Definition. Convolution kernels with positive coefficients [fdip-007I]

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For a finite or compact Lie group \(G\), use the normalized Haar measure, irreducibles and dimensions of the character expansion. For convolution constructions, consider real continuous central kernels \(k\) with the separate hypotheses \[\begin {gathered} k(h)>0,\qquad k(h^{-1})=k(h),\\ k(h)=\sum _{\rho \in \widehat G}d_\rho a_\rho \chi _\rho (h), \qquad a_\rho >0,\qquad \sum _{\rho \in \widehat G}d_\rho ^2a_\rho <\infty . \end {gathered}\] The last condition gives uniform absolute convergence; inversion symmetry and reality impose the corresponding agreement of conjugate coefficients. The convention is \(c_\rho =d_\rho a_\rho \), not \(c_\rho =a_\rho \). These are assumptions on a kernel, not consequences of the Peter–Weyl theorem. In particular, pointwise positivity does not imply the coefficient condition, as the circle counterexample shows.