Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot. 



A book about almost periodic functions
What do you mean "a new", of what years?
Classics are always new-the books of Bohr, Levitan and Zhikov, Besikovich himself.
There is also a book of Corduneanu with standard name "almost periodic function" of 1989. Look also for Diagana 2013, Albrecht Böttcher et al 2002, Gaston M. N’Guerekata 2001, Wolfgang Schwarz 1994, S. Zaidman 1985 ...


