Harmonics
The simplest periodic functions of the form
![]() |
These functions are encountered in the study of many oscillatory processes. The number is known as the amplitude,
is known as the frequency,
is known as the initial phase, and
is the oscillation period. The functions
are, respectively, the second, third, etc., higher harmonics with respect to the fundamental harmonic. In addition to the harmonics themselves, their sums
![]() | (*) |
are also considered, since a very broad class of functions can be expanded in series of the form (*) in the study of various processes.
Comments
More generally, if is a compact group,
is a closed subgroup of
and if the regular representation of
on
decomposes uniquely into irreducible subrepresentations, then the functions on the homogeneous space
belonging to irreducible subspaces of
are called harmonics, cf. [a1]. For
,
, one finds the classical harmonics.
References
[a1] | H. Weyl, "Harmonics on homogeneous manifolds" Ann. of Math. , 35 (1934) pp. 486–499 |
[a2] | E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint (1955) |
Harmonics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonics&oldid=11587