Establish the HSS structure and ranks of the normal operator

Establish that the matrix representing the normal operator \(\fiomat^{\herm}\fiomat\) associated with a discrete Fourier integral operator has the hierarchically semiseparable (HSS) property, and analyze the corresponding HSS ranks to provide a rigorous foundation for the proposed algorithm’s complexity estimates.

Background

The proposed approximate inversion method relies on compressing the normal operator $\fiomat^{\herm}\fiomat$ with an HSS representation and then applying a ULV factorization. The complexity analysis assumes bounded or explicitly controlled HSS ranks, but the paper does not provide a theoretical proof that the normal operator possesses the required HSS structure.

A rigorous characterization of the HSS property and rank growth of $\fiomat^{\herm}\fiomat$ would justify the stated offline and online complexity bounds and clarify how those bounds depend on the dimension and on the amplitude and phase functions of the discrete Fourier integral operator.