Harmonic Analysis in Mathematics James Arthur Summary: Harmonic Analysis in Mathematics James Arthur Harmonic analysis in musicis the studyof chords,and of how they are used in combination to create musical effects.Harmonic analysis in mathematics takes on a somewhat different meaning. It too has roots in music,or at least in the mathematical analysis of sound.However,the term can also mean a kind of universal duality that runs throughout mathematics,as well as other sciences. Differentmusical instruments make different kinds of sound, even when they play the samenote.The analysis of this phenomenon can be very complicated,but to a first approximation,it is the shape of the instrument that determines the sound it creates.Shape is an obviousgeometric property.Sound is an example of what is often called aspectralphenomenon.The term spectralhas a precise mean- ing in mathematics,but we can also think of it in the everyday sense of the word-ghostlike, hard to pin down-the very opposite of geometric.We thus have an example of somethinggeometric- the shapeof a musical instrument-which correspondsto something spectral-the sound produced by the instrument. Harmonicanalysis . in the broadest sense refers to a general principlein mathematics Collections: Mathematics