# Tag Info

Accepted

### 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 $\phi$ ...
• 4,288
Accepted

### 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 ...
• 45.8k
Accepted

### 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,...
• 45.8k
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\...