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, …

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 …

Oct 27, 2019 · What are some real world applications of Fourier series? Particularly the complex Fourier integrals?

Fourier transform commutes with linear operators. Derivation is a linear operator. Game over.

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.

You seem to be assuming that it is an either/or situation. It isn't. A wave can be all three: odd (OR even), have half wave symmetry, and also have quarter wave symmetry. All your examples …

May 6, 2017 · Does that mean that the function is valued 2π−−√ 2 π at all points in the frequency domain? I think this is reasonable because such function i.e. f(t) = 1 f (t) = 1 in the time domain …

Dec 15, 2012 · The Fourier transform projects functions onto the plane wave basis - basically a collection of sines and cosines. A Fourier series is also a projection, but it's not continuous - …

The conditions are "not necessary" because no one proved a theorem that if the Fourier series of a function f(x) f (x) converge pointwise then the function satisfies the Dirichlet conditions.