Fourier Series theorem is relevant to Ballooning representation.
Theorem 21.2 (Fourier Series Theorem)
.
- Let
be a function which is piecewise continuous on ![$ [-\pi , \pi ]$](http://www.math.ohio-state.edu/~gerlach/math/BVtypset/img673.png)
.Its Fourier series is given by
![$\displaystyle \frac{1}{2\pi}\int^\pi_{-\pi}f(t)\,dt+\frac{1}{\pi}\sum^\infty_{n=1} \int^\pi_{-\pi} f(t)\cos n(t-x)\,dt = \frac{1}{2}[f(x^-)+f(x^+)] $](http://www.math.ohio-state.edu/~gerlach/math/BVtypset/img674.png)
at each point 
where the one sided derivatives 
and 
both exist.
2. If 
is continuous the result is
![$\displaystyle \frac{1}{2\pi}\sum^\infty_{n=-\infty}\int^\pi_{-\pi} e^{in(x-t)} f(t)\,dt =\frac{1}{2}[f(x^-)+f(x^+)]=f(x)\quad \forall\, f\in C[-\pi ,\pi ]\,. $](http://www.math.ohio-state.edu/~gerlach/math/BVtypset/img676.png)
Excerpt from http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node27.html#Fourier_series_theorem
Ref. A. Thyagaraja, J. Plasma Physics 59, 367 (1998)
