Calculating Spectra by Sequential High-Pass Filtering

We expand on the method of sequential filtering for calculating spectra of inhomogeneous fields. Sadek&Aluie [Phys. Rev. Fluids, 3, 124610 (2018)] showed that the kernel has to have at least $p$ vanishing moments to extract a power-law spectrum $k^{-\alpha}$ with $\alpha<p+2$ by low-pass filtering. Here, we show that sequential high-pass filtering allows for extracting steeper spectra with $\alpha<2p+3$ using the same $p$-th order kernel. For example, any spectrum of a field that is shallower than $k^{-5}$ can be extracted by sequential high-pass filtering using any 1st order kernel such as a Gaussian or top-hat. Finally, we demonstrate how second-order structure functions fail to capture spectral peaks because they cannot detect scaling that is too shallow.