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convolution of a rotational function with a rotational or spherical function

1. Convolution of two rotational functions

If there are two SO3Fun $$f \colon {}_{S_f^L } \backslash SO(3) /_{S_f^R} \to \mathbb{C}$$ where $$S_f^L$$ is the left symmetry and $$S_f^R$$ is the right symmetry and $$g: {}_{S_g^L} \backslash SO(3) /_{S_g^R} \to \mathbb C$$ given. Then the convolution $$f {*}_L g \colon {}_{S_f^L} \backslash SO(3) /_{S_g^R} \to \mathbb C$$ is defined by

$(f {*}_L g)(R) = \frac{1}{8\pi^2} \int_{SO(3)} f(q) \cdot g(q^{-1}\,R) \, dq$

and the convolution $$f {*}_R g \colon {}_{S_g^L}\backslash SO(3) /_{S_f^R} \to \mathbb{C}$$ is defined by

$(f {*}_R g)(R) = \frac1{8\pi^2} \int_{SO(3)} f(q) \cdot g(R\,q^{-1}) \, dq$

with $$vol(SO(3)) = \int_{SO(3)} 1 \, dR = 8\pi^2$$. The convolution $$*_L$$ is used as default. The convolution of matrices of SO3Functions with matrices of SO3Functions works elementwise.

2. Convolution of a rotational function with a spherical function

The convolution of an SO3Fun $$f: {}_{S_f^L} \backslash SO(3) /_{S_f^R} \to \mathbb{C}$$ with an S2Fun $$h \colon \mathbb S^2 /_{S_h} \to \mathbb C$$ yields the S2Fun $$f*h \colon \mathbb S^2/_{S_f^L} \to \mathbb C$$ with

$(f * h)(\xi) = \frac1{8\pi^2} \int_{SO(3)} f(q) \cdot h(q^{-1}\,\xi) \, dq$

3. Convolution of a rotational function with a kernel function

In particular we convolute an SO3Fun with an SO3Kernel similar to the first case. Therefore the Right and Left sided convolution are equivalent. The convolution of an SO3Fun with an S2Kernel works analogue to case 2.%

## Input

 SO3F1, SO3F2 SO3Fun psi convolution SO3Kernel sF1 S2Fun phi convolution S2Kernel

## Output

 SO3F SO3FunHarmonic sF2 S2FunHarmonic