Periodic Fourier series

The Fourier series associated with an integrable $\mathbb Z^d$ -periodic function $f$ is

\sum_{m\in\mathbb Z^d}\widehat f(m)e^{2\pi i m\cdot x}.

For example, rectangular partial sums retain the coefficients with $|m_i|\le N$ for every $i$ . Other summation methods must be specified.

Convergence is an additional assertion

The definition supplies coefficients and a series, but not pointwise equality with $f$ . Smooth periodic functions have convergence with every derivative. An $L^2$ function has convergence in $L^2$ for rectangular partial sums. Merely continuous functions can have divergent pointwise Fourier partial sums.

References

Leonid Ryzhik, Lecture notes for Math 205A, Chapters 8–9.