Diagonal operators#
Operators diagonal in the spherical harmonic basis (Sec. VII B).
Harmonic multiplication of Kennedy et al. and the sifting convolution of Roddy & McEwen multiply each coefficient by \(g_{\ell m}\) or \(g^*_{\ell m}\) (Definition III.9). They commute with every polar rotation, and are equivariant under all rotations exactly when \(g_{\ell m}\) does not depend on \(m\), in which case they are the lifted convolution \(\mathcal{C}_{0\to0}[\Psi_T]\) with \(c_\ell = g_{\ell0}\) (Proposition VII.2).
- s2conv.convolutions.diagonal.harmonic_multiplication(flm, glm) Array[source]#
Harmonic multiplication \((g \odot_{\mathrm{K}} f)_{\ell m} = g_{\ell m} f_{\ell m}\) (Definition III.9).
- Parameters:
flm – Scalar harmonic coefficients,
[..., L, 2L - 1].glm – Scalar harmonic coefficients of the filter,
[..., L, 2L - 1].
- Returns:
Scalar harmonic coefficients,
[..., L, 2L - 1].- Return type:
jnp.ndarray
- s2conv.convolutions.diagonal.sifting_convolution(flm, glm) Array[source]#
Sifting convolution \((f \circledcirc g)_{\ell m} = f_{\ell m} g^*_{\ell m}\) (Definition III.9).
- Parameters:
flm – Scalar harmonic coefficients,
[..., L, 2L - 1].glm – Scalar harmonic coefficients of the filter,
[..., L, 2L - 1].
- Returns:
Scalar harmonic coefficients,
[..., L, 2L - 1].- Return type:
jnp.ndarray