Dec 29, 2019 · What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow.

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.

0 One could derive the formula via dual numbers and using the time shift and linearity property of the Fourier transform.

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 …

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 5, 2015 · Here is my biased and probably incomplete take on the advantages and limitations of both Fourier series and the Fourier transform, as a tool for math and signal processing.

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Explore related questions functional-analysis analysis fourier-analysis fourier-transform See similar questions with these tags.

Aug 30, 2020 · This might be a good approach. However, the Fourier inversion theorem is valid only for a subset of functions, so it seems that more caution is required.

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