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.

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.

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 would …

Sep 9, 2016 · One approach could have been to see that the ramp function is the convolution of $2$ Heaviside step functions (at $0$). Hence, its Fourier transform should have been the product of the …

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 have half wave …

Oct 30, 2021 · One can consider 2D Fourier transform as a sequence of two 1-dimensional discrete Fourier transforms: applied to the first variable and then to the second. The properties follow …

Sep 4, 2020 · From what I currently understand about this topic the equation above should be the Fourier representation of the Dirac's Delta Function, however I don't see how to prove it. …