For a periodic function with
fundamental period , the corresponding Fourier Series representation is given by:
The Fourier Coefficients
and can be determined from the following integrals:
where can be interpreted as
the average value of over the interval .
Also Note that the above integration interval from - to
can actually be any interval of length
, such as from 0 to ,
which may be more convenient in some cases.
Mean Value Convergence Theorem:
If a periodic function with period is piecewise continuous over the interval , the Fourier Series of converges to the mean value at point where both the left-hand and right-hand first derivatives of exist.
Important: For non-periodic functions, one can argue that they are periodic with
an infinite period, that is, . The Fourier Series then
becomes the Fourier Integral.