  harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of fourier series and fourier transforms (i.e. an extended form of fourier analysis). in the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience. the term "harmonics" originated as the ancient greek word, "harmonikos," meaning "skilled in music."http://www.etymonline.com/index.php?term=harmonic it arose in physical eigenvalue-problems to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. the classical fourier transform on rn is still an area of ongoing research, particularly concerning fourier transformation on more general objects such as tempered distributions. for instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the fourier transform of f. the paleyâwiener theorem is an example of this. the paleyâwiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its fourier transform is never compactly supported. this is a very elementary form of an uncertainty principle in a harmonic analysis setting. see also: convergence of fourier series. fourier series can be conveniently studied in the context of hilbert spaces, which provides a connection between harmonic analysis and functional analysis. ==abstract harmonic analysis== one of the most modern branches of harmonic analysis, having its roots in the mid-twentieth 
Harmonic_analysis