A fast approximate method for variable-width broadening of spectra

Spectral data is routinely broadened in order to improve appearance, approximate a higher sampling level or model experimental measurement effects. While there has been extensive work in the signal processing field to develop efficient methods for the application of fixed-width broadening functions, these are not suitable for all scientific applications -- for example, the instrumental resolution of inelastic neutron scattering measurements varies along the energy-transfer axis. Na\"ive application of a kernel to every point has $O(N \times M)$ complexity and scales poorly for a high-resolution spectrum over many data points. Here we present an approximate method with complexity $O(N + W\times M \log M)$, where $W$ scales with the range of required broadening widths; in practice the number and cost of mathematical operations is drastically reduced to $N$ polynomial evaluations and a modest number of discrete Fourier transforms. Applications are demonstrated for Gaussian interpolation of density-of-states data and to instrumental resolution functions. We anticipate that these performance improvements will assist application of resolution functions inside fitting procedures and interactive tools.