Fourier analysis associated to a vector measure on a compact group

In this paper, we introduce and study the Fourier transform of functions which are integrable with respect to a vector measure on a compact group (not necessarily abelian). We also study the Fourier transform of vector measures. We also introduce and study the convolution of functions from $L^p$-spaces associated to a vector measure. We prove some analogues of the classical Young's inequalities. Similarly, we also study convolution of a scalar measure and a vector measure.