Fourier Transform

Very broadly speaking, the Fourier transform is a systematic way to decompose “generic” functions into a superposition of “symmetric” functions. These symmetric functions are usually quite explicit (such as a trigonometric function sin(nx) or cos(nx)), and are often associated with physical concepts such as frequency or energy. What “symmetric” means here will be left vague, but it will usually be associated with some sort of group G, which is usually (though not always) abelian. Indeed, the Fourier transform is a fundamental tool in the study of groups (and more precisely in the representation theory of groups, which roughly speaking describes how a group can define a notion of symmetry). The Fourier transform is also related to topics in linear algebra, such as the representation of a vector as linear combinations of an orthonormal basis, or as linear combinations of eigenvectors of a matrix (or a linear operator).