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For a compact group $G$ we define the Beurling-Fourier algebra $A_\omega(G)$on $G$ for weights $\omega$ defined on the dual $\what G$ and taking positivevalues. The classical Fourier algebra corresponds to the case $\omega$ is theconstant weight 1. We study the Gelfand spectrum of the algebra realizing it asa subset of the complexification $G_{\mathbb C}$ defined by McKennon andCartwright and McMullen. In many cases, such as for polynomial weights, thespectrum is simply $G$. We discuss the questions when the algebra $A_\omega(G)$is symmetric and regular. We also obtain various results concerning spectralsynthesis for $A_\omega(G)$.

Beurling-Fourier algebras on compact groups: spectral theory