An analytic expression for the Green function of the one-dimensional periodic Helmholtz operator is used to analyze the behaviour of the local density of states (LDOS). We provide numerical evidence that in the n-th band the LDOS has at least n minima and maxima. Near the band edge wc, the LDOS exhibits strong fluctuations, with its mimima approaching zero and the maxima diverging to infinity. For the lowest band, the ratio of the maximum to the minimum of the LDOS is shown to diverge as (wc- w)-1 as w approaches the band edge. Since, in the case of electromagnetic waves, the LDOS governs the spontaneous emission of light of embedded atoms and molecules, this may have profound implications for technology using photonic crystals.