Bohr Compactification
Albrecht Böttcher, Yuri I. Karlovich, Ilya M. Spitkovsky · Birkhäuser Basel eBooks · 2002
In this chapter we record some results from harmonic analysis on locally compact Abelian groups. These results will be needed in the following chapters. In particular, we need the fact that an almost periodic function with Bohr-Fourier spectrum in some set E can be uniformly approximated by almost periodic polynomials whose Bohr-Fourier spectrum is supported in E. This result will be proved by making use of the Bochner-Fejér operators, which are the almost periodic analogues of the Fejér-Cesàro means we know from approximation by trigonometric polynomials. Furthermore, the abstract context allows us to interpret Wiener-Hopf and almost periodic factorization as AP(G) factorizations corresponding to the groups G = T and G = R , respectively. Finally, we introduce the Besicovitch space B 2 . When studying Wiener-Hopf operators with symbols inW N×N it is often useful to think of these operators as acting not only on,(R) but also on W N . The role of W N is played by, when considering Wiener-Hopf operators with symbols in AP N×N . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.