# Struggling to interpret spherical harmonic transforms from FastTransforms.jl

I am trying to compute the spherical harmonic transform of a function that is given by `sin(θ)cos(ϕ)` on the face of the sphere facing us, and zero on the other face that is away from us. I am attempting to use `FastTransforms.jl` ans follow this example as

``````julia> f_test(θ, ϕ) = (π/2 <= ϕ <= 3π/2) ? zero(Float64) : Float64(sin(θ)cos(ϕ))
f_test (generic function with 1 method)

julia> function threshold!(A::AbstractArray, ϵ)
for i in eachindex(A)
if abs(A[i]) < ϵ A[i] = 0 end
end
A
end
threshold! (generic function with 1 method)

julia> lmax = 3
3

julia> N = lmax + 1
4

julia> θ = ((0.5:N-0.5)/N) .* pi
0.39269908169872414:0.7853981633974483:2.748893571891069

julia> M = 2N-1
7

julia> ϕ = ((0:M-1)/M) .* 2pi
0.0:0.8975979010256552:5.385587406153931

julia> F = [f_test(θ,ϕ) for θ in θ, ϕ in ϕ]
4×7 Array{Float64,2}:
0.382683  0.238599  0.0  0.0  0.0  0.0  0.238599
0.92388   0.576029  0.0  0.0  0.0  0.0  0.576029
0.92388   0.576029  0.0  0.0  0.0  0.0  0.576029
0.382683  0.238599  0.0  0.0  0.0  0.0  0.238599

julia> using FastTransforms

julia> P = plan_sph2fourier(F);

julia> PA = plan_sph_analysis(F);

julia> V = PA*F;

julia> threshold!(P\V, 400*eps())
4×7 Array{Float64,2}:
0.888525  0.0  1.0394  0.0  0.417051   0.0  -0.0697631
0.0       0.0  0.0     0.0  0.0        0.0   0.0
-0.259657  0.0  0.0     0.0  0.0571638  0.0  -0.018645
0.0       0.0  0.0     0.0  0.0        0.0   0.0
``````

However I do not understand this result. Going by the storage of the spherical harmonic coefficients, the last column should only have the top row filled. The element at index `[3,7]` should be zero (at least within floating-point tolerance). Also, why is there no conjugate symmetry?

Am I misunderstanding something here? Some help would be welcome!

Have you tried to use a single spherical harmonic in `f_test` and see if your script recovers that as a single number in your final array?

1 Like

This is what is demonstrated in the example that I was trying to follow, and it does seem to work.

The discussion is continued in the issue https://github.com/JuliaApproximation/FastTransforms.jl/issues/120.