Harmonic analysis on homogeneous spaces is a far-reaching generalization
of the classical theory of Fourier series and Fourier integrals. It is a branch of
functional analysis which is vigorously developing now. The principal contents
is closely connected with group representation theory in infinite-dimensional
spaces. On the other hand, it interacts with such diverse fields as algebra,
algebraic geometry, spectral theory of operators, number theory, Hamiltonian
mechanics, quantum mechanics.