On the Fourier analysis of operators on the torus

In this paper we will discuss the version of the Fourier analysis and pseudodifferential operators on the torus. Using the toroidal Fourier transform we will show several simplifications of the standard theory. We will also discuss the corresponding toroidal version of Fourier integral operators. To distinguish them from those defined using the Euclidean Fourier transform, we will call them Fourier series operators. The use of discrete Fourier transform will allow to use global representation of these operators, thus eliminating a number of topological obstructions known in the standard theory. We will prepare the machinery and describe how it can be further used in the calculus of Fourier series operators and applications to hyperbolic partial differential equations. In fact, the form of the required discrete calculus is not a-priori clear, for example, the form of the discrete Taylor’s theorem best adopted to the calculus. We will develop the corresponding version of the periodic analysis similar in formulations to the standard Euclidean theory. It was realised already in the 1970s that on the torus, one can study pseudodifferential operators globally using Fourier series expansions, in analogy to Euclidean pseudodifferential calculus. These periodic pseudodifferential operators were treated e.g. by Agranovich [1, 2]. Contributions have been made by many authors, and the following is a non-comprehensive list of the research on the torus: Agranovich, crediting the idea to Volevich, proposed the Fourier series representation of pseudodifferential operators. Later, he proved the equivalence of the Fourier series representation and Hörmander’s definition for (1, 0)-symbol classes; the case of classical pseudodifferential operators on the circle had been treated by Saranen