[dlfivefifty]

Adds ComplexZernike(a, b): the same ordering and radial parts as Zernike(a, b), but with sin(|m|θ) replaced by exp(-im*|m|*θ) and cos(|m|θ) by exp(im*|m|*θ). So the entry of degree ℓ and mode m is

r^|m| * jacobip((ℓ-|m|) ÷ 2, b, |m|+a, 2r^2-1) * exp(im*m*θ)

Following Laurent, the first type parameter is the (complex) element type, with a and b stored as reals: ComplexZernike() isa ComplexZernike{ComplexF64,Float64}.

Supported: evaluation (complexzernikez, normalizedcomplexzernikez), Normalized, transforms/expand, x .* C and y .* C, conversion to and from Zernike, conversion between parameters, lowering with Weighted, laplacian (weighted and unweighted), abslaplacian and sum. Plotting is not supported.

Implementation

• New AbstractZernike supertype. Operators that act on each Fourier mode through the radial part (conversion, lowering, Laplacians, sum) are shared between Zernike and ComplexZernike. They now build Jacobi with real parameters even for complex element types.
• Zernike and ComplexZernike are related by a block-diagonal matrix that mixes the ±|m| entries of each block. Between the normalized bases it is unitary.
• Transforms reuse ZernikeTransform and then mix the Fourier modes of the coefficients.
• Jacobi matrices are the Zernike ones conjugated by the mixing matrix. For Normalized(ComplexZernike()), X is real symmetric and Y is Hermitian.

Fixes to Zernike

• AbstractQuasiArray{T}(::Zernike) dropped the parameters, so Zernike(0.1,0.2) \ f hung for complex f.

Notes

• The mixing matrix stores its bands as an adjoint, with conjugated entries, to work around copy of a transpose of a BlockBroadcastArray conjugating its entries. That is fixed in Fix transpose copy JuliaLinearAlgebra/LazyBandedMatrices.jl#166, after which the workaround can go.
• This is an alternative to Add ComplexZernike #98 from 2021, which also added ComplexDerivative.

test/test_disk.jl passes locally, including a new ComplexZernike testset.

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[codecov]

Codecov Report

❌ Patch coverage is 98.11321% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 96.99%. Comparing base (d5106e6) to head (dfcb26a).

Files with missing lines	Patch %	Lines
src/disk.jl	98.11%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #234      +/-   ##
==========================================
+ Coverage   96.90%   96.99%   +0.08%     
==========================================
  Files           8        8              
  Lines        1100     1165      +65     
==========================================
+ Hits         1066     1130      +64     
- Misses         34       35       +1     

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