# Questions tagged [fourier-transform]

For question related to Fourier transforms.

4,728 questions
Filter by
Sorted by
Tagged with
59 views

### calculating inverse fourier transform of $\cosh(t\sqrt{1-k^2})$

So i am trying to calculate the following integral $$\int_{-\infty}^{\infty}e^{-ikx}\cosh(t\sqrt{1-k^2})dk$$ I think this should be related to modified Bessel function of ...
• 141
1 vote
38 views

31 views

29 views

### Fourier transform on Schwartz space leads to identity.

Let $f$ be a function in the Schwartz space and $F(f) = \left(\frac{1}{2 \pi}\right)^{n/2} \int_{R^n} e^{i \langle x,y \rangle } f(x) dx$ the Fourier transform. Is it true that $F^4(f) = f$? I know ...
• 21
192 views

### Question about computing the Fourier transform of a product

[After further searching around, I decided to modify the question slightly] Let $A$ be an invertible real matrix and $g(\mathbf{x}) = e^{ i \mathbf{x}^T A \mathbf{x} }$. Let $w$ be a real valued ...
• 2,877
45 views

1 vote
70 views

### Smoothness of Fourier transform

I am trying to understand, because the Fourier transform of the function $f(x) = e^{ -\sqrt{ \lvert x \rvert } }$ is smooth. My question: Under which conditions is the Fourier transform of an $L^1$ or ...
1 vote
46 views

### Showing that Sobolev Space $H^m$ is in $L^\infty$

I'm very new to Fourier analysis/Sobolev spaces and am stuck on this exercise. I found proofs of more general embedding theorems for Sobolev spaces and some similar questions on here, but they are too ...
122 views

### Calculating the Integral: $\int_{-\infty}^{\infty} e^{j(w_0-w)t} dt$ [duplicate]

I have encountered the following integral: $$\int_{-\infty}^{\infty} e^{j(w_0-w)t} dt$$ where $w_0$ and $w$ are constants. I know that the result of this integral is $2\pi\delta(w-w_0)$, where $\delta$...
• 35
54 views

### Fourier transform of $g(x - \frac{xy}{f}, y)$

Let $g(x,y)$ have Fourier transform $G(k_x, k_y)$. I am interested in the Fourier transform of $g(x - \frac{xy}{f}, y)$ for some $f > 0$. I've gotten part of the way there using the separability ...
• 269
1 vote
205 views

### Why isn't the fourier transform defined differently?

I never really understood why the amplitude of the Fourier transform does not represent the amplitude of a sin/cos signal of a specific frequency? The way the Fourier transform is typically defined, ...
• 44
1 vote
28 views

### Approximating the surface measure of spheres via Fourier transform

Recently I'm reading a paper by Magyar-Stein-Wainger, Discrete analogues in harmonic analysis:Spherical averages, Annals of Mathematics, 155 (2002), 189–208. A key ingredient in the proof of their ...
22 views

• 7,255
1 vote