A combined theoretical and experimental approach is used to study the directed assembly of a lamellae-forming block copolymer on chemically patterned substrates. The period of the pattern is lower than that of the copolymer, whose characteristic morphology is then used to interpolate the features of the substrate. The pattern considered in this study consists of stripes of width W repeated over the background substrate with period L-P = 2L(0), where L-0 is the copolymer natural period. The stripe and background areas are characterized by their affinities Lambda(s) and Lambda(b) for the blocks of the polymer. Using theoretically informed Monte Carlo simulations of a coarse-grained model, we investigate in a systematic manner the influence of the pattern parameters W, Lambda(s), and Lambda(b) on the morphology of copolymer thin films. Thermodynamic integration is used to compute the free energy difference and the relative stability of competing morphologies. It is found that the parameter space considered here is dominated by nonbulk and often metastable morphologies. The conditions that yield successful interpolation of lamellae are identified. Consistent with theoretical predictions, experiments on patterned substrates with carefully controlled interfacial characteristics reveal new, three-dimensional morphologies that do not arise in the bulk. The sought-after vertical lamellae, which are desirable for pattern interpolation and lithography, are found to occur only when the interaction between one of the blocks and the background area is relatively weak.