The electronic structure of the Penrose lattice is studied using a tight-binding model. The local densities of states are calculated for 15 vertices having distinct nearest neighbors and the averaging is done exactly. The density of states shows a strong dependence on the local environment of a vertex and consists of several peaks and gaps but shows no self-similarity behavior. In conformity with a recent study we also find a-function-like localized state lying in a gap at the center of the band. {\textcopyright} 1987 The American Physical Society.