## LMSpace with complex signal space partners

LMSpace stores only one half of the complex-conjugated pairs one gets if the signal field was real valued. In this case the dot product between a=(a_-1, a_0, a_1) and b=(...) can be computed by simply taking a_0*b*0 + 2*(a_1.real*b_1.real + a_1.imag*b_1.imag) as a_-1 and a_1 are complex-conjugated.

If the signal space field was complex, one gets something with cc-symmetry for the real and the imaginary part of this field respectively; modulo i. FFT is linear -> the result has no cc-symmetry anymore.

Solution: Store the full lm spectrum for LMSpace. Apply the action of libsharp (transform, smooth) on the half sum and half difference of a field with its transposed as this reconstructs the cc-symmetric parts comming from the real (imaginary, resp.)-part of the signal field.