# Questions tagged [fourier-transform]

The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

4,023 questions
Filter by
Sorted by
Tagged with
26 views

### Solving $\widehat{f}=f$ with $f\in L^2$.

The following question arose. What functions do they satisfy? $\widehat{f}=f$ with $f\in L^2(\mathbb{R})$? Only the function $f=0$?
9 views

### Fractional Fourier Transform of $\sqrt{c} x( c(t - \tau))$

I am trying to figure out what the Fractional Fourier Transform of the signal $\sqrt{c} x(c(t-\tau))$ would be with respect to that of $x(t)$. According to the paper "The Fractional Fourier ...
1 vote
17 views

### Solving the discrete Fourier Transform of $\sin(x)+\sin(2x)$

I'm new to the discrete Fourier Transform and have managed to do a few examples by hand, (for example the Fourier Transform of $\sin(x)$). I now want to try the transform on signals that are made up ...
39 views

14 views

### Inverting Fourier Transform for Heat Kernel

I am trying to show that the Fourier Transform of the heat kernel is given by a simple Gaussian but I have no idea how to handle the prefactor 1/sqrt(2pi). The first line is from my lecture notes. ...
14 views

1 vote
41 views

28 views

39 views

### How to compute the discrete Fourier transform of f(x) = 1?

I am learning about the discrete Fourier transform in the context of digital image processing and not in the context of complex analysis so I might have some slightly different notation/formulas. Our ...
1 vote
171 views

38 views

### Function in $H^1(\mathbb{R})$

Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$. The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$. I ...
19 views

### How to prove $|-\Delta_{x} ((-x)^{\alpha} \phi(x))|\leq A_{j,\alpha}(1+|x|)^{-n-1}$

How to prove $|-\Delta_{x} ((-x)^{\alpha} \phi(x))|\leq A_{j,\alpha}(1+|x|)^{-n-1}$ Hi all, i am reading Pseudo-differential Operators, singularities, applications. Y Egorov, on page 2. The inequality ...
30 views

### Dispersive Waves on a Semi-Infinite String

I am having an insane amount of trouble figuring out this problem that I solved probably ten years ago. Googling leads to solutions that make use of D'Alembert's formula, but that doesn't work for ...
10 views

### Solve $g_1(x) \mathcal{F}[e^{-c_1 f(x)}] = g_2(x)\mathcal{F}[e^{-c_2 f(x)}]$ for $f(x)$

Is there any way to approach such a problem? Let $f:\mathbb{R} \to [0,\infty]$ and $g_1, g_2:\mathbb{R} \to \mathbb{R}$ be smooth, compactly supported functions, and let $c \in [0,\infty]$. Denote the ...
20 views

### Frequency peak always appearing at half of Nyquist frequency in Fourier transform

When taking the FT of a signal I always get a sharp peak at exactly half the Nyquist frequency. My signal is shown here: and its FT here: The Nyquist frequency is 36.7 KHz. As can been seen in the ...
10 views

26 views

### Finding solutions to $\sin\left(n \pi \frac{T_p}{T}\right) - \sin\left(n \pi \frac{T_0}{T}\right) = \sin\left( \frac{{n \pi}}{2}\right)$

I'm trying to figure out how I can find a solution(s) to the equation below. $$\sin\left(n \pi \frac{T_p}{T}\right) - \sin\left(n \pi \frac{T_0}{T}\right) = \sin\left( \frac{{n \pi}}{2}\right)$$ ...
48 views

1 vote
25 views

### Poisson Summation at Half Integers

In the proof of Poisson Summation, for a Schwarz function $f$, you define $$F(x)=\sum_{n\in\mathbb{Z}}f(x+n)$$ and you show that $$F(x)=\sum_{n\in\mathbb{Z}}\hat{f}(n)e^{2\pi inx}$$ Then plugging ...
1 vote
### Fourier transform of $\dfrac{e^{-ax}}{x}$
I am required to find the Fourier transform of the following function: $$f(x) = \dfrac{e^{-ax}}{x}$$ Where $a$ is an arbitrary constant. This is what I've tried, but I'm not quite sure: The Fourier ...