Almost Periodic Functions on Topological Groups

We study the definition and properties of almost periodic functions on topological groups. We show the equivalence between Bochner’s and Bohr’s definitions of almost periodicity. We discuss Weil’s construction of Bohr compactification b(G) and study its properties. Using Peter-Weyl’s density theorem we show that a function f in Cb(G) is almost periodic if and only if it is the uniform limit of linear combinations of coefficients of the finite-dimensional irreducible unitary representations of G. We show the existence of a unique invariant mean on the space of almost periodic functions. We investigate the Fourier series of almost periodic functions, and show that it extends the classical Fourier series of 2-periodic functions