Pseudo-differential operators on are operators of the form
where is the Fourier transform of and is called the symbol of
It may look complicated but in fact, roughly,
every linear continuous operator taking functions to functions is a pseudo-differential operator!
This is very easy to see, even rigorously, in the case of the torus For a smooth function its Fourier coefficients are defined by
The Fourier inversion formula then gives
Let now be a linear continuous operator, and let us define its so-called toroidal symbol by
Then using (3) and the fact that is linear and continuous we have
where we used (4) in the last line. Thus, we have shown an analogue of (1) on the torus:
which is called the toroidal quantization on the torus
A more or less comprehensive analysis of this toroidal quantization, its properties and relations to (1) can be read here:
Ruzhansky M., Turunen V., Quantization of pseudo-differential operators on the torus, J. Fourier Anal. Appl., 16 (2010), 943-982. download from arxiv, link
You can also read a bit more on pseudo-differential operators at nLab or in Wikipedia.