DFT-Commuting Matrix With Arbitrary or Infinite Order Second Derivative Approximation

Recently, Candan introduced higher order DFT-commuting matrices whose eigenvectors are better approximations to the continuous Hermite-Gaussian functions (HGFs). However, the highest order 2k of the O(h2k) NtimesN DFT-commuting matrices proposed by Candan is restricted by 2k+1 les N. In this paper, we remove this order upper bound restriction by developing two methods to construct arbitrary-order DFT-commuting matrices. Computer experimental results show that the Hermite-Gaussian-like (HGL) eigenvectors of the new proposed DFT-commuting matrices outperform those of Candan. In addition, the HGL eigenvectors of the infinite-order DFT-commuting matrix are shown to be the same as those of the n2 DFT-commuting matrix recently discovered in the literature.