Harmonic Representation of 1 dimensional Spherical Functions edit page

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 \(\hat f_{-2} = 0\)

fun = S1FunHarmonic([0;3;1;0;4])

clf
plot(fun)
fun = S1FunHarmonic
  bandwidth: 2
 
ans = 
  Line with properties:

              Color: [0 0.4470 0.7410]
          LineStyle: '-'
          LineWidth: 0.5000
             Marker: 'none'
         MarkerSize: 10.5000
    MarkerFaceColor: 'none'
          ThetaData: [0 0.0175 0.0349 0.0524 0.0698 0.0873 … ] (1×361 double)
              RData: [8 7.9971 7.9884 7.9740 7.9538 7.9278 … ] (1×361 double)

  Use GET to show all properties

More practically, periodic functions appear after density estimation from circular data, e.g. of the azimuth angle of three dimensional vectors

% some random directions
v = vector3d.rand(1000);

% perform density estimation of the azimuth angle
fun = calcDensity(v.rho,'periodic')

clf
plot(fun,'linewidth',2)
fun = S1FunHarmonic
  bandwidth: 61
  isReal: true
 
ans = 
  Line with properties:

              Color: [0 0.4470 0.7410]
          LineStyle: '-'
          LineWidth: 2
             Marker: 'none'
         MarkerSize: 10.5000
    MarkerFaceColor: 'none'
          ThetaData: [0 0.0175 0.0349 0.0524 0.0698 0.0873 … ] (1×361 double)
              RData: [1.0290 1.0853 1.1403 1.1874 1.2226 … ] (1×361 double)

  Use GET to show all properties