Rapidly decaying Fourier-like bases.

In many applications it is natural to seek to extract a characteristic scale for a function's variations by reference to a frequency spectrum. Although the moments of a spectrum appear to promise simple options to make such a connection, standard Fourier methods fail to yield finite moments when the function's domain is itself finite. We investigate a family of Fourier-like bases with rapidly decaying spectra that yield well-defined moments for such cases. These bases are derived by considering classes of functions for which a normalised mean square derivative is stationary. They are shown to provide precisely the type of spectrum needed to complete a recent investigation of mid-spatial frequency structure on optical surfaces [K. Liang, Opt. Express 27, 3390-3408 (2019)].