__Fourier series motivation__A Fourier transform is the continus limit of a Fourier series.

Think of a function f(x) which satisfies some boundary conditions f(0)=0 and f(pi)=0. I have drawn some example functions in blue. The function sin(nx) satisfies the boundary conditions for n=integer. Any sum of sin(nx) will also satisfy the boundary conditions.

The inverse is true, that any continuous function satisfying the boundary conditions can be written as a sum of sin(nx) with coefficients a_{n}.

There are 2 ways to describe the function f(x):

- Tell me the value of f(x) for every x
- Tell me the Fourier coefficients a_{1}, a_{2}, a_{3}....

__Waves in a box__One particular wavelength or distance X, corresponds to a single value in K space. Unless we have an infinite decomposition, our K spectrum is discretised.

To fill in K space values for a continuous spectrum you have to count waves that go into the box a non-integer number of times. eg what wavelength x fits in the box 1.1 times, 1.2, 1.3, 1.4 etc and all the waves in between.

The size of the box in X space limits the largest wavelength mode you can see. The size of the pixels in X space limits the smallest wavelength mode you can see, which corresponds to the highest frequency mode you can resolve in K space.

(Text reads: Lowest frequency mode you can resolve

Highest frequency mode you can resolve

Can't resolve this very high frequency mode)

Highest frequency mode you can resolve

Can't resolve this very high frequency mode)

Size of box in X space not equal to size of box in K space, but you do get same number of pieces of information. Highest mode in K space is inverse to smallest spacing in X. Longest wavelength mode in X is inverse to smallest spacing in K. Cutting the box decreases resolution in K space and information about the longest wavelength modes is lost.

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