What is Smoothness?

תקציר מקורי באנגליתarXiv:2609.03246v1 Announce Type: cross Abstract: Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of the irreducible representations of $G$, but for non-abelian $G$, no ordering is canonical. Given a symmetric generating set $S$, the Laplacian of the associated Cayley graph is block diagonal over the dual, and we order the irreps by the mean of the eigenvalues in each block. This produces an ordering function $\omega:\widehat{G}\to\mathbb{R}$ that depends only on the pair $(G,S)$. This function is bounded between zero and two, vanishing only at the trivial represe