Fourier character on the standard torus

For $m\in\mathbb Z^d$ , the Fourier character is the periodic function

e_m(x)=\exp(2\pi i\,m\cdot x),\qquad x\in\mathbb R^d.

It is unchanged by every integer translation, so it defines a function on $\mathbb R^d/\mathbb Z^d$ . It obeys $e_m(x+y)=e_m(x)e_m(y)$ and $|e_m|=1$ .

Algebra and normalization

The exponential identity gives $e_me_n=e_{m+n}$ and $\overline{e_m}=e_{-m}$ . In an angular coordinate $\theta$ of period $2\pi$ , the corresponding characters are $e^{im\theta}$ . They are concrete examples of unitary characters; specifying the coordinate period fixes where the factor $2\pi$ occurs.

References

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