HarmonicOrthogonalPolynomials.jl

A Julia package for working with spherical harmonic expansions and harmonic polynomials in balls.

A harmonic polynomial is a multivariate polynomial that solves Laplace's equation. Spherical harmonics are restrictions of harmonic polynomials to the sphere. Importantly they are orthogonal. This package is primarily an implementation of spherical harmonics (in 2D and 3D) but exploiting their polynomial features.

Currently this package focusses on support for 3D spherical harmonics. We use the convention of FastTransforms for real spherical harmonics:

julia> θ,φ = 0.1,0.2 # θ is polar, φ is azimuthal (physics convention)

julia> sphericalharmonicy(ℓ, m, θ, φ)
0.07521112971423363 + 0.015246050775019674im

But we also allow function approximation, building on top of ContinuumArrays.jl and ClassicalOrthogonalPolynomials.jl:

julia> S = SphericalHarmonic() # A quasi-matrix representation of spherical harmonics
SphericalHarmonic{Complex{Float64}}

julia> S[SphericalCoordinate(θ,φ),Block(ℓ+1)] # evaluate all spherical harmonics with specified ℓ
5-element Array{Complex{Float64},1}:
 0.003545977402630546 - 0.0014992151996309556im
  0.07521112971423363 - 0.015246050775019674im
    0.621352880681805 + 0.0im
  0.07521112971423363 + 0.015246050775019674im
 0.003545977402630546 + 0.0014992151996309556im

julia> 𝐱 = axes(S,1) # represent the unit sphere as a quasi-vector
Inclusion(the 3-dimensional unit sphere)

julia> f = 𝐱 -> ((x,y,z) = 𝐱; exp(x)*cos(y*sin(z))); # function to be approximation

julia> S \ f.(𝐱) # expansion coefficients, adaptively computed
∞-blocked ∞-element BlockedArray{Complex{Float64},1,LazyArrays.CachedArray{Complex{Float64},1,Array{Complex{Float64},1},Zeros{Complex{Float64},1,Tuple{InfiniteArrays.OneToInf{Int64}}}},Tuple{BlockedOneTo{Int,ArrayLayouts.RangeCumsum{Int64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}:
        4.05681442931116 + 0.0im                   
 ──────────────────────────────────────────────────
      1.5777291816142751 + 3.19754060061646e-16im  
  -8.006900295635809e-17 + 0.0im                   
      1.5777291816142751 - 3.539535261006306e-16im 
 ──────────────────────────────────────────────────
      0.3881560551355611 + 5.196884701505137e-17im 
  -7.035627371746071e-17 + 2.5784941810054987e-18im
    -0.30926350498081934 + 0.0im                   
   -6.82462130695514e-17 - 3.515332651034677e-18im 
      0.3881560551355611 - 6.271963079558218e-17im 
 ──────────────────────────────────────────────────
     0.06830566496722756 - 8.852861226980248e-17im 
 -2.3672451919730833e-17 + 2.642173739237023e-18im 
     -0.0514592471634392 - 1.5572791163000952e-17im
  1.1972144648274198e-16 + 0.0im                   
    -0.05145924716343915 + 1.5264133695821818e-17im
                         ⋮

julia> f̃ = S * (S \ f.(𝐱)); # expansion of f in spherical harmonics

julia> f̃[SphericalCoordinate(θ,φ)] # approximates f
1.1026374731849062 + 4.004893695029451e-16im