Almost-Periodic Functions and Functional Equations

The theory of almost-periodic functions with complex values created by H. Bohr [1] in his two classical papers published in Acta Mathematica in 1925 and 1926 has been developed by many authors and has had note- worthy we recall the works of Weyl De la Vallee Poussin Bochner Stepanov Wiener Besicovic Favard Delsarte Maak Bogoliu- bov Levitan. This subject has been widely treated in the monographs by Bohr [2] Favard [1] Besicovic [1] Maak [1] Levitan [1] Cinquini [1] Corduneanu [1] [2]. An important class of almost-periodic functions was studied at the beginning of the century by Bohl and Esclangon. Bohrs theory has been extended by Muckenhoupt [1] in a particular case and subsequently by Bochner [1] and by Bochner and Von Neumann [1] to very general abstract spaces. The extension to Banach spaces is in particular of great interest in view of the fundamental importance of these spaces in theory and application.