Translation

Abstract

For each real t define the translation operator: T _t f(x) = f(x−t) for all functions f defined on R. Translation operators on Z and on T are defined similarly. N. Wiener began the study of closed subspaces of L¹(R) and of L²(R) that are invariant under all translations T _t . A. Beurling studied subspaces that are invariant under translations in just one direction on R or Z; these subspaces axe more complicated and interesting. Translation is carried by the Fourier transform to multiplication by exponentials. Thus much of Chapter 4 was about such subspaces. The first objective of this chapter is to characterize the closed subspaces of L²(R) invariant under all translations, or under translations to the right. These results are analogous to theorems of Chapter 4 on the circle.