I am interested in finding the functions $f:\mathbb{R}^n \to \mathbb{R}^n$ for which $f \circ U = U \circ f$ for all orthogonal transformations $U:\mathbb{R}^n \to \mathbb{R}^n$. Note that $f$ need not be linear. Any ideas on the conditions on $f$ ?
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Welcome to MSE. Please use MathJax. – José Carlos Santos Jan 16 '18 at 11:52
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3See my answer + comments at https://math.stackexchange.com/questions/2591815/let-v-be-spherically-symmetric-and-w-i-i-d-gaussian-then-evvw-t-g-t/2592127#2592127 – Dap Jan 16 '18 at 11:54
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1Also here: https://math.stackexchange.com/questions/2533891/what-are-some-interesting-functions-that-are-equivariant-under-rotations-in-so3/2534273#2534273 – Qiaochu Yuan Jan 16 '18 at 20:04