Rapidly decaying Fourier-like bases

Kevin Liang (1) , G. Forbes (2) , Miguel Alonso (3, 4, 1, 5, 6)

1 University of Rochester [USA]
2 Macquarie University
3 MOSAIC - MOSAIC
4 AMU - Aix Marseille Université
5 ECM - École Centrale de Marseille
6 FRESNEL - Institut FRESNEL

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)].

Domaines: Optique [physics.optics]

Kevin Liang, G. Forbes, Miguel Alonso. Rapidly decaying Fourier-like bases. Optics Express, 2019, 27 (22), pp.32263. ⟨10.1364/OE.27.032263⟩. ⟨hal-02342134⟩