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Harmonics

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The simplest periodic functions of the form

$$A\sin(\omega x+\phi).$$

These functions are encountered in the study of many oscillatory processes. The number $A$ is known as the amplitude, $\omega$ is known as the frequency, $\phi$ is known as the initial phase, and $T=2\pi/\omega$ is the oscillation period. The functions $\sin(2\omega x+\phi),\sin(3\omega x+\phi),\dots,$ are, respectively, the second, third, etc., higher harmonics with respect to the fundamental harmonic. In addition to the harmonics themselves, their sums

\begin{equation}a_0+a_1\sin(\omega x+\phi)+a_2\sin(2\omega x+\phi)+\dots\label{*}\end{equation}

are also considered, since a very broad class of functions can be expanded in series of the form \eqref{*} in the study of various processes.


Comments

More generally, if $G$ is a compact group, $K$ is a closed subgroup of $G$ and if the regular representation of $G$ on $L_2(G/K)$ decomposes uniquely into irreducible subrepresentations, then the functions on the homogeneous space $G/K$ belonging to irreducible subspaces of $L_2(G/K)$ are called harmonics, cf. [a1]. For $G=O(2)$, $K=O(1)$, 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)
How to Cite This Entry:
Harmonics. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Harmonics&oldid=43448
This article was adapted from an original article by A.I. Barabanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article