Harmonic Representation of 1 dimensional Spherical Functions
Harmonic Representation of 1 dimensional Spherical Functions
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Functions on the circle are periodic functions. Hence they may be represented as weighted sums of sines and cosines (Fourier series). A spherical function \(f\) can be written as series of the form
\[ f(x) = \sum_{k=-N}^N \hat f_k e^{-ikx} \]
with respect to Fourier coefficients \(\hat f_k\). Note that \(f\) is \(2\pi\)-periodic.
Within the class S1FunHarmonic spherical functions are represented by their Fourier coefficients which are stored in the field fun.fhat. As an example lets define a Fourier series which Fourier coefficients \(\hat f_0 = 1\), \(\hat f_1 = 0\), \(\hat f_{-1} = 3\), \(\hat f_2 = 4\) and \(\hat f_{-2} = 0\)
More practically, periodic functions appear after density estimation from circular data, e.g. of the azimuth angle of three dimensional vectors