Linked Questions

381
votes
14answers
334k views

Fourier transform for dummies

What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. ...
17
votes
6answers
20k views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
7
votes
2answers
16k views

Difference between Fourier integral and Fourier transform

What is the difference between Fourier integral and Fourier transform? I know that for Fourier integral, the function must satisfy $\int_{-\infty}^\infty |f(t)| dt < \infty$, but what if I have a ...
1
vote
1answer
1k views

Why Fourier series has summation and Fourier transform has integration symbol in their respective formulae?

Fourier transform for aperiodic signal is given by $$ X(\omega) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j \omega t} dt. \quad (1) $$ Fourier series for periodic signal is given by $$ y(t) = \...
2
votes
2answers
369 views

Is Fourier transform still writing a function as a series of sines and cosines?

In the Fourier series we write a function as a series of sines and cosines. Fourier transform seems to me to be totally different, we are not finding a series but rather a function $\hat f(w)$. So ...
2
votes
1answer
865 views

Integral definition of delta function and Kronecker symbol

I know the following two definitions for the delta function and Kronecker delta, respectively: (1) $\int_{-\infty}^{\infty}\frac{e^{iwt}}{2\pi}\mathrm{d}t = \delta(w)$ (2) $\int_{-\pi}^{\pi}\frac{e^{...
2
votes
1answer
844 views

Separation of Variables vs Fourier Transform (for PDE)

I would like to know how can I know if I have to solve a PDE (Heat Equation, Laplace Equation, Wave Equation, etc.) using Separation of Variables or Fourier Transform. Which boundary conditions do I ...
0
votes
1answer
434 views

Summary of different Fourier Transforms / Fourier Series

I am currently writing my PhD and would like to display a table summing up the different kind of Fourier Transforms and Fourier Series. Here is the table I got from my different readings (mostly ...
1
vote
0answers
65 views

Is it possible to use FFT to derive a Fourier series fitting to data?

I want to do something like what is done in this question about fitting , ie find a Fourier series that approximates a continuous but complicated function. However I want to know whether it is ...
1
vote
1answer
36 views

Frequency multiples in Fourier Series expansions

When we expand signal with Fourier series, why angular frequencies are in multiples? I.e. why $$x(t)=\sum_{k=-\infty}^{\infty}z_k*e^{jk\omega_0t}=z_0+z_1e^{j\omega_0t}+z_2e^{j2\omega_0t}+z_3e^{j3\...