Quantum-mechanical evolution of systems with periodic time-modulated Hamiltonians is often described by effective interactions. Such average Hamiltonians, calculated as few terms of an expansion in powers of the interaction, are sometimes difficult to relate to experimental observations. We propose a frequency-domain approach to this problem, which offers certain advantages and produces an approximate solution for the density matrix, better linked to measurable quantities. The formalism is suitable for calculating the intensities of narrowed spectral peaks. Fast magic-angle-spinning NMR spectra of solids are used to experimentally illustrate the method.