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...
While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas...
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
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...
Then the Fourier transform of the Dirac delta-function (well, actually it's not a function, but the calculations work anyways) is $$ \mathcal {F}\ {\delta (x)} = \int_ {-\infty}^ {\infty} \delta (x) , e^ {-ikx} , dx = 1. $$
Both Fourier transform and Taylor series are means to represent functions in a different form. What is the connection between these two? Is there a way to get from one to the other (and back agai...
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.
While saz has already answered the question, I just wanted to add that this can be seen as one of the simplest examples of the Uncertainty Principle found in quantum mechanics, and generalizes to something called Hardy's uncertainty principle. 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 ...