  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.
