Periodic and Discrete Analysis

Abstract

In this chapter we will review basics of the periodic and discrete analysis which will be necessary for development of the theory of pseudo-differential operators on the torus in Chapter 4. Our aim is to make these two chapters accessible independently for people who choose periodic pseudo-differential operators as a starting point for learning about pseudo-differential operators on ℝⁿ. This may be a fruitful idea in the sense that many technical issues disappear on the torus as opposed to ℝⁿ. Among them is the fact that often one does not need to worry about convergence of the integrals in view of the torus being compact. Moreover, the theory of distributions on the torus is much simpler than that on ℝⁿ, at least in the form required for us. The main reason is that the periodic Fourier transform takes functions on $\mathbb{T}^n$ = ℝⁿ/ℤⁿ to functions on ℤⁿ where, for example, tempered distributions become pointwise defined functions on the lattice ℤⁿ of polynomial growth at infinity. Also, on the lattice ℤⁿ there are no questions of regularity since all the objects are defined on a discrete set. However, there are many parallels between Euclidean and toroidal theories of pseudo-differential operators, so looking at proofs of similar results in different chapters may be beneficial. In many cases we tried to avoid overlaps by presenting a different proof or by giving a different explanation.