  Pseudo-differential and Fourier series operators on the n-torus are analyzed
by using global representations by Fourier series instead of local
representations in coordinate charts. Toroidal symbols are investigated and the
correspondence between toroidal and Euclidean symbols of pseudo-differential
operators is established. Periodization of operators and hyperbolic partial
differential equations is discussed. Fourier series operators, which are
analogues of Fourier integral operators on the torus, are introduced, and
formulae for their compositions with pseudo-differential operators are derived.
It is shown that pseudo-differential and Fourier series operators are bounded
on $L^2$ under certain conditions on their phases and amplitudes.
