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...
I like this question. You're absolutely right to be cautious about the claim that continuity alone implies normal convergence of a Fourier series, that is not true in general. Let’s construct a continuous function on $ [-\pi, \pi]$, periodic with period $2\pi$, whose Fourier series does not converge normally. Let’s consider: $$ f (x) = \begin {cases} 0, & x = 0 \ \frac {\sin x} {x}, & x ...
Fourier had to fight to get others to believe that he might be correct in his belief that such expansion could be general. Many still unfairly accuse Fourier of not having been precise at all. To Fourier's credit, the Dirichlet kernel integral expression for the truncated trigonometric Fourier series was in Fourier's original work.
What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. An...
While saz has already answered the question, I just wanted to add that this can be seen as one of the simplest examples of the Uncertainty Principle found in quantum mechanics, and generalizes to something called Hardy's uncertainty principle. In the QM context, momentum and position are each other's Fourier duals, and as you just discovered, a Gaussian function that's well-localized in one ...
Then the Fourier transform of the Dirac delta-function (well, actually it's not a function, but the calculations work anyways) is $$ \mathcal {F}\ {\delta (x)} = \int_ {-\infty}^ {\infty} \delta (x) , e^ {-ikx} , dx = 1. $$