The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions ( sines and cosines ) are used to represent functions on a circle via Fourier series.
22.
On the applications side : quantum mechanics-- linear algebra; co-ordinate transformation locally-- linear algebra ( globally too, if it's flat ); basis function decompositions-- Fourier theory, Laplace theory, wavelets, functional analysis, orthogonal function theory-- all linear algebra.
23.
The Pacific Decadal Oscillation "'index "'is the leading empirical orthogonal function ( EOF ) of monthly sea surface temperature anomalies ( "'SST "'- A ) over the North Pacific ( poleward of 20?N ) after the global average sea surface temperature has been removed.
24.
Where, Yi ( t ) is the ith Principal Component and Vi ( t ) the ith Empirical orthogonal function ( EOF ) pattern and K is the smaller one of M and N . PC and EOFs are often obtained by solving the eigen-value / eigen-vector problem of either temporal co-variance matrix or spatial co-variance matrix based on which dimension is smaller.
