What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. An...

Oct 26, 2012 · The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by …

In the QM context, momentum and position are each other's Fourier duals, and as you just discovered, a Gaussian function that's well-localized in one space cannot be well-localized in the other.

18 The Fourier Transform is a very useful and ingenious thing. But how was it initiated? How did Joseph Fourier composed the Fourier Transform formula and the idea of a transformation between periodic …

Apr 26, 2019 · So, yes, we expect a $\mathrm {e}^ {\mathrm {i}kx_0}$ factor to appear when finding the Fourier transform of a shifted input function. In your case, we expect the Fourier transform of the …

Jun 27, 2013 · 0 One could derive the formula via dual numbers and using the time shift and linearity property of the Fourier transform.

I've been practicing some Fourier transform questions and stumbled on the following one. To start off, I defined the Fourier transform for this function by taking integral from $-\\tau$ to $0$ and $...

Let us consider the Fourier transform of $\\mathrm{sinc}$ function. As I know it is equal to a rectangular function in frequency domain and I want to get it myself, I know there is a lot of material...

11 Fourier Transform is a function. Fast Fourier Transform is an algorithm. It is similar to the relationship between division and long division. Division is a function, long division is a way to compute the function.

The theory of Fourier transforms has gotten around this in some way that means that integral using normal definitions of integrals must not be the true definition of a Fourier transform.