Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in $L_{2}$, where it always exists and converges pointwise to the function: Carleson's theorem.

## Ancestors

## Incoming links

- Bessel function
- Carleson's theorem
- Complete basis
- Fourier series
- Fourier transform
- Lebesgue integral vs Riemann integral
- Riesz-Fischer theorem
- Separation of variables
- Solving partial differential equations with the Fourier series
- Solving the Schrodinger equation with the time-independent Schrödinger equation