Spherical harmonics

of degree $k$

A restriction of a homogeneous harmonic polynomial $h^{(k)}(x)$ of degree $k$ in $n$ variables $x=(x_1,\dots,x_n)$ to the unit sphere $S^{n-1}$ of the Euclidean space $E^n$, $n\geq3$. In particular, when $n=3$, the spherical harmonics are the classical spherical functions.

Let $x\in E^n$, $x\neq0$, $r=|x|$, $x'=x/r\in S^{n-1}$. The basic property of spherical harmonics is the property of orthogonality: If $Y^{(k)}(x')$ and $Y^{(l)}(x')$ are spherical harmonics of degree $k$ and $l$, respectively, with $k\neq l$, then

$$\int\limits_{S^{n-1}}Y^{(k)}(x')Y^{(l)}(x')dx'=0.$$

The simplest spherical harmonics are the zonal spherical harmonics. For any $t'\in S^{n-1}$ and any $k>0$, a zonal spherical harmonic $Z_{t'}^{(k)}(x')$ exists which is constant on any parallel of the sphere $S^{n-1}$ that is orthogonal to the vector $t'$. The zonal spherical harmonics $Z_{t'}^{(k)}(x')$ differ from the Legendre polynomials $P_k^{(\lambda)}$, when $n=3$, or from the ultraspherical polynomials $P_k^{(\lambda)}$, when $n>3$, only by a constant factor:

$$Z_{t'}^{(k)}(x')=c(k,n)P_k^{(\lambda)}(x't'),$$

where the polynomials $P_k^{(\lambda)}$ are defined, when $n\geq3$, by the generating function

$$(1-2st+s^2)^{-\lambda}=\sum_{k=0}^\infty P_k^{(\lambda)}(t)s^k,$$

$0\leq|s|<1$, $|t|=1$, $\lambda=(n-2)/2$. The polynomials $P_k^{(\lambda)}$, $k=0,1,\dots,$ are orthogonal with weight $(1-t^2)^{\lambda-1/2}$ and form an orthogonal basis of the space $L_2([-1,1];(1-t^2)^{\lambda-1/2})$. If $f$ is a function in $L_2(S^{n-1})$ with $\int_{S^{n-1}}f(x')dx'=0$, then there is a unique set of spherical harmonics $Y^{(k)}$ such that

$$f(x')=\sum_{k=1}^\infty Y^{(k)}(x'),$$

where the series converges in the norm of $L_2(S^{n-1})$.

Expansions in spherical harmonics are largely analogous to expansions in Fourier series, of which they are essentially a generalization. The homogeneous harmonic polynomials $h^{(k)}(x)$ are sometimes called spatial spherical harmonics. By virtue of the homogeneity

$$h^{(k)}(x)=|x|^kY^{(k)}(x'),$$

spherical harmonics are sometimes also called surface spherical harmonics.

References

[1]	P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)
[2]	E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)

References

[a1]	I.M. Gel'fand, R.A. Minlos, Z.Ya. Shapiro, "Representations of the rotation group and the Lorentz group, and their applications" , Macmillan (1963) (Translated from Russian)
[a2]	N.Ya. Vilenkin, "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian)
[a3]	N.Ya. Vilenkin, A.U. Klimyk, "Special functions, group representations, and integral transforms" , 1 , Kluwer (1991) (Translated from Russian)