt$ and $\int_{-\infty}^{\infty}\cos(at)e^{-\frac{t^2}{2}}\,dt$

Yes, these proofs are just fine. As for as hints to calculate the integrals, let's look at $$ \mathcal{I}(x) := \int_{-\infty}^{\infty} e^{ixt} \cos(at) e^{-\frac{t^2}{2}} \, \mathrm{d}t $$ I assume,...
PrincessEev's user avatar
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2 votes
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Bound of an integral function

Your bound works on $[0,U]$. As for a bound on $\mathbb R_+$, the answer is no. Two changes of variable allow you to write $$ f(r) = \int_{\mathbb R^d} \left| \int_{\mathbb R^d} e^{2i\pi \langle x,y\...
Thomas Lehéricy's user avatar

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