An Introduction to Harmonic Analysis 3Edition (Yitzhak Katznelson)

Harmonic analysis is the study of objects (functions, measures, etc.), defined on topological groups. The group structure enters into the study by allowing the consideration of the translates of the object under study, that is, by placing the object in a translation-invariant space. The study consists of two steps. First: finding the "elementary components" of the object, that is, objects of the same or similar class, which exhibit the simplest behavior under translation and which "belong" to the object under study (harmonic or spectral analysis); and second: finding a way in which the object can be construed as a combination of its elementary components (harmonic or spectral synthesis).