Failure of Orthogonality of Rounded Fourier Bases

TL;DR

This paper investigates the orthogonality failure of rounded Fourier bases, providing estimates for sums involving sign functions of cosines, and explains observed patterns in signed discrete cosine transforms.

Contribution

It offers new estimates for sums of sign functions of cosines in prime dimensions, revealing conditions under which orthogonality fails in rounded Fourier bases.

Findings

Large sums occur only when a^{-1}b is close to 1 in the finite field.

Explains line patterns in the matrix A^T A for signed discrete cosine transforms.

Results suggest broader applicability at a general level.

Abstract

The purpose of this note is to prove estimates for $$ \left| \sum_{k=1}^{n} \mbox{sign} \left( \cos \left( \frac{2\pi a}{n} k \right) \right) \mbox{sign} \left( \cos \left( \frac{2\pi b}{n} k \right) \right)\right|,$$ when $n$ is prime and $a,b \in \mathbb{N}$. We show that the expression can only be large if $a^{-1}b \in \mathbb{F}_n$ (or a small multiple thereof) is close to $1$. This explains some of the surprising line patterns in $A^T A$ when $A \in \mathbb{R}^{n \times n}$ is the signed discrete cosine transform. Similar results seem to exist at a great level of generality.