Convolution theorem
Periodic convolution (Fourier series coefficients)

Consider P {\displaystyle P} -periodic functions u P {\displaystyle u_{_{P}}} and v P, {\displaystyle v_{_{P}},} which can be expressed as periodic summations:

In practice the non-zero portion of components u {\displaystyle u} and v {\displaystyle v} are often limited to duration P, {\displaystyle P,} but nothing in the theorem requires that.

The Fourier series coefficients are:

where F {\displaystyle {\mathcal {F}}} denotes the Fourier series integral.

is also P {\displaystyle P} -periodic, and is called a periodic convolution.

The corresponding convolution theorem is:
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