The basic problem of harmonic analysis is the decomposition of elements in some topological vector space of functions in some linear basis which is typically distinguished by some nice representation theoretical properties. This decomposition can be a sum, and a basis a topological basis, but more general it is a decomposition in the sense of an integral. The elements of the distinguished bases were in historical examples thought of as “basic waves” or “harmonics”. Some standard examples are Fourier analysis on locally compact abelian groups, wavelet analysis?, quantum group Fourier transform etc. In some cases the elements of the “basis” are not linearly independent, e.g. in the case of decomposition into coherent states.
Edwin Hewett, Kenneth Ross, Abstract Harmonic Analysis
Volume I: Structure of Topological Groups, Integration Theory, Group Representations, Grundlehren der mathematischen Wissenschaften 115, Springer 1979 (doi:10.1007/978-1-4419-8638-2)
Volume II: Structure and Analysis for Compact Groups, Analysis on Locally Compact Abelian Groups, Grundlehren der mathematischen Wissenschaften 152, Springer 1970 (doi:10.1007/978-3-662-26755-4)
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. gBooks
Elias Stein, Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press 1971
See also:
Wikipedia, Harmonic analysis
MSRI, Categorical Structures in Harmonic Analysis, Nov 2014. (videos)