Fourier transforms appear when limits are applied to Fourier series, which are usually a series representation of functions on a circle. Fourier transform can best be applied to handle problems with non-periodic functions. There are a lot of ordinary and partial differential equation problems which require the knowledge of Fourier transform to simplify their solutions.
We define the Fourier transform of a piecewise continuous and absolutely integrable function by
and the inverse Fourier transform by
Here must converge.
Now, let's look at an example.
Let Compute the Fourier transform of
Solution
By definition
Note that
The generalization to high dimensions:
https://see.stanford.edu/materials/lsof ... /chap8.pdf
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