We study the basic properties of the Fourier and Herz spaces $A(K)$ and $A_p(K)$ of a compact hypergroup $K$ and associate them with subspaces of the Cartesian product of matrix algebras on the dual hypergroup. When $K$ is a tensor hypergroup we show that $A(K)$ is a regular Banach algebra whose spectrum is $K$. We also compute some of the corresponding multiplier algebras.